Answers & Explanations

PSAT/NMSQT Prep 2019 - Princeton Review 2019


Answers & Explanations

Chapter 1

PRACTICE

8. C/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: The Kaplan Strategy for Translating English into Math will work well here. Watch out for extra information that may confuse you. Let a be the average of Test One (92); b, the average of Test Two (77); c, the average of Test Three (what you need to find); and d, the overall test average (84). The average is the sum of the terms divided by the number of terms: . Plugging in the values you know yields . Multiply both sides by 3 and then subtract to solve for c, which gives 83. This corresponds to (C).

9. D/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: This is another question suited for the Kaplan Strategy for Translating English into Math. Let b represent the total number of candies in the box. Then write each candy quantity as a fraction of b.

Putting the pieces together in an equation gives . From here, you can find a common denominator, but multiplying through by a common multiple (such as 12) will be easier. Solving for b yields 36, but you’re not done yet. Plug 36 back into the expressions for peppermints (12) and chocolates (9), then multiply together to get 108, which is (D).

PERFORM

10.C/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: You know there will be a flat fee of $50, and for text messages you’ll be looking for 0.1t. This eliminates answer choice D. If you’re stuck on the data plan cost, plug in some numbers. For g = 2, you wouldn’t expect there to be an additional fee. For g = 3, you’d expect to see an $8 charge, and for g = 4, you’d expect to see a $16 charge. The only answer choice that reflects this is (C).

11.1/3/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: Notice that n − 2 is in both the equation and the desired expression. Find that first and plug it into the expression. Determine the value of n — 2 by dividing both sides of 3(n — 2) = 6 by 3. Now you know that n — 2 = 2, so plug 2 into the numerator of the desired expression. To find the value of the denominator, continue simplifying the given equation to find that n = 4. Plug 4 into n + 2, which leaves . Grid in and move on to the next question.

12.B/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: Follow the Kaplan Strategy for Translating English into Math. Assemble equations and set them equal to each other. Let d = Die-Hard Package, p = Personal Package, and v = number of visits. Using the information about each package, you know that d = 250 and p = 130 + 4v. Now, solve for v:

This is an exact match for (B).

ON YOUR OWN

1. D/Difficulty: Easy

Category: Heart of Algebra / Linear Equations

Getting to the Answer: The total bill consists of a flat tax and a percentage of annual income. The flat per capita tax is a one-time fee that does not depend on the taxpayer’s income and therefore should not be multiplied by i. This means that 25 is the per capita tax. The other expression in the equation, 0.02i, represents the percentage income tax times the annual income (which the question tells you is i). Therefore, 0.02 must represent the amount of the income tax as a percentage, which is (D).

2. A/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: Start by simplifying the numerators. Don’t forget to distribute the negative to both terms inside the parentheses on the right side of the equation:

Next, cross multiply and solve for n using inverse operations:

This matches (A).

3. C/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: If a linear equation has no solution, the variables cancel out, leaving two numbers that are not equal to each other. Start by simplifying the left side of the equation:

The variable term on the left is 12x. Because the variable terms must cancel, the right side of the equation must also have a 12x, so it must be that c = 6, which is (C).

4. B/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: Use the information in the question to write your own equation, then look for the answer choice that matches. Simplify your equation only if you don’t find a match. Start with the cost, not including tax or the environmental impact fee. If Jenna rents a car for d days at a daily rate of $54.95, the untaxed total is 54.95d. There is a 6% tax on this amount, so multiply by 1.06 to get 1.06(54.95d). The $10 environmental impact fee is not taxed, so simply add 10 to your expression. The total cost is c = 1.06(54.95d) + 10, which matches (B), so you do not need to simplify.

5. A/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: The answer choices are given in slope-intercept form, so start by finding the slope. To do this, substitute two pairs of values from the table into the slope formula, . Keep in mind that the projected number of cans sold depends on the price, so the price is the independent variable (x) and the projected number is the dependent variable (y). Using the points (0.75, 10,000) and (1.0, 5,000), the slope is:

This means that (A) must be correct because it is the only one that has a slope of —20,000. Don’t let D fool you—the projected number of cans sold goes down as the price goes up, so there is an inverse relationship, and the slope must be negative.

6. B/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: The x-axis represents the number of golf balls, so find 1 on the x-axis and trace up to where it meets the graph of the line. The y-value is somewhere between $1 and $2, so the only possible correct answer choice is $1.67, which is (B).

You could also find the unit rate by calculating the slope of the line using two of the points shown on the graph. The graph rises 5 units and runs 3 units from one point to the next, so the slope is , or 1.67.

7. C/Difficulty: Hard

Category: Heart of Algebra / Linear Equations

Getting to the Answer: The key to answering this question is to determine how many arrows hit each circle. If there are 12 arrows total and a hit the inner circle, the rest, or 12 — a, must hit the outer circle. Now, write the expression in words: points per inner circle (8) times number of arrows in inner circle (a) plus points per outer circle (4) times number of arrows in outer circle (12 — a). Next, translate the words into numbers, variables, and operations: 8a + 4(12 — a). This is not one of the answer choices, so simplify the expression by distributing the 4 and then combining like terms: 8a + 4(12 — a) = 8a + 48 — 4a = 4a + 48, so the equation is p = 4a + 48. Rearrange the order of the terms on the right side to arrive at (C).

8. D/Difficulty: Hard

Category: Heart of Algebra / Linear Equations

Getting to the Answer: In a real-world scenario, the y-intercept of a graph usually represents a flat fee or an initial value. The slope of the line represents a unit rate, such as the cost per pound. The y-intercept of the graph is 12, so the flat fee is $12. To find the cost per pound (the unit rate), substitute two points from the graph into the slope formula. Using the points (0, 12) and (2, 17), the unit rate is , which means $2.50 per pound will be added to the cost. The total cost to ship a 25-pound box is $12 + 2.50(25) = $12 + $62.50 = $74.50, which is (D).

9. D/Difficulty: Easy

Category: Heart of Algebra / Linear Equations

Getting to the Answer: Quickly examine the numbers in the equation and choose the best strategy to answer the question. Distributing the will result in messy calculations, so clear the fraction instead. To do this, multiply both sides of the equation by the reciprocal of , which is :

Choice (D) is correct. Note that you could also clear the fraction by multiplying both sides of the equation by the denominator (3) and then using the distributive property, but this is not the most time-efficient strategy.

10.C/Difficulty: Easy

Category: Heart of Algebra / Linear Equations

Getting to the Answer: When writing a linear equation, a flat rate is a constant while a unit rate is always multiplied by the independent variable. You can identify the unit rate by looking for words like per or for each. Because the amount Sandy gets paid per day, $70, is a flat rate that doesn’t depend on the number of tires she sells, it should be the constant in the equation. The clue “for each” tells you to multiply $14 by the number of tires she sells, so the equation is pay = 14 × number of tires + 70, or y = 14x + 70. This matches (C).

11.C/Difficulty: Easy

Category: Heart of Algebra / Linear Equations

Getting to the Answer: It is reasonable to assume that the value of the car is determined by multiplying the depreciation rate by the number of years the car is owned, which is given in the equation as 0.15x, and subtracting it from the car’s initial value. This means 27,000, which is the constant in the equation, is the value of the car when the number of years is 0, or in other words, the purchase price of the car, choice (C).

12.A/Difficulty: Easy

Category: Heart of Algebra / Linear Equations

Getting to the Answer: Read the axis labels carefully. The y-intercept is the point at which x = 0, which means the number of rides is 0. The y-intercept is (0, 8). This means the cost is $8 before riding any rides, and therefore 8 most likely represents a flat entrance fee, (A).

13.A/Difficulty: Easy

Category: Heart of Algebra / Linear Equations

Getting to the Answer: The slope of the line is the rise over the run from one point to another (always left to right). Put your pencil on the y-intercept, (0, 6), and move to the point (3, 1). The line moves down 5 units and to the right 3 units, so the slope is , which is (A).

You could also substitute two points, such as (0, 6) and (3, 1), into the slope formula to find the slope of the line.

14.C/Difficulty: Easy

Category: Heart of Algebra / Linear Equations

Getting to the Answer: The graph in (C) does not represent a linear relationship because the slope on the left side of the y-axis is negative (the graph is decreasing) while the slope on the right side of the y-axis is positive (the graph is increasing). The graph of a linear relationship does not change direction. Choice (C) is correct.

15.C/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: When a linear equation is written in the form y = mx + b, the variable m represents the slope of the line, and b represents the y-intercept of the line. Quickly scan the answer choices—they involve inequalities, so you’ll need to translate them into something that makes more sense to you. Use the fact that “< 0” means negative and “> 0” means positive. Now, look at the graph—the line is increasing (going up from left to right), so the slope is positive (m > 0). This means you can eliminate A and B. Finally, look at the y-intercept—it is below the x-axis and is therefore negative (b < 0), which means (C) is correct.

16.B/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: Use the graph to identify the y-intercept and the slope of the line. Then write an equation in slope-intercept form, y = mx + b. The line crosses the y-axis at (0, 6), so the y-intercept (b) is 6. The line falls vertically 1 unit for every 2 units that it runs to the right, so the slope (m) is . The equation of the line is , which matches (B).

Resist the temptation to graph each answer choice on your calculator—such a strategy is too time-consuming.

17.C/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: Look for a way to make the math easier, such as clearing the fractions first. To do this, multiply both sides of the equation by 6, then solve for h using inverse operations:

Because the variable terms in the equation cancel out, and 15 does not equal 48, the equation has no solution. In other words, there is no value of h that satisfies the equation, so (C) is correct.

18.B/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: Write the equation in words first, then translate from English to math. The total cost, c, is the weight of the watermelon in pounds, p, multiplied by the sale price since the purchase is made on Monday: $0.60 × 80% = 0.6 × 0.8 = 0.48. This gives the first part of the expression: 0.48p. Now add the cost of four sweet potatoes, 0.79 × 4 = 3.16, to get the equation c = 0.48p + 3.16, which matches (B).

You could also use the Picking Numbers strategy: pick a number for the weight of the watermelon and calculate how much it would cost (on sale). Then add the cost of four sweet potatoes. Finally, find the equation that gives the same amount.

19.A/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: In this scenario, the total cost is the dependent variable and is calculated by multiplying the per person rate by the number of people attending and then adding the room rental fee. Therefore, the total cost is represented by y. Because the room rental fee and the per person rate are fixed amounts (determined by the hotel), they should be represented by constants in the equation—325 and 15, respectively. The total cost depends on the number of people attending, so the number of people is the independent variable and is represented by x. Choice (A) is correct.

20.B/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: There is not a lot of information to go on here, so start by determining the relationship between the number given in the problem, 3,600 bolts per day, and the number in the equation, 150. Because 3,600 ÷ 150 = 24 and there are 24 hours in a day, 150 is the number of bolts the machine can produce in 1 hour. If the machine can produce 150 in 1 hour, then it can produce 150 times x in x hours. This means the equation y = 150x represents the number of bolts the machine can produce in x hours, which is (B).

21.B/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: The slope-intercept form of a linear equation is y = mx + b, where m represents the slope. If the slope is 0, that means m = 0. Substitute 0 for m and simplify y = mx + b:

The only answer that matches this form is (B), y = 2. You could also memorize that x = a represents a vertical line with an undefined slope, and y = b represents a horizontal line with a 0 slope.

22.A/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: Quickly skim the answer choices to get the right context. Because the number of bacteria depends on the temperature of the room, the temperature would be graphed on the x-axis and the number of bacteria on the y-axis. Add these labels to the graph. Then describe what you see using your labels: as the temperature increases, the number of bacteria also increases. Unfortunately, this is not one of the answer choices, so think of the scenario in terms of the type of relationship between the variables: there is a direct variation between them, or in other words, when one goes up, the other goes up. This can also be stated as when one goes down, the other goes down; when the temperature decreases, the number of bacteria decreases, and (A) is correct. Don’t let D fool you—the change in the number of bacteria is constant, but the actual number of bacteria is not constant.

23.A/Difficulty: Hard

Category: Heart of Algebra / Linear Equations

Getting to the Answer: There is not a lot of information to go on, so put the equation in slope-intercept form, and see what conclusions you can draw:

The slope is negative, so the line is decreasing (going down) from left to right. This means you can eliminate C and D because those lines are increasing. The y-intercept is positive, but this doesn’t help because the y-intercepts in A and B are both positive. But don’t forget, you are given that c = b, so rewrite the y-intercept as . The y-intercept is 1, so (A) is correct.

24.D/Difficulty: Hard

Category: Heart of Algebra / Linear Equations

Getting to the Answer: The graph is increasing, so the slope must be positive. This means you should put each of the answer choices in slope-intercept form and check the sign of the slope. However, you can eliminate some of the answer choices fairly quickly using the other piece of information—the graph has a negative x-intercept, so when y = 0, the resulting value of x is negative. Quickly check each equation. You don’t need to find the exact value of x, simply whether it is positive or negative.

Choice A: 4x + 3(0) = 1 4x = 1, so x is positive.

Choice B: —x + 2(0) = −8 x = −8, so x is positive.

Choice C: 2x + 3(0) = −9 2x = −9, so x is negative.

Choice (D): 3x − 5(0) = −10 3x = −10, so x is negative.

Eliminate A and B and write C in slope-intercept form:

Eliminate choice C because it is decreasing (has a negative slope). Choice (D) must be correct.

25.A/Difficulty: Hard

Category: Heart of Algebra / Linear Equations

Getting to the Answer: Sometimes drawing a diagram is the quickest way to answer a question. Sketch a quick graph of any linear equation. Change the signs of the slope and the y-intercept, then sketch the new graph on the same coordinate plane. Pick a simple equation that you can sketch quickly, like y = −2x + 3, and then change the signs of m and b. The new equation is y = 2x — 3. Sketch both graphs. The second graph is a perfect reflection of the first graph across the x-axis, which is (A). The graph that follows illustrates this:

Two lines graphed on a coordinate plane, one that is solid and one that is dashed. The solid line represents the equation y equals negative 2x plus 3. It begins at the point zero comma 3 and falls from left to right. The dashed line represents the equation y equals 2x minus 3. It begins at the point zero comma negative 3 and rises from left to right. The dashed line is a reflection of the solid line across the x-axis.

You could also graph your equations in a graphing calculator and compare them. If you’re not convinced, try another pair of equations.

26.D/Difficulty: Hard

Category: Heart of Algebra / Linear Equations

Getting to the Answer: When solving a linear equation that has infinitely many solutions, the variable terms will all cancel out, leaving a number that is equal to itself. Start by looking for an equation that, when simplified, has the same variable term on the left side of the equation as on the right side. Choice (D) is correct because:

27.A/Difficulty: Hard

Category: Heart of Algebra / Linear Equations

Getting to the Answer: Always pay careful attention to units of measure. The units change from hours to days. The mouse receives one pellet every 2.5 hours, which is equivalent to pellets per day. Over the course of d full days, the mouse will be fed pellets, which is (A).

28.B/Difficulty: Hard

Category: Heart of Algebra / Linear Equations

Getting to the Answer: This question sounds more difficult than it is. Try to match each number or term in the expression with a piece of information given in the question. Start with 2f. You are given that 2 is the number of points earned for fish with a purple dot and that Karla catches f of these fish, so 2f is the total number of points she earns for fish with purple dots. Move on to the second term, 5(8 — f). You are given that 5 is the number of points earned for fish with a green dot. If Karla catches f fish with a purple dot, then she must catch (the number of attempts minus f) fish with a green dot. Because the term in parentheses is 8 — f, Karla must have made a total of 8 attempts. Each attempt costs 50 cents, so she spent $0.50(8) = $4 playing the game. Choice (B) is correct.

29.D/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: Write an equation in words first, then translate from English to math. Finally, rearrange your equation to find what you’re interested in, which is the initial amount. Call the initial amount A. After you’ve written your equation, solve for A.

Amount now (x) = initial amount (A) minus y plus 6:

This is the same as y + x — 6, so (D) is correct.

You could also use Picking Numbers to answer this question.

30.C/Difficulty: Hard

Category: Heart of Algebra / Linear Equations

Getting to the Answer: The birth rate starts high on the y-axis, while the gross domestic product (indicating wealth) starts low on the x-axis. At the other end of the graph, the birth rate is lower, while the gross domestic product is higher. This tells you that there is an inverse relationship between birth rate and wealth. In other words, as one variable increases, the other variable decreases. The only answer choice that describes this type of inverse relationship is (C).

31.14/Difficulty: Easy

Category: Heart of Algebra / Linear Equations

Getting to the Answer: Eliminate the fractions to simplify the math. To do this, multiply both sides of the equation by 4, then solve for y using inverse operations:

32.24/Difficulty: Medium

Category: Heart of Algebra / Linear Equations

Getting to the Answer: The place where the line crosses the y-axis is the y-intercept, or b when the equation is written in slope-intercept form (y = mx + b), so rewrite the equation in this form:

The y-intercept is 24.

Because the y-intercept of a graph is always of the form (0, y), you could also substitute 0 for x in the original equation and solve for y.

Chapter 2

PRACTICE

7. A/Difficulty: Medium

Category: Heart of Algebra / Systems of Linear Equations

Getting to the Answer: Use the Kaplan Strategy for Translating English into Math to make sense of the situation. First, define your variables: t for texts and p for pictures are good choices. Breaking apart the question, you know this student sent a total of 75 texts and pictures. You’re also told each text costs $0.10 and each picture is $0.15, as well as the fact that the bill is $8.90. You’ll have two equations: one relating the number of texts and pictures and a second relating the costs associated with each:

Multiply the top equation by —0.1 to solve for p using combination:

Divide both sides of the resulting equation by 0.05 and you’ll find that p = 28. But you’re not done yet; you’re asked for the difference between the text and picture count. Substitute 28 for p in the first equation and then solve for t to get t = 47. Subtracting 28 from 47 yields 19, which is (A).

8. 51/Difficulty: Medium

Category: Heart of Algebra / Systems of Linear Equations

Getting to the Answer: Creativity is key to getting the correct answer to this question. Rather than multiplying just one equation by a factor, you’ll need to multiply both by a factor to use combination. Which factors do you pick? It depends on which variable you want to eliminate. Suppose you want to eliminate x. The coefficients of the x terms are 2 and 5, so you need to multiply the equations by numbers that will give you —10 and 10 as your new x term coefficients. To do this, multiply the first equation by —5 and the second equation by 2:

—5(2x + 5y = 49)

2(5x + 3y = 94)

Add the resulting equations:

Solving for y gives you 3. Next, plug 3 back in for y in either equation and solve for x, which equals 17. Multiplying x and y together yields 51, the correct answer.

PERFORM

9. A/Difficulty: Hard

Category: Heart of Algebra / Systems of Linear Equations

Getting to the Answer: A system of linear equations that has no solution should describe two parallel lines. This means the coefficients of the variables should be the same (so the slopes of the lines are the same). Only the constant should be different (so the y-intercepts are not the same). The easiest way to make the coefficients the same is to look at the second equation. Clearly, multiplying it by 40 would make the coefficients of x the same: 8x + 40zy = 20. Now equate the coefficients of y to get 4y = 40zy. Solve for z, to reveal that z = , which is (A).

10.14/Difficulty: Medium

Category: Heart of Algebra /Systems of Linear Equations

Getting to the Answer: Be on the lookout for “sets” of equations that aren’t really systems that require substitution or combination. Here, notice that there is only one variable in each equation, so you can solve for each variable to find that x = 17 — 6 = 11 and y = 12 — 9 = 3. Therefore, the value of x + y is 11 + 3 = 14. Grid in 14 and move on to the next question.

11.C/Difficulty: Medium

Category: Heart of Algebra / Systems of Linear Equations

Getting to the Answer: Use the Kaplan Strategy for Translating English into Math to extract what you need. First, define the variables using letters that make sense. Use c for children and a for adults. Now break the word problem into shorter phrases: children’s tickets sold for $8 each; adult tickets sold for $12 each; 60 people attended the concert; $624 was collected in ticket money. Translating each phrase into a math expression will produce the components needed:

Children’s tickets (c) cost $8 each 8c

Adult tickets (a) cost $12 each 12a

60 people attended the concert c + a = 60

$624 was collected in ticket money  Total $ = 624

Now put the expressions together:

Solving for the variables using combination or substitution gives a = 36 and c = 24. Remember, the question asks for the product of the number of children and the number of adults, so the correct answer is 36 × 24 = 864, which corresponds to (C).

ON YOUR OWN

1. C/Difficulty: Easy

Category: Heart of Algebra / Systems of Linear Equations

Getting to the Answer: Translate English into math. One equation should represent the total number of meals ordered, while the other equation should represent the cost of the meals.

The number of people who ordered chicken plus the number who ordered vegetarian equals the total number of people, 62, so one equation is c + v = 62. This means you can eliminate A. Now write the cost equation: cost per chicken dish (12.75) times number of dishes (c) plus cost per vegetarian dish (9.5) times number of dishes (v) equals the total bill (725.25). The cost equation should be 12.75c + 9.5v = 725.25. Together, these two equations form the system in (C).

2. A/Difficulty: Easy

Category: Heart of Algebra / Systems of Linear Equations

Getting to the Answer: Quickly compare the two equations. The system is already set up perfectly to solve using elimination, so add the two equations to cancel —4y and 4y. Then solve the resulting equation for x. Remember, the question asks for the y-coordinate of the solution, so you will need to substitute x back into one of the original equations and solve for y:

Choice (A) is correct.

3. D/Difficulty: Medium

Category: Heart of Algebra / Systems of Linear Equations

Getting to the Answer: Because x has a coefficient of 1 in the second equation, solve the system using substitution. First, solve the second equation for x. Then substitute the resulting expression for x into the first equation and solve for y:

Next, substitute this value back into x = 6y + 10 and simplify:

Finally, subtract xy to find that (D) is correct:

4. A/Difficulty: Medium

Category: Heart of Algebra / Systems of Linear Equations

Getting to the Answer: Translate English into math to write a system of equations with r being the cost of the radio in dollars and t equaling the cost of the television in dollars. A television costs $25 less than twice the cost of the radio, or t = 2r — 25; together, a radio and a television cost $200, so r + t = 200.

The system is:

The top equation is already solved for t, so substitute 2r — 25 into the second equation for t:

The radio costs $75, so the television costs 2(75) — 25 = 150 — 25 = $125. This means the TV costs $125 — $75 = $50 more than the radio, which is (A).

5. C/Difficulty: Medium

Category: Heart of Algebra / Systems of Linear Equations

Getting to the Answer: Translate English into math to write a system of equations with t being the cost of a turkey burger and w equaling the cost of a bottle of water. The first statement is translated as 2t + w = $3.25 and the second as 3t + w = $4.50. The system is:

You could solve the system using substitution, but combination is quicker in this question because subtracting the first equation from the second eliminates w and you can solve for t:

Substitute this value for t in the first equation and solve for w:

Two bottles of water would cost 2 × $0.75 = $1.50, which is (C).

6. A/Difficulty: Hard

Category: Heart of Algebra / Systems of Linear Equations

Getting to the Answer: One way to answer this question is to think about the graphs of the equations. Graphically, a system of linear equations that has no solution indicates two parallel lines or, in other words, two lines that have the same slope. Write each of the equations in slope-intercept form (y = mx + b), and set their slopes (m) equal to each other to solve for w.

First equation:

Second equation:

Set the slopes equal:

This matches (A).

7. B/Difficulty: Hard

Category: Heart of Algebra / Systems of Linear Equations

Getting to the Answer: A system of linear equations has infinitely many solutions if both lines in the system have the same slope and the same y-intercept (in other words, they are the same line). Write each of the equations in slope-intercept form (y = mx + b). Their slopes should be the same. To find c, set the y-intercepts (b) equal to each other and solve. Before rewriting the equations, multiply the first equation by 6 to make it easier to manipulate.

First equation:

Second equation:

Set the y-intercepts equal:

Choice (B) is correct.

8. C/Difficulty: Hard

Category: Heart of Algebra / Systems of Linear Equations

Getting to the Answer: Create a system of two linear equations where t represents tables with 2 chairs and f represents tables with 4 chairs. The first equation should represent the total number of tables, each with 2 or 4 chairs, or t + f = 25. The second equation should represent the total number of chairs. Because t represents tables with 2 chairs and f represents tables with 4 chairs, the second equation should be 2t + 4f = 86. Now solve the system using substitution. Solve the first equation for either variable, and substitute the result into the second equation:

There are 18 tables with 4 chairs each, (C). This is all the question asks for, so you don’t need to find the value of t.

9. C/Difficulty: Easy

Category: Heart of Algebra / Systems of Linear Equations

Getting to the Answer: Graphically, the solution to a system of linear equations is the point where the lines intersect. Jot down the coordinates of the point on the graph where the two lines intersect, (—1, 2). The question asks for the sum of x + y, so add the coordinates to get —1 + 2 = 1. Choice (C) is correct.

Chapter 3

PRACTICE

8. D/Difficulty: Easy

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Miles per gallon is a rate, so use the DIRT equation. Determine the fuel used during each leg of Jack’s trip. Plugging in values for the first leg, you get 180 mi = 40 mi/gal × tt = 4.5 gallons. The second leg: 105 mi = 35 mi/gal × tt = 3 gallons. Added together, you get 7.5 gallons of fuel used, which matches (D).

9. B/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Remember to avoid merely adding the percentages together. Find each change individually. You’re not given a concrete share count in the question, so assume the client starts with 100. To save a step with each change, calculate the shares left instead of the shares gained or lost. The first change is —25%; the number of shares left is 75% × 100 → 0.75 × 100 = 75. The second change is +10%, which corresponds to 110% × 75 → 1.1 × 75 = 82.5 shares. The final change is +50%, which means there are now 150% × 82.5 → 1.5 × 82.5 = 123.75 shares. The percent change is ; rounded to the nearest percent, you get 24%, which is (B).

10.432/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Determine the number of times the pattern repeats, and then find the corresponding number of green tiles. The question states that the ratio of green to blue tiles is 5:3 and that the pattern appears once per square foot. There are 12 × 18 or 216 square feet in the mosaic, meaning there are 5 × 216 or 1,080 green tiles and 3 × 216 or 648 blue tiles. Taking the difference gives 1,080 - 648 = 432 more green tiles than blue.

11.324/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Use the blue tile count from the previous question to determine the new green tile count. The green tile count changes in this question, so determine that first. The blue tile count stays constant, so you can just multiply 2 × 216 to get 432 green tiles needed with the new pattern. The ratio of red to green tiles should be 3:4; you can use this in conjunction with the new green tile count in a proportion to find the number of red tiles needed: . Use cross multiplication to find that 1,296 = 4r. Solving for r yields 324 red tiles.

PERFORM

12.A/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Draw a diagram to make sense of the given situation. Your diagram should look similar to what’s shown:

The two triangles are similar, which means you can use a proportion to answer the question. First, find the correct height by taking 25% of 40, which is 10, and deducting that from 40 to give 30. Keeping the heights on the left of your proportion and the hypotenuses on the right, you have . Reduce the left side to get , then cross multiply to eliminate the fractions: 225 = 4x. Solving for x yields 56.25. But don’t stop yet. The question asks for the difference in escalator length, not the new length. Subtract 56.25 from 75 to get 18.75, which matches (A).

13.3.5/Difficulty: Easy

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Whenever you’re asked for the cost of something per a set measurement, think unit rates. First, determine the price per ounce (a.k.a. the unit rate) for the smaller can: .

Next, do the same for the larger can: .

Take the difference to get $0.035, which is 3.5 cents. Pay careful attention to the way the question is worded—grid in 3.5, not .035, because the question asks how many more cents.

14.330/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: First, find the price at 40% off using the three-part percent formula: p = 0.6 × $2,200 = $1,320. Repeat for the extra 25%: p’ = 0.75 × $1,320 = $990. Notice we used the difference between 100% and the percent off (instead of the percent off itself) to avoid an extra calculation with each step. Lastly, find the difference between what the first customer paid and what the second customer paid: $1,320 — $990 = $330.

15.1,318/Difficulty: Hard

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Keep your percentages (and whether each is an increase or decrease) for your math steps straight. For this question, you’ll need the prices including tax. Using 1, which is equal to 100%, the price of the home theater system, plus the tax rate instead of the tax rate alone will again save you a calculation in each step. The full price with tax is 1.0635 × $2,200 = $2,339.70. After the two price reductions, the total sale price is 1.03175 × $990 = $1,021.43. The difference in price is $2,339.70 — $1,021.43 = $1,318.27. Rounding to the nearest dollar, you get $1,318.

ON YOUR OWN

1. B/Difficulty: Easy

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: To answer a question that says “directly proportional,” set two ratios equal to each other and solve for the missing amount. Don’t forget—match the units in the numerators and in the denominators on both sides.

Let c equal the number of cars that can safely pass through a light that lasts 24 seconds. Set up the proportion and solve for c, to find that (B) is correct.

2. B/Difficulty: Easy

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: When ratios involve large numbers, simplify if possible to make the calculations easier.

Let b equal the number of snowblowers produced. Set up a proportion and solve for b. Be sure to match the units in the numerators and in the denominators on both sides:

This matches (B)

3. B/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Pay careful attention to the units. As you read the question, decide how and when you will need to convert units. The answer choices are given in hours and minutes, so start by converting the rate from words per second to words per minute:

Next, find the number of words in a 45-page chapter:

Finally, let m be the number of minutes it takes Jorge to read the whole chapter. Set up a proportion and solve for m:

Because 75 minutes is not an answer choice, convert it to hours and minutes: 75 minutes = 1 hour, 15 minutes, (B).

4. C/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Find the percent increase using the formula: Percent change = amount of change divided by original amount. Then apply the same percent increase to the amount for 2013. The amount of increase is 30,100 — 15,800 = 14,300, so the percent increase is 14,300 ÷ 15,800 = 0.905 = 90.5% over 25 years. If the total percent increase over the next 25 years is the same, the average cost of tuition and fees will be 30,100 × 1.905 = 57,340.50, or about $57,300, which is (C).

5. A/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Don’t let all the technical words in this question overwhelm you. Solve it step-by-step, examining the units as you go. First, use the factor-label method to determine the number of megabits the computer can upload in 1 night (8 hours):

Next, convert this amount to gigabits (because the information about the scans is given in gigabits, not megabits):

Finally, each scan produces about 3.6 gigabits of data, so divide this number by 3.6 to determine how many scans the computer can upload to the remote server: 56.25 ÷ 3.6 = 15.625 scans. You should round this number down to 15, (A), as the computer cannot complete the 16th scan in the time allowed.

6. D/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: You need to find the ratio of milk to water. You’re given two ratios: milk to eggs and water to eggs. Both of the given ratios contain eggs, but the egg amounts (2 and 3) are not identical. To directly compare them, find a common multiple (6). Multiply each ratio by the factor that will make the number of eggs equal to 6:

Milk to eggs: (5:2) × (3:3) = 15:6

Water to eggs: (4:3) × (2:2) = 8:6

Now that the number of eggs needed is the same in both ratios, you can merge the two ratios to compare milk to water directly: 15:6:8. Therefore, the proper ratio of milk to water is 15:8, (D).

7.  680/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Break the questions into steps. First, find how long it took Verona to mulch one flowerbed. Then use that amount to find how long it took her to mulch all the flowerbeds.

According to the figure, she started the first house at 8:00 and the fifth house at 9:20, so it took her 1 hour and 20 minutes, or 80 minutes, to mulch 4 flowerbeds (she hadn’t mulched the fifth flowerbed yet). This gives a unit rate of 80 ÷ 4 = 20 minutes per flowerbed. Count the houses in the figure—there are 34. Multiply the unit rate by the number of houses to get 20 × 34 = 680 minutes to mulch all of the flowerbeds.

8.  25/Difficulty: Hard

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: The total acreage of all the flowerbeds in the community is 34 × 0.006 = 0.204 acres. Convert this amount to square feet using the conversion ratio given in the question:

Each bag of mulch covers 24 square feet, so divide to find that Verona needs 8,886.24 ÷ 24 = 370.26, or about 370, bags to mulch all the flowerbeds. The wagon holds 15 bags, so Verona will need to fill it 370 ÷ 15 = 24.7, or 25 times.

9. C/Difficulty: Easy

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: When a question involves several rates, break the situation into separate, manageable pieces and deal with each in turn. Visitors must pay the café a flat $25.00 fee to join regardless of the number of visits. The first 50 visits cost $0.30 each, or 50($0.30) = $15.00. The remaining 72 — 50 = 22 visits cost $0.10 each, or 22($0.10) = $2.20. The total cost for 72 visits is therefore $25.00 + $15.00 + $2.20 = $42.20, which is (C).

10.D/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: The total budget can be represented by 100%, so start there. The percent of the budget spent on lunch is 100% — 40% — 25% — 20% = 15%. You’re told that the club plans to spend $225 on lunch. Let x be the total amount of the budget in dollars. Then 15% of x is 225, so 0.15x = 225. Solving this equation for x yields 0.15x = 225 → x = 1,500. The total budget is $1,500. Of this amount, 40% was budgeted for a guest speaker, or 0.4 × $1,500 = $600. Choice (D) is correct.

11.D/Difficulty: Easy

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: According to the pie chart, 20% of the beads are blue. The percentage of beads that are not blue is 100% — 20% = 80%. Now be sure to answer the right question. This question asks for the number (not the percent) of beads that are not blue, not the number that are blue. The number of beads that are not blue is 80% of 120, or 0.8 × 120 = 96 beads, (D).

12.C/Difficulty: Easy

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Break the question into short steps and solve each step, checking units as you go. First, find the total number of milliliters of milk Drew will need for 5 loaves:

Second, convert the total number of milliliters needed to cups:

Drew will need about 3.8 or cups of milk, (C).

13.C/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Skim through the first couple of sentences to get the context, and then break the actual question into short steps. First, find the number of businesses that were required to have inspections. There are 2,625 businesses. The ratio of exempt to covered businesses is 5:2, so 5 parts exempt (don’t need inspections) plus 2 parts covered (do need inspections) equals a total of 2,625. Write this as 5x + 2x = 2,625, where x represents one part. Next, simplify and solve the equation to find that x = 375. Now multiply this by 2 because the ratio states that 2 parts are covered by OSHA: 375 × 2 = 750. Now use that number to find the number of businesses that had violations: 12% of 750 or 0.12 × 750 = 90.

Finally, find the number of businesses that did not have violations by subtracting 90 from 750 to get 750 — 90 = 660 businesses that did not have to address any OSHA safety issues. Choice (C) is correct.

14.B/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Start with the smallest possible green light length, 8 seconds. The next length must be at least 25% larger, so multiply by 1.25 to get 10 seconds. This is the next light length. Now multiply 10 by 1.25 to get 12.5 seconds. Then round up to the next whole second (which is not necessarily the nearest second) to meet the last requirement. You must always round up because rounding down would make the subsequent length less than 25% longer than the one before it. Only (B) works.

15.A/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Let the units in this question guide you to the solution. The speeds of the cars are given in miles per hour, but the question asks about the number of miles each car can travel in 30 seconds, so convert miles per hour to miles per second, and then multiply by 30 seconds:

Consumer car:

Race car:

The race car can travel 1.75 — 1 = 0.75 miles farther in 30 seconds, which is the same as miles, (A).

16.D/Difficulty: Hard

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Let h be the number of hours Jared spent using the Internet during the month. The first hour costs k dollars, and the remaining hours (h — 1) are charged at the rate of m dollars per hour. Therefore, the total charge for a month is k + (h — 1)m. Set this equal to the amount Jared paid and solve for h. Note that you’re not going to get a numeric answer because the question doesn’t give you the actual rates:

This expression matches (D). Note that you could also use the Picking Numbers strategy to answer this question.

17.C/Difficulty: Hard

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: It’s easiest to compare two amounts when they are written in the same units, so start by converting the car’s pints to quarts and then go from there. The conversion from pints to quarts is straightforward:

Next, find the amount of zinc in each car using the percent formula: Percent × whole = part. Write the percents as decimals and multiply:

Car: 0.09 × 4 quarts = 0.36 quarts of zinc

Truck: 0.04 × 6 quarts = 0.24 quarts of zinc

Finally, compare the amount in the car to the amount in the truck: . The car has 1.5 times as much zinc in the oil pan as the truck, which is (C).

18.C/Difficulty: Hard

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Draw a chart or diagram detailing the various price reductions for each 30 days. Determine the percent change and new price for each date.

Date

% of Most Recent Price

Resulting Price

Jul 15

100 — 40% = 60%

$1,050 × 0.6 = $630

Aug 15

100 — 20% = 80%

$630 × 0.8 = $504

Sep 15

100 — 20% = 80%

$504 × 0.8 = $403.20

You can stop here because the item was sold on October 5, which is not 30 days after September 15. The final selling price was $403.20, (C).

19.D/Difficulty: Hard

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: This is another question where the units can help you find the answer. Use the number of vehicles owned to find the total number of miles driven to find the total number of gallons of gas used. Then you can write an equation relating the tax rate to the amount of taxes paid. The starting quantity is 1.75 vehicles; to relate this quantity to the tax rate, it must first be converted to gallons of gasoline:

The tax rate was 18.4 cents per gallon, (D).

20.B/Difficulty: Hard

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Pay careful attention to the units. You need to convert all of the dimensions to inches, then set up and solve a proportion. There are 12 inches in 1 foot, so the real plane’s length is (220 × 12) + 6 = 2,640 + 6 = 2,646 inches; the length of the wingspan on the real plane is 176.5 × 12 = 2,118 inches; the length of the amusement park ride is 36 feet, 9 inches, or 441 inches; the length of the wingspan on the ride is unknown. Set up a proportion. Try writing the proportion in words first:

The ride’s wingspan should be 353 inches, which is equal to 29 feet, 5 inches, (B).

21. 42/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Questions that involve distance, rate, and time can almost always be solved using the formula Distance = rate × time. Use the speed, or rate, of the small passenger plane (200 mph) and its distance from FLL (110 mi) to determine when it arrived. You don’t know the time, so call it t:

This means it took 0.55 hours for the plane to arrive. This is less than a full hour, so multiply 0.55 by 60 to find the number of minutes it took: 60 × 0.55 = 33 minutes. Now determine how long it took the cargo plane. It left at 8:00 A.M. and arrived at 9:15 A.M., so it took 1 hour and 15 minutes, or 75 minutes. This means the small passenger plane arrived 75 — 33 = 42 minutes before the cargo plane.

22. 1036/Difficulty: Hard

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: To get started, you’ll need to find the distance for each part of the trip—the question only tells you the total distance. Then use the formula Distance = rate × time to find how long the plane flew at 300 mph and then how long it flew at 500 mph.

First part of trip:

Second part of trip:

There are two possible approaches: 825 = 550 miles OR 825 mi — 275 mi = 550 miles. Now solve for t:

This means the plane flew for a total of 55 + 66 = 121 minutes. Next, add the time the plane circled overhead: 121 + 35 = 156. The total trip took 156 minutes (2 hours and 36 minutes), which means the plane landed at 8:00 + 2 hours = 10:00 + 36 minutes = 10:36. Enter the answer as 1036.

23. 1500/Difficulty: Easy

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Find the amount of the annual fee for each portfolio, multiply these amounts by 4, and then subtract to find the difference.

Lower-risk portfolio:

0.035 × 100,000 = 3,500

3,500 × 4 = 14,000

Higher-risk portfolio:

0.0125 × 250,000 = 3,125

3,125 × 4 = 12,500

Difference:

14,000 — 12,500 = 1,500

24. .12/Difficulty: Hard

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: This question requires multiple steps and multiple formulas, so make a plan before you dive in. Percent gain is the same as percent increase, so you’ll need that formula:

This tells you that you need the final amount of the higher-risk portfolio, which will depend on the total fee for the lower-risk portfolio. You’ll need to use the percent formula (Percent × whole = part) to determine what whole would be required at 1.25% for the fees to be equal.

The annual fee for the lower-risk portfolio is 0.035 × 100,000 = 3,500. Use this amount to find the final amount for the higher-risk portfolio:

Now use the percent increase formula:

The percent increase needed is 12%, but be careful—the directions given in the question tell you to enter the number as a decimal, so grid your answer in as .12.

25. 53.6/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Break the question into short steps: Find the in-store price (no percent off and add sales tax), find the infomercial price as one of the first 250 callers (10% off, then 20% off that amount, with no sales tax), and subtract to find the savings.

With tax, the in-store price would be $160 × 1.055 = $168.80. The infomercial price (including the 10% off) would be $160 × 90% = $160 × 0.9 = $144. As one of the first 250 callers (20% more off), the final price would be $144 × 80% = $144 × 0.8 = $115.20. You would save $168.80 — $115.20 = $53.60. Grid in this amount as 53.6.

26. 126/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: When answering questions that share information, you can often save yourself some time by using amounts you found in the first question to answer the second. You already found that the price for 1 item for the first 250 callers is $115.20. Malik buys 5 items at this price for a total of $115.20 × 5 = $576. You also already found that the price for 1 item after the first 250 callers is $144. Malik buys 3 items at this price for a total of $144 × 3 = $432. This means he buys 8 items for a total cost of $576 + $432 = $1,008. To find the average price he paid, divide the total amount ($1,008) by the total number of items (8) to arrive at the answer, $126.

27. 28/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Use the formula:

To use the formula, find the part of the mass represented by the oxygen:There is 1 mole of oxygen, and it has a mass of 15.9994 grams. Next, find the whole mass of the mole of acetone: 1 mole oxygen (15.9994 g) + 3 moles carbon (3 × 12.0107 = 36.0321 g) + 6 moles hydrogen (6 × 1.00794 = 6.04764 g) = 15.9994 + 36.0321 + 6.04764 = 58.07914. Now use the formula:

Before you grid in your answer, make sure you follow the directions—round to the nearest whole percent, which is 28.

28. 48/Difficulty: Hard

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: This part of the question contains several steps. Think about the units given in the question and how you can use what you know to find what you need. Be careful—there are lots of calculations that involve decimals, and you shouldn’t round until the final answer.

Start with grams of acetone: the chemist starts with 1,800 and uses 871.1871, so there are 1,800 — 871.1871 = 928.8129 grams left. From the previous question, you know that 1 mole of acetone has a mass of 58.07914 grams, so there are 928.8129 ÷ 58.07914 = 15.9922, or about 16 moles of acetone left. Don’t grid in this amount because you’re not finished yet! The question asks for the number of moles of carbon, not acetone. According to the table, each mole of acetone contains 3 moles of carbon, so there are 16 × 3 = 48 moles of carbon left.

29. 1.25/Difficulty: Medium

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: Try writing an equation in words first and then translate into math. Let p represent the number of British pounds; 1,512 is the number of British pounds needed, including the 5% fee, to get 1,800 euros. This means 1,512 = 105% times p, or 1,512 = 1.05p. Divide by 1.05 to find that the number of British pounds needed (before the fee) is 1,440. Next, determine the exchange rate by setting up a proportion (e euros to 1 pound equals 1,800 euros to 1,440 pounds):

The exchange rate is 1.25 euros to 1 British pound.

30. 1.3/Difficulty: Hard

Category: Problem Solving and Data Analysis / Rates, Ratios, Proportions, and Percentages

Getting to the Answer: From the previous question, you know that Rosslyn lost 1,512 — 1,440 = 72 pounds because of the fee she paid to the bank in France. The question states that she lost a total of 74 pounds, so only 2 of those pounds were because of the increase in the exchange rate. Set up a proportion using what you know so far. Label as much as possible so you can keep track of exactly what is being converted:

There are two variables (e and x) in the equation, so there is not enough information to solve it yet. However, you also know that Rosslyn would have received 2 more pounds (or x + 2) at the old exchange rate, which was 1.25. Set up another proportion using this information:

Now, solve for x:

This is not the answer. The question asks for the new exchange rate, so substitute 50 back into the first proportion and solve for e:

The new exchange rate is 1.3 euros to 1 British pound.

Chapter 4

PRACTICE

8. A/Difficulty: Hard

Category: Passport to Advanced Math / Exponents

Getting to the Answer: The goal here is to solve for the variable e. You start with ; begin by squaring both sides to eliminate the radical to get . Isolating e2 gives ; take the square root of both sides to get , which is (A).

9. 5/Difficulty: Medium

Category: Passport to Advanced Math / Exponents

Getting to the Answer: Use polynomial long division to find the remainder:

The question asks for the remainder, so the correct answer is 5.

10.B/Difficulty: Medium

Category: Passport to Advanced Math / Exponents

Getting to the Answer: Start by separating the radicals:. All the terms are being multiplied, so you can cancel a and to get . Rationalize the denominator, then simplify: . Choice (B) is correct.

PERFORM

11.C/Difficulty: Medium

Category: Passport to Advanced Math / Exponents

Getting to the Answer: Convert the wavelength from nanometers into meters; using scientific notation will help (150 nm = nm) because you can use exponent rules: 1.5 × 102 nm m = m. Plug this into the given equation along with the speed of light to get m/s = m × ν.

Solving for ν gives s—1. You’ll see that 1.5 divides evenly into 3 and the bases of 10 abide by exponent rules, so s—1. Choice (C) is correct.

12.A/Difficulty: Hard

Category: Passport to Advanced Math / Exponents

Getting to the Answer: The conjugate of the denominator, , is . Multiply the original expression by to get , then distribute carefully. The numerator becomes:

Repeat with the denominator:

The answer is , which is (A).

13. 256/Difficulty: Medium

Category: Passport to Advanced Math / Exponents

Getting to the Answer: If , then . Simplify the given expression via exponent rules and then plug —2 in for n:

ON YOUR OWN

1. B/Difficulty: Easy

Category: Passport to Advanced Math / Exponents

Getting to the Answer: Raising 9 to the 4th and 5th powers is very time-consuming without a calculator. Instead, start by using exponent rules to simplify expressions written in the same base. Then raise any remaining numbers to the resulting powers. The power of 9 is larger in the denominator, so subtract the exponents there. Then simplify further if possible:

Choice (B) is correct.

2. D/Difficulty: Easy

Category: Passport to Advanced Math / Exponents

Getting to the Answer: Distribute each term in the first factor to each term in the second factor, paying careful attention to the signs:

Don’t let B fool you—the question asks for the coefficient of x, not x2, so the correct answer is (D).

3. C/Difficulty: Medium

Category: Passport to Advanced Math / Exponents

Getting to the Answer: First, move the term with the negative exponent from the denominator to the numerator. Don’t flip the exponent; just move the whole term and make the exponent positive: .

Now, rewrite the expression as a radical expression using the rule “power over root.” The numerator tells you the power of the variable before the root is applied—here, the power of the variable is 2. The denominator tells you the degree of the root—here, it is a 5th root because the denominator is 5. The number 3 is simply being multiplied by the variable, so it should be outside the radical. The result is: . Choice (C) is correct.

4. C/Difficulty: Medium

Category: Passport to Advanced Math / Exponents

Getting to the Answer: Before you can use the rules of exponents, you need to write the radical as a fraction exponent. Write as and then subtract the exponents because the terms are being divided:

5. B/Difficulty: Medium

Category: Passport to Advanced Math / Exponents

Getting to the Answer: A fraction is the same as division, so you can use polynomial long division to simplify the expression:

The simplified expression is 2x + 3, choice (B). As an alternate method, you could factor the numerator of the expression and cancel common factors:

Use whichever method gets you to the correct answer in the shortest amount of time.

6. A/Difficulty: Medium

Category: Passport to Advanced Math / Exponents

Getting to the Answer: Substitute 160 for P and simplify using your calculator. Don’t try to enter the entire expression into the calculator all at one time—you’re likely to miss a parenthesis somewhere. Instead, divide 160 by 0.02 to get 8,000. Then take the cube root of 8,000 using the cube root function or by raising it to the one-third power. The result is 20, so the wind velocity is 20 miles per hour. Choice (A) is correct.

7. A/Difficulty: Medium

Category: Passport to Advanced Math / Exponents

Getting to the Answer: The goal here is to solve the equation for L. First, divide both sides of the equation by 2π to isolate the radical. Then you’ll need to square both sides to remove the radical:

This matches (A).

8. B/Difficulty: Hard

Category: Passport to Advanced Math / Exponents

Getting to the Answer: Find the volume of each box and then subtract.

Box 1: V = x(x + 1)(x — 1) = x(x 2 — 1) = x 3 — x

Box 2: Substitute 2x for x in the volume you already found:

(2x)3 — 2x = 23x 3 — 2x = 8x 3 — 2x

Difference: (8x 3 — 2x) — (x 3x) = 8x 3x 3 — 2x + x = 7x 3x, (B).

Alternatively, you could also pick a value for x (e.g., x = 2, in which case y = 2(2) = 4) to determine the difference in volume and then check the answer choices against your result.

9. D/Difficulty: Easy

Category: Passport to Advanced Math / Exponents

Getting to the Answer: To subtract the expressions, simply combine like terms. Be careful—don’t forget to distribute the negative to all the terms of N:

This matches (D).

10.B/Difficulty: Easy

Category: Passport to Advanced Math / Exponents

Getting to the Answer: When you find the language confusing, try to put it in concrete terms. If you wanted to know how much more 9 was than 7, what would you do? You would subtract 9 — 7 = 2 more. So you need to subtract these two algebraic expressions to get (6x + 5) — (6x — 1) = 6x + 5 — 6x + 1 = 6, which is (B).

11.A/Difficulty: Easy

Category: Passport to Advanced Math / Exponents

Getting to the Answer: When asked to simplify a rational expression, start by finding the greatest common factor that can be divided out of the numerator and the denominator. Factor a 2 from both the numerator and the denominator, and then cancel the :

This matches (A).

12.B/Difficulty: Easy

Category: Passport to Advanced Math / Exponents

Getting to the Answer: First, write the question as a subtraction problem. Pay careful attention to which expression is being subtracted: .

The terms in the expression have the same denominator, x + 3, so their numerators can be subtracted. Simply combine like terms and keep the denominator the same. Don’t forget to distribute the negative to both 3x and 1:

This matches (B).

13.C/Difficulty: Hard

Category: Passport to Advanced Math / Exponents

Getting to the Answer: Because this is a no-calculator question, you need to rewrite the exponent in a way that makes it easier to evaluate. Use exponent rules to rewrite as a unit fraction raised to a power. Then write the expression in radical form and simplify:

Choice (C) is correct.

14.C/Difficulty: Hard

Category: Passport to Advanced Math / Exponents

Getting to the Answer: Always pay careful attention to units. In this question, the equation is defined in terms of inches and minutes, but the diameter of the puddle is given in feet.

Substitute the diameter (3 feet = 36 inches) for d in the equation and then solve for m. Before dealing with the radical, divide both sides of the equation by 1.25:

The liquid has been leaking for about 830 minutes, which is (C).

15.D/Difficulty: Medium

Category: Passport to Advanced Math / Exponents

Getting to the Answer: Write the first factor as repeated multiplication, expand it using FOIL, and then multiply by the second factor to find the product. Pay careful attention to signs:

This matches (D).

16.C/Difficulty: Hard

Category: Passport to Advanced Math / Exponents

Getting to the Answer: Multiply each term in the first expression by and each term in the second expression by . Then add the two polynomials by combining like terms:

This matches (C).

17.A/Difficulty: Hard

Category: Passport to Advanced Math / Exponents

Getting to the Answer:This is a great question to choose a value for x and see what happens. Take a peek at the answer choices to see that you’ll want to pick a value of x that is a perfect square (so you can take the square root). The question states that x > 1, and the next perfect square is 4, so let x = 4.

Try (A): , which is less than 0, so (A) is most likely correct.

Try B: , which is not less than 1, so eliminate B.

Try C: , which is equal to but not less than 4, so eliminate C.

Try D: , which is not greater than 42, so eliminate D.

Choice (A) is correct.

18.13/Difficulty: Medium

Category: Passport to Advanced Math / Exponents

Getting to the Answer: Treat the square root as a single quantity until you isolate it on one side of the equation. Then square both sides to get rid of the square root:

Chapter 5

PRACTICE

8. C/Difficulty: Medium

Category: Passport to Advanced Math / Functions

Getting to the Answer: Because the slopes of all the answer choices are different, you can use the slope formula to determine which choice is correct. The number of colonies depends on the time, so start by writing the information given as ordered pairs in the form (time, number of colonies). Using the ordered pairs (21, 8) and (35, 10), the slope is .

The only choice with this slope is (C).

9. A/Difficulty: Easy

Category: Passport to Advanced Math / Functions

Getting to the Answer: The notation r(x) = 0 means that the function is crossing the x-axis (has a y-value of 0), so look for the x-intercepts. The function r(x) intersects the x-axis at x = −2, 2, and 5. You can only grid in a positive answer, so use 2 or 5.

PERFORM

10.C/Difficulty: Easy

Category: Passport to Advanced Math / Functions

Getting to the Answer: This is a composition of functions, so start with the innermost set of parentheses, which is g(1). According to the table on the right, g(1) = 0; this becomes the input for f. Now find f (0); the table on the left states this is equal to 4, which is (C).

11.D/Difficulty: Medium

Category: Passport to Advanced Math / Functions

Getting to the Answer: Identifying the type of relationship between h and f (h) is key to solving this problem. The f (h) values (shingle counts) are increasing at a variable rate, which rules out a linear function. The increase in f (h) is not significant with each increase in h, so an exponential relationship is not likely. A quadratic relationship is a good bet. Start with f (h) = h2. That gives f (1) = 12 = 1, f (2) = 22 = 4, and f (3) = 32 = 9. Obviously this doesn’t match the pattern, so ask yourself how much more f (h) needs to increase after squaring h; you’ll see you need to add 11 to h2 to get the corresponding value of f (h). You’ll find that f (h) = h2 + 11 accurately depicts the relationship between h and f (h). You’re asked for the shingle count for the seventh house, so plug 7 into your function: f (7) = 72 + 11 = 49 + 11 = 60, which matches (D).

12.A/Difficulty: Hard

Category: Passport to Advanced Math / Functions

Getting to the Answer: To determine the transformed graph, pick a point on the original function f(x) = (x — 4)2, for example, (3,1). To get from x — 4 to x + 2, 6 units are added; thus, the x-coordinate is shifted 6 units left to —3: (—3,1). The plus 3 outside the parentheses indicates the y-coordinate is shifted 3 units up: (—3,4). The negative in front of the x indicates a reflection over the y-axis, so the x-coordinate is now 3: (3,4). The new point falls on the graph of (A), the correct answer. You can check your answer by plugging in values for x in the g(x) = (—x + 2)2 + 3 function to get other points on the graph: (—1, 12), (0, 7), (1, 4), (2, 3).

ON YOUR OWN

1. D/Difficulty: Easy

Category: Passport to Advanced Math / Functions

Getting to the Answer: The domain of a function represents the possible input values. In this function, the input values are represented by n, which is the number of eggs laid by the fish over a given period of time. Because there cannot be a negative number of eggs laid or a fraction of an egg laid, the list in (D) is the only one that could represent part of the function’s domain.

2. C/Difficulty: Easy

Category: Passport to Advanced Math / Functions

Getting to the Answer: The notation ’h(x)’ is read h of x and represents the range value (output) of the function that corresponds to a given domain value (input). Therefore, h(5) means the output value of the function when 5 is substituted for the input (x), and h(2) means the output value of the function when 2 is substituted for the input (x). Substitute 5 and 2 into the equation, one at a time, and then subtract the results:

Choice (C) is correct. Caution—this is not the same as subtracting 5 — 2 and then substituting 3 into the function.

3. B/Difficulty: Medium

Category: Passport to Advanced Math / Functions

Getting to the Answer: Compare each answer choice, one at a time, to the graph. Be careful—you’re looking for the statement that is not true, so cross out true statements as you go. The statement in A is true (and therefore not correct) because at x = 0, y is also 0. Move on to B. When x is 2, q(x), or y, is somewhere between 1 and 2, not equal to 4, so this statement is not true, making (B) the correct answer. The statements in C and D are both true because the x- (domain) and y-values (range) are all greater than or equal to 0.

4. D/Difficulty: Medium

Category: Passport to Advanced Math / Functions

Getting to the Answer: In this question, you are given that the value of f (x) is 5, and you are asked for the value of x that produces this result. This means you are solving for x, not substituting for x. Set the function equal to 5 and solve using inverse operations:

5. C/Difficulty: Medium

Category: Passport to Advanced Math / Functions

Getting to the Answer: The question tells you that the function is linear, which means you need to know the slope (rate of change in the height of the solution) and the y-intercept (height of the solution when there are 0 pennies) to pick the correct function.

You already know the height of the solution when there are 0 pennies—it’s 5 inches. This means you can eliminate B and D. To determine the rate of change in the height of the solution, write what you know as ordered pairs, and then use the slope formula:

At 0 pennies, the height is 5 inches → (0, 5).

At 50 pennies, the height is 8.5 inches → (50, 8.5).


The rate of change in the height of the solution is . This means the correct function is h(p) = 0.07p + 5, which is (C).

6. C/Difficulty: Medium

Category: Passport to Advanced Math / Functions

Getting to the Answer: The notation (g(h(x)) indicates a composition of two functions that can be read “g of h of x.” It means that the output when x is substituted into h(x) becomes the input for g(x). First, use the h(x) table to find h(—2); you’ll see it equals 4. Use this as the domain in the g(x) table to find g(4), which is 2. Choice (C) is correct.

7. A/Difficulty: Hard

Category: Passport to Advanced Math / Functions

Getting to the Answer: Transformations that are grouped with the x in a function shift the graph horizontally and therefore affect the x-coordinates of points on the graph. Transformations that are not grouped with the x shift the graph vertically and, therefore, affect the y-coordinates of points on the graph. Remember, horizontal shifts are always the opposite of what they look like.

Start with (x + 5). This shifts the graph left 5 units, so subtract 5 from the x-coordinate of the given point: (—2, 6) → (—2 — 5, 6) = (—7, 6). Next, apply the negative in front of R, which makes the y-coordinate negative: (—7, 6) → (—7, —6). Finally, the + 1 is not grouped with x, so add 1 to the y-coordinate: (—7, —6) → (—7, —6 + 1) = (—7, —5), which is (A).

You could also plot the point on a coordinate plane, perform the transformations (left 5, reflect vertically over the x-axis, and then up 1), and find the resulting point.

8.  5/Difficulty: Medium

Category: Passport to Advanced Math / Functions

Getting to the Answer: Evaluate the function at t = 20 and at t = 10, and then subtract the results. Make sure you follow the correct order of operations as you simplify:

The question asks how many more parts per million are in a patient’s bloodstream after 20 hours than after 10 hours, so subtract 22 — 17 = 5.

9. B/Difficulty: Easy

Category: Passport to Advanced Math / Functions

Getting to the Answer: If a relationship represents a function, then each input can have only one corresponding output. When a function is presented as a set of ordered pairs, this means that no number in the domain (the first value in an ordered pair) can be repeated. Choice (B) is correct because each input (x-value) has only one output (y-value). All other choices have multiple outputs for a single input.

10.D/Difficulty: Easy

Category: Passport to Advanced Math / Functions

Getting to the Answer: To determine the domain, look at the x-values. To determine the range, look at the y-values. For the domain, the graph is continuous and has arrows on both sides, so the domain is all real numbers. This means you can eliminate choices A and B. For the range, the function’s maximum is located at (0, 5), which means the highest possible value of f (x) is 5. The graph is continuous and opens downward, so the range of the function is f (x) ≤ 5, making (D) correct.

11.D/Difficulty: Medium

Category: Passport to Advanced Math / Functions

Getting to the Answer: Skim the answer choices. They are all asking about the sign (positive or negative) of g(x) because g(x) < 0 means negative and g(x) > 0 means positive. When using function notation, g(x) is the same as y, so you are trying to decide whether (and when) the y-values of the function are negative or positive. The quickest way to do this is to draw a sketch of the function, such as the one shown here:

image

From the graph, you can see that part of the line will eventually be below the y-axis (or negative) and part above the y-axis (or positive), so you can eliminate choices A and B. To choose between C and D, you’ll need to examine the graph more carefully. There are some values of x that are less than 0 (to the left of the vertical axis) for which the y-values are greater than 0, so C is not true either, which means choice (D) must be correct. For all values of x greater than 0, the y-values are in fact all positive (or g(x) > 0).

12.A/Difficulty: Medium

Category: Passport to Advanced Math / Functions

Getting to the Answer: Graphically, the notation f (3) means the y-value when x is 3. Pay careful attention to which graph is which. It may help to draw dots on the graph. Find x = 3 on the x-axis, then mark each function at this x-value on the graph as shown here:

image

Now read the y-coordinates from the graph and subtract: f (3) is —2 and g(3) is 1, so f (3) — g(3) = −2 — 1 = −3, (A).

13.A/Difficulty: Hard

Category: Passport to Advanced Math / Functions

Getting to the Answer: The notation (f º g)(x) indicates a composition of two functions that can also be written as f (g(x)) and is read “f of g of x.” It means that the output when x is substituted into g(x) becomes the input for f (x). Substitute for x in g(x), simplify, and then substitute the result into f (x):

Therefore, , (A).

14.A/Difficulty: Hard

Category: Passport to Advanced Math / Functions

Getting to the Answer: Think about how each transformation affects the graph of g(x), and draw a sketch of k(x) on the same grid. Compare the new graph to each of the answer choices until you find one that is true. The graph of k(x) = —g(x) — 1 is a vertical reflection of g(x) over the x-axis that is then shifted down 1 unit. The graph looks like the dashed line here:

image

Now compare the dashed line to each of the answer choices. The range of k(x) is the set of y-values from lowest to highest (based on the dashed line). The lowest point occurs at the new point B and has a y-value of —5, and the highest value occurs at both ends of the graph and is 1, so the range is —5 ≤ y ≤ 1. This means (A) is correct and you can move on to the next question. Don’t waste valuable time checking the other answer choices unless you are not sure about the range. (Choice B: The minimum value of k(x) is —5, not —4. Choice C: The coordinates of point A on k(x) are (—2, —4), not (2, 0). Choice D: The graph of k(x) is decreasing, not increasing, between x = 0 and x = 1.)

15.D/Difficulty: Hard

Category: Passport to Advanced Math / Functions

Getting to the Answer: When dealing with a composition, the range of the inner function becomes the domain of the outer function, which in turn produces the range of the composition. In the composition f (g(x)), the function g(x) = x2 is the inner function. Every value of x, when substituted into this function, will result in a non-negative value (because of the square on x). This means the smallest possible range value of g(x) is 0. Now look at f (x). Substituting large positive values of x in the function will result in large negative numbers. Consequently, substituting the smallest value from the range of g, which is 0, results in the largest range value for the composition, which is —0 + 5 = 5. Because 9 > 5, it is not in the range of f (g(x)), making (D) correct.

Chapter 6

PRACTICE

· D/Difficulty: Medium

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: You’ll need to find the y-coordinate of the vertex to discover the maximum height of the projectile. The x-coordinate is given by . Now use that to get the y-coordinate by plugging 4 into the height function:

Move on to the time it takes for the projectile to hit the ground. You know this will occur when f (t) = 0. Plug in and factor to solve:

Because you need a positive solution, t = 10. Now find the product of 10 and 576 to reveal that (D) must be correct.

· B/Difficulty: Medium

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: The y-coordinate will be 0 when the function crosses the x-axis, so plug in points to quickly eliminate A and C. Now FOIL the binomials of the solutions to get a quadratic:

Calculate the x-coordinate of the vertex of this quadratic: . Because (B) and D are in vertex form, simply match to the appropriate function. Recall that form is , so (B) is correct.

PERFORM

· A/Difficulty: Easy

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: Rearrange the equation first so you can factor 4 out. From there, divide by 4, then factor as usual:

Keep in mind that while z is equal to —10 or 2, the question only asks for the positive value, which is (A). This is very tricky, so read carefully.

· B/Difficulty: Hard

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: Because you have variables, you might think to use the Picking Numbers strategy. Unless you coincidentally pick the “right” numbers, you won’t actually get an answer choice, but you’ll be able to confirm what happens with those numbers. Given the rules of the problem, you might pick a = −2 and b = 3. Setting up the binomials and performing FOIL, you get:

Unfortunately, this isn’t one of the answer choices. What makes this difficult is that when you naturally set up the binomials, you assume the parabola is opening up. If you multiply the entire function by —1, it reflects about the x-axis, but because the solutions are on the x-axis, they will stay the same. The correct answer is (B).

· 0/Difficulty: Easy

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: All the question is really asking you to do is solve for the zeros and subtract them:

The quadratic only has one unique solution, 4, so the positive difference between them is actually 0.

· C/Difficulty: Medium

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: An axis of symmetry splits the parabola in half and travels through the vertex. All you need to do is solve for the x-coordinate of the vertex, which can be accomplished using the formula for h (the quadratic formula without the square root portion):

Thus, the answer is (C).

ON YOUR OWN

· C/Difficulty: Easy

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: Taking the square root is the inverse operation of squaring, and both sides of this equation are already perfect squares, so take their square roots. Then solve the resulting equations. Remember, there will be two equations to solve:

Now simplify each equation: and . Choice (C) is correct.

· B/Difficulty: Easy

Category: Passport to Advanced Math/Quadratics

Getting to the Answer: From the factored form of the equation, you can see that the values of —2 and 7 would make y equal 0, so the x-intercepts are —2 and 7. This means you can eliminate A and C. From the standard form of the equation, you can see that the y-intercept is —14 because 02 — 2(0) — 14 = −14, so (B) is correct.

· D/Difficulty: Medium

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: Quadratic equations can be written in several forms, each of which reveals something special about the graph. For example, the vertex form of a quadratic equation (y = a(xh)2 + k) gives the minimum or maximum value of the function (it’s k), while the standard form (y = ax 2 + bx + c) reveals the y-intercept (it’s c). The factored form of a quadratic equation reveals the solutions to the equation, which graphically represent the x-intercepts. Choice (D) is the only equation written in factored form and therefore must be correct. You can set each factor equal to 0 and quickly solve to find that the x-intercepts of the graph are and x = 2, which agree with the graph.

· B/Difficulty: Medium

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: When finding solutions to a quadratic equation, always start by rewriting the equation to make it equal 0 (unless both sides of the equation are already perfect squares). Then take a peek at the answer choices—if they are all nice numbers, then factoring is probably the quickest method for solving the equation. If the answers include messy fractions or square roots, then using the quadratic formula may be a better choice. To make the equation equal 0, subtract 48 from both sides to get x 2 + 8x — 48 = 0. The answer choices are all integers, so factor the equation. Look for two numbers whose product is —48 and whose sum is 8. The two numbers are — 4 and 12, so the factors are (x — 4) and (x + 12). Set each factor equal to 0 and solve to find that x = 4 and x = −12. The question states that x > 0, so x must equal 4. Before selecting an answer, don’t forget to check that you answered the right question—the question asks for the value of x — 5, not just x, so the correct answer is 4 — 5 = −1, (B).

· C/Difficulty: Medium

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: Equations that are equivalent have the same solutions, so you are looking for the equation that is simply written in a different form. You could expand each of the equations in the answer choices, but unless you get lucky, this will use up quite a bit of time. The answer choices are written in vertex form, so use the method of completing the square to rewrite the equation given in the question stem. First, subtract the constant, 17, from both sides of the equation. To complete the square on the right-hand side, find , and add the result to both sides of the equation:

Next, factor the right-hand side of the equation (which should be a perfect square trinomial), and rewrite it as a square. Finally, solve for y:

This matches (C).

· B/Difficulty: Medium

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: Even though one of the equations in this system is not linear, you can still solve the system using substitution. You already know that y is equal to 3x, so substitute 3x for y in the second equation. Don’t forget that when you square 3x, you must square both the coefficient and the variable:

The question asks for the value of x 2, not x, so there is no need to take the square root of 36 to find the value of x. Choice (B) is correct.

· A/Difficulty: Medium

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: The roots of an equation are the same as its solutions. The equation is already written in the form y = ax 2 + bx + c and the coefficients are fairly small, so using the quadratic formula is probably the quickest method. Jot down the values that you’ll need: a = 1, b = 8, and c = −3. Then substitute these values into the formula and simplify:

This is not one of the answer choices, which tells you that you’ll need to simplify the radical, but before you do, you can eliminate C and D because is —4. To simplify the radical, look for a perfect square that divides into 76 and take its square root:

This matches (A).

· A/Difficulty: Hard

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: To get a picture of what is going on, you could draw a quick sketch using the information given in the question and what you know about quadratic equations. First, determine the solutions of the equation by thinking about where the x-intercepts, or the zeros, of its graph would be. Because the lead ball is thrown from ground level, the graph would begin at the origin, so one solution is 0. The ball returns to the ground, or has a height of 0 again, at 150 feet, so the other solution is 150. Write the solutions as factors: x — 0 and x — 150. Then write the equation as the product of the two factors and use the distributive property (or FOIL) to multiply them together: y = (x — 0)(x — 150) = x(x — 150) = x 2 — 150x. The general equation is y = a(x 2 — 150x). To find the value of a, use another point that satisfies the equation or, in other words, lies on its graph. Because the graph of a parabola is symmetrical, the ball reaches its maximum height, 45 feet, exactly halfway between the two zeros, 0 and 150, which is 75. Therefore, another point on the graph is (75, 45). Substitute the x- and y-values into the equation, and solve for a:

The equation is y = −0.008(x 2 — 150x) or y = −0.008x 2 + 1.2x.

If you have time, you could also graph each equation in your graphing calculator and find the one that has a maximum value of 45. Choice (A) is the only equation for which this is true.

· B/Difficulty: Medium

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: According to the graph, one x-intercept is to the left of the y-axis, and the other is to the right. This tells you that one value of x is positive, while the other is negative, so you can immediately eliminate choices A and C (both factors have the same sign). To choose between choices (B) and D, find the x-intercepts by setting each factor equal to 0 and solving for x. In choice (B), the x-intercepts are 7 and —3. In choice D, the x-intercepts are 1 and —10. Choice (B) is correct because the x-intercepts are exactly 10 units apart, while the x-intercepts in choice D are 11 units apart.

· B/Difficulty: Medium

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: There are no coefficients (numbers) in the equation, so you’ll need to think about how the values of a, b, and c affect the graph. You’ll also need to recall certain vocabulary. For example, increasing means rising from left to right while decreasing means falling from left to right, and zero is another way of saying x-intercept. Compare each statement to the graph to determine whether it is true, eliminating choices as you go. Remember, you are looking for the statement that is not true. The parabola opens downward, so the value of a must be negative, which means you can eliminate A. When a quadratic equation is written in standard form, c is the y-intercept of the parabola. According to the graph, the y-intercept is above the x-axis and is therefore positive, so the statement in (B) is false, making it the correct answer. Move on to the next question. (Choice C is true because the graph rises from left to right until you get to x = 3, and then it falls. Choice D is true because the zeros are the same as the x-intercepts, and the graph does intersect the x-axis at —2 and 8.)

· C/Difficulty: Medium

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: When a quadratic equation is written in vertex form, y = a(xh)2 + k, the minimum value (or the maximum value if a < 0) is given by k, and the axis of symmetry is given by the equation x = h. The question states that the minimum of the parabola is —3, so look for an equation where k is —3. You can eliminate choices A and B because k is +2 in both equations. The question also states that the axis of symmetry is 2, so h must be 2. Be careful—this is tricky. The equation in choice D is not correct because the vertex form of a parabola has a negative before the h, so (x + 2) actually means (x — (—2)), and the axis of symmetry would be —2. This means (C) is correct.

You could also graph each equation in your graphing calculator to see which one matches the criteria given in the question, but this is likely to use up valuable time on Test Day.

· A/Difficulty: Medium

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: When a quadratic equation is written in vertex form, y = a(xh)2 + k, the axis of symmetry is given by the equation x = h. Don’t let the different letters in the equation confuse you. The letter p is simply being used in place of h. Because the h (in the vertex form of a quadratic) has a negative in front of it, the value of h is the opposite sign of the operation performed on h. Here, h (and therefore p) is —5 because (x — (—5)) = (x + 5). So the axis of symmetry is x = −5, which is (A).

· D/Difficulty: Medium

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: Look for an equation that has a maximum value that is less than the one shown in the graph (because Meagan’s ball did not go as high). You do not need to graph the equations to determine this. The maximum value shown in the graph is about 31 feet. When a quadratic equation is written in vertex form, y = a(xh)2 + k, the maximum value is given by k, so check C and D first because they will be the easiest to compare to the graph. In C, k is 35, which is greater than 31 and therefore not correct. In (D), k is 28, which is less than 31 and therefore the correct answer. You do not need to examine the other two equations. Note: If all of the answer choices had been given in standard form, you would have needed to convert each one to vertex form, or you could have substituted the result of finding into each equation to find the maximum value.

· A/Difficulty: Hard

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: To answer this question, you need to recall just about everything you’ve learned about quadratic graphs. The equation is given in vertex form (y = a(xh)2 + k), which reveals the vertex (h, k), the direction in which the parabola opens (upward when a > 0 and downward when a < 0), the axis of symmetry (x = h), and the minimum/maximum value of the function (k).

Start by comparing each answer choice to the equation, y = −2(x — 6)2 + 5. The only choice that you cannot immediately compare is A because vertex form does not readily reveal the y-intercept, so start with B. Don’t forget, you are looking for the statement that is not true. Choice B: The axis of symmetry is given by x = h, and h is 6, so this statement is true and therefore not correct. Choice C: The vertex is given by (h, k), so the vertex is indeed (6, 5) and this choice is not correct. Choice D: The value of a is —2, which indicates that the parabola opens downward, so this choice is also not correct. That means (A) must be the correct answer. To confirm, you could substitute 0 for x in the equation to find the y-intercept:

The y-intercept is (0, —67), not (0, 5), so the statement is false.

· D/Difficulty: Hard

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: You are not expected to solve each system. Instead, think about it graphically. The solution to a system of equations is the point(s) where their graphs intersect, so graph each pair of equations in your graphing calculator, and look for the ones that intersect at x = −8 and x = −3. The graphs of the equations in A don’t intersect at all, so you can eliminate A right away. The graphs in B and C intersect, but one of the points of intersection for each pair is positive, which means (D) must be correct. The graph looks like the following:

image

· 5/Difficulty: Hard

Category: Passport to Advanced Math / Quadratics

Getting to the Answer: Unfortunately, this is a non-calculator question, so you’ll need to solve the system using substitution. Substitute the first equation for y into the second. Before you solve for x, multiply the whole equation by 2 to remove the fractions. If you factor along the way, don’t forget that you must first set the whole equation equal to 0:

Now set each factor equal to 0 and solve to find that x = −1 and x = 5. The question only asks for a, which is the x-coordinate of the solution, so you do not need to substitute x back into an equation and solve for y. The two possible values of a are —1 and 5. Because the question specifies that a > 0, the answer must be 5.

Chapter 7

PRACTICE

· C Difficulty: Easy

Category: Additional Topics in Math / Geometry

Getting to the Answer: The measures of the interior angles of a triangle must add up to 180, so write an equation that reflects this: . Simplify the equation to , which reduces to . But you’re not done yet; this isn’t the angle requested. You need an angle that is supplementary to ∠ACB. The measure of ∠ACB is (x + 20)º = (65 + 20)º = 85º, so the supplementary angle is 180º — 85º = 95º. This matches (C).

· 52 Difficulty: Medium

Category: Additional Topics in Math / Geometry

Getting to the Answer: First, draw a rectangle to represent the wall space. Next draw another rectangle inside the first to represent the TV. Label the diagram with the information the question provides:

To find the dimensions of the TV, subtract 6 from the length and width of the wall space. Note that many students might erroneously subtract only 3 inches from each dimension, yielding TV dimensions of 25.2550. However, the space adds 3 inches on all four sides of the TV, so you should actually subtract 6 inches from the length and width of the wall space. Therefore, the correct TV dimensions are 22.25 × 47.

sat_c13_cb_13.eps

With the TV dimensions in hand, you can use the Pythagorean theorem to determine the size (a.k.a. the diagonal) of the largest TV that Whitney can fit in this wall space:

Grid in 52.

· 70 Difficulty: Medium

Category: Additional Topics in Math / Geometry

Getting to the Answer: The shaded area of the circle is a sector, and x marks a central angle. You’re given a vital piece of information about the shaded area: it is of the total circle. This is your area ratio for a circle proportion. You know that a circle contains 360°, so you can find the central angle of the shaded area. Use a variable other than x for this calculation to avoid confusion:

The sector marked by A and B has a central angle of 90°. Because is a diameter, you know this sector combined with the shaded sector and the one marked by x must add up to 180°. That means .

· D Difficulty: Easy

Category: Additional Topics in Math / Geometry

Getting to the Answer: Pick a pair of manageable numbers to represent the radii when doing your calculations. Suppose the first sphere has a radius of 1 and the second sphere has a radius of 2. That means the volume of the first sphere is , and the volume of the second sphere is . The ratio of the volumes is therefore :, which simplifies to 1:8, making (D) the correct answer.

You can also use r and 2r in your volume calculations: the corresponding volumes of the spheres would be and . The r  3 and components cancel, leaving 1:8 as the ratio.

PERFORM

· Difficulty: Medium

Category: Additional Topics in Math / Geometry

Getting to the Answer: Although you won't have graph paper on Test Day, drawing a quick sketch is still the quickest path to the correct answer. The coordinates of the vertices are fairly small, so draw a quick grid, plot the vertices of ΔSNO, and then follow the steps described in the question to determine where ΔICE is located. The following coordinate plane shows both:

Two right triangles drawn on a coordinate plane. The coordinates of the vertices of the top triangle are: I, zero comma 5; C, negative 4 comma 5; and E, zero comma 2. The coordinates of the vertices of the bottom triangle are: S, zero comma zero; N, 4 comma zero; and O, zero comma negative 3.

Right away you can see that the y-intercept of the hypotenuse of ΔICE is 2, so eliminate B and D. Calculate the slope of the line that forms the hypotenuse: . Choice (A) has the correct equation.

· Difficulty: Medium

Category: Additional Topics in Math / Geometry

Getting to the Answer: The sides of a 30-60-90 triangle are in the ratio x:x:2x, so label the triangle accordingly:

sat_c13_cb_19.eps

The area of a triangle is , and you’re told that the area of this triangle is . Substitute this for A and use the side lengths from the ratio. Then solve for x:

Length must be positive, so the value of x is 4. This matches A, but you're not done yet! You solved for x, which is the length of . The question asks for the length of , which is twice , so the correct answer is 8, which is (D).

· Difficulty: Medium

Category: Additional Topics in Math / Geometry

Getting to the Answer: You can use Picking Numbers, but the math is simple enough that straight algebra will be quicker. First, find the area of the rectangle You're told that and are tangent to the circle, so they touch the circle only once, meaning there is no space between the circle and rectangle. Therefore, the circle’s diameter equals the width of the rectangle (2x), and the radius must be half that, or x. The area of the circle is . This makes the area of the shaded portion , which is (A).

· Difficulty: Hard

Category: Additional Topics in Math / Geometry

Getting to the Answer: Identify familiar 2-D shapes present in the solid that will aid you. The base of the paperweight is a square with a diagonal of length centimeters. The diagonal of a square cuts the square into two 45-45-90 triangles, so each side of the square base is 2 centimeters. The height of the paperweight is 50% larger than this; therefore, it is 1.5 × 2 = 3 centimeters high. Plug these values into the volume formula for a right pyramid: cubic centimeters. The volume of water in the rectangular box is 2 × 4 × 6 = 48 cubic centimeters. The paperweight displaces 4 cubic centimeters of this water, so the answer is 48 — 4 = 44.

ON YOUR OWN

· Difficulty: Easy

Category: Additional Topics in Math / Geometry

Getting to the Answer: The sides of a 30-60-90 triangle are in the ratio x:x:2x. Don’t forget—this ratio is provided for you on the formula page at the beginning of each math section of the PSAT. Because 17 feet is opposite the 30° angle, the hypotenuse will be twice its length, or 17 × 2 = 34 feet, (D).

· Difficulty: Medium

Category: Additional Topics in Math / Geometry

Getting to the Answer: First, use the coordinate grid to figure out the dimensions of the triangle. Point B has the same x-coordinate as A and the same y-coordinate as C, so its coordinates are (—3, —2). The length of is 5 — (—2) = 5 + 2 = 7, and the length of is 4 — (—3) = 4 + 3 = 7.

Right triangle A B C drawn on a coordinate plane. Vertex A has coordinates negative three comma five. Vertex C has coordinates four comma negative two. Vertex B is at the right angle of the triangle and has coordinates negative three comma negative two. There is a vertical measure line along side A B showing that it has length 7. There is a horizontal measure line along side B C showing that it has length 7 as well.

Because the triangle is a right triangle with equal leg lengths, it must be a 45-45-90 triangle, so the measure of angle ACB is 45 degrees, or (C).

· Difficulty: Medium

Category: Additional Topics in Math / Geometry

Getting to the Answer: Many geometry questions require adding lines to the figure given. If you’re not sure where to get started on a geometry question, see if you can add a line to make a right triangle or other familiar figure:

A path consisting of three connected line segments. The path begins at point W and goes 7 miles directly north to point X. The path then goes from point X directly east for 24 miles to point Y. Finally, the path goes from point Y directly north again for 3 miles to point Z. Three dashed lines forming a right triangle have been added to the diagram. One dashed line connects points W and Z and is the hypotenuse of the right triangle. A second dashed line is below and parallel to line segment X Y, has a length of 24, and is one leg of the right triangle. The other leg of the right triangle consists of line segment Z Y, which has length three, and a dashed line that is parallel and equal in length to segment X W. The total length of this leg is three plus seven equals 10.

First you need to find the straight-line distance from W to Z. Think of as the hypotenuse of a right triangle. If you can determine the lengths of the legs of the triangle, then you can find the length of the hypotenuse. The length of one leg of the triangle is 24 miles, the horizontal distance that Gina travels. The length of the other leg is 7 + 3 = 10 miles, the vertical distance that Gina travels. The two legs of the right triangle are 10 and 24. Now, apply the Pythagorean theorem:

The route shown in the figure is 7 + 24 + 3 = 34 miles long. The direct route is 26 miles, so if Gina were able to travel from W to Z directly, her trip would be 34 — 26 = 8 miles shorter, which is (A).

· Difficulty: Hard

Category: Additional Topics in Math / Geometry

Getting to the Answer: Recall that the sides of a 30-60-90 triangle are in the ratio x:x:2x. Because = 2 and JKM is a 30-60-90 triangle, . The question tells you that M is the midpoint of , so . The question is asking for , the hypotenuse of JKL, so use the Pythagorean theorem:

This matches (D).

· Difficulty: Hard

Category: Additional Topics in Math / Geometry

Getting to the Answer: In a right triangle, one leg is the base, and the other is the height. Use x and x + 4 to represent the lengths of these two legs. Use the area formula and set the equation equal to the given area:

Now, subtract 96 to make the equation equal 0 and then factor it. The factors are (x + 12) and (x — 8), which means x = −12 and x = 8. Lengths cannot be negative, so the shorter leg must have a length of 8. This means the longer leg has a length of 8 + 4 = 12. Now use the Pythagorean theorem to find the length of the hypotenuse:

Choice (B) is correct.

· Difficulty: Medium

Category: Additional Topics in Math / Geometry

Getting to the Answer: Corresponding sides of similar triangles are proportional. This means the perimeters of similar triangles are also proportional and in the same proportion as each pair of corresponding side lengths.

Find the perimeter of ΔCAT and compare it to the perimeter of ΔDOG. ΔCAT has a perimeter of 7 + 12 + 15 = 34. The perimeter of ΔDOG is times larger than the perimeter of ΔCAT, so each side length of ΔDOG must be 1.5 times longer than the corresponding side of ΔCAT. The longest side of ΔCAT is 15, so the longest side of ΔDOG is 15 × 1.5 = 22.5, which is (D).

· Difficulty: Hard

Category: Additional Topics in Math / Geometry

Getting to the Answer: To find the area of a triangle, you need to know the lengths of its base and height. Try drawing a sketch to determine these lengths.

image

Similar triangles have congruent angles. Based on the sketch, ΔPQR is a right triangle with leg lengths of 5 and 5, making it a 45-45-90 triangle. Because ΔSTU is similar, it must also be a 45-45-90 triangle. has length 9, so must also have length 9. In a right triangle, the legs form the base and the height, so the area of the triangle is .

· Difficulty: Hard

Category: Additional Topics in Math / Geometry

Getting to the Answer: There are two right triangles in the diagram—the smaller one that has a base of 3 feet and a height of 4.5 feet (the height of the boy converted to feet) and the larger one that has a base of 3 + 21 = 24 feet and an unknown height (the height of the pole). The two triangles share one angle (the small angle on the left side), and each has a 90° angle (where the boy and the pole meet the ground), making the third pair of corresponding angles also congruent. This means the triangles are similar by AAA.

Because the triangles are similar, their sides are in proportion. Writing the proportion in words first will help you stay organized:

The height of the telephone pole is 36 feet.

· Difficulty: Easy

Category: Additional Topics in Math / Geometry

Getting to the Answer: Use the relationship to answer this question. To help you remember this relationship, just think .

The unknown in this question is the arc length, so call it a. You need to find the circumference of the circle before you set up the relationship. The question tells you that the radius ( ) is 9, so use the formula C = 2πr to find that the circumference is 2π(9) = 18π. The answers are given in terms of π, so don’t simplify any further. You also know that the central angle has a measure of 100°, so you’re ready to set up and solve the relationship:

Choice (B) is correct.

· Difficulty: Medium

Category: Additional Topics in Math / Geometry

Getting to the Answer: Add any information you have to the figure. For instance, must be 90 degrees, since it is one corner of a square. This means the shaded sector is equal to 360° ÷ 90° = of the total area of the circle.

Now, find the length of one side of the square. The perimeter of a square is 4 times the length of a side. Because the perimeter of this square is 24, a side of this square has length 24 ÷ 4 = 6. One side is equal to the radius of the circle, so the radius is also 6. This means the area of the circle is π(6)2 = 36π. So, the area of the shaded portion is 36π ÷ 4 = 9π, which is (A).

· Difficulty: Medium

Category: Additional Topics in Math / Geometry

Getting to the Answer: The equation of a circle takes the form (xh)2 + (yk)2 = r  2, where (h, k) is the center of the circle and r is the length of the radius. This means that to find the equation of a circle, you need the radius and the x- and y-coordinates of the center point. Look at the figure—Q and R lie on a diameter of the circle, so the center must lie halfway between them on the x-axis. The diameter of the circle is the length of , or 1.5 — (—2.5) = 4. Therefore, the radius is 2 and the x-coordinate of the center point is —2.5 + 2 = −0.5. This means the center is (—0.5, 0), and the equation of the circle is (x — (—0.5))2 + (y — 0)2 = 22, or (x + 0.5)2 + y 2 = 4. That's choice (B).

· Difficulty: Hard

Category: Additional Topics in Math / Geometry

Getting to the Answer: When the equation of a circle is in the form (xh)2 + (yk)2 = r  2, the r is the radius. To get the equation into this form, complete the square. You already have an x 2 and a y 2 in the given equation and the coefficients of x and y are even, so completing the square is fairly straightforward—there are just a lot of steps. Start by grouping the x’s and y’s together. Then, take the coefficient of the x term and divide it by 2, square it, and add it to the two terms with x variables. Do the same with the y term. Don’t forget to add these amounts to the other side of the equation as well. This creates a perfect square of x terms and y terms, so take the square root of each:

The equation tells you that r  2 is 81, which means that the radius is 9 and the diameter is twice that, or 18. Choice (B) is correct.

· Difficulty: Medium

Category: Additional Topics in Math / Geometry

Getting to the Answer: After it’s poured, the volume of the milk in the cylinder will still be the same volume as the rectangular container. The volume of the rectangular container is 4 × 9 × 10, or 360 cubic inches. The volume of a cylinder equals the area of its base times its height, or πr 2h. Because the diameter is 6 inches, the radius, r, is 3 inches.

Now we’re ready to set up an equation to solve for h (which is the height of the milk):

This matches (C).

· Difficulty: Easy

Category: Additional Topics in Math / Geometry

Getting to the Answer: Don’t forget to refer to the reference page provided for you on Test Day rather than trying to memorize all the area and volume formulas that you might encounter. The volume of a regular pyramid is given by , where B is the area of the base and h is the height. Here, the base is a square, so write the formula as , where s represents the length of one side of the base.

Because you’re given the volume and the height of the pyramid, plug them into the formula and solve for s:

Therefore, the length of one side of the base is 30 meters, or (B).

· Difficulty: Medium

Category: Additional Topics in Math / Geometry

Getting to the Answer: Be careful—the question asks for the volume in cubic feet, but the figure shows the dimensions of the building in yards. You must first convert the yards to feet (by multiplying each dimension by 3); then you can find the volume in cubic feet using the formula V = l × w × h. Write the conversions on the figure so you don’t forget.

To find the volume of the building, imagine splitting the figure into two rectangular prisms as shown in the figure:

A building that is composed of two rectangular prisms. The dimensions are given in feet and there is a dashed line dividing the building into two parts. The shorter prism on the left has height 150 feet, width 120 feet, and length 90 feet. The taller prism on the right has height 300 feet, width 120 feet, and length 120 feet.

Then use the volume formula, V = l × w × h, to find the volume of each prism. The shorter prism on the left has a length of 210 ft — 120 ft = 90 feet, a width of 120 feet, and a height of 150 feet, so the volume of this prism is 90 × 120 × 150 = 1,620,000 cubic feet. The taller prism on the right has a length of 120 feet, a width of 120 feet, and a height of 300 feet, so the volume of this prism is 120 × 120 × 300 = 4,320,000 cubic feet. The total volume of space inside the building is 1,620,000 + 4,320,000 = 5,940,000 cubic feet, which is (C).

· Difficulty: Medium

Category: Additional Topics in Math / Geometry

Getting to the Answer: Find the diameter of each sphere using the volume formula. Be careful—the formula gives the radius, not the diameter, so after you solve for r, be sure to multiply it by 2 to find the diameter. Start with the smaller sphere:

Now, the large sphere:

Therefore, the diameter of the larger sphere is 6 — 2 = 4 units longer than the smaller sphere, (B).

Chapter 8

PRACTICE

Suggested Passage Map notes:

· ¶1: panda classification controversy

· ¶2: homologous trait categorization

· ¶3: analogous trait categorization

· C/Difficulty: Medium

Category: Inference

Getting to the Answer: Review your notes from the second and third paragraphs to determine how these two paragraphs differ. Your notes for the second paragraph should focus on homologous traits; the third paragraph focuses on analogous traits. Choice (C) accurately depicts the difference between the paragraphs.

· B/Difficulty: Easy

Category: Vocab-in-Context

Getting to the Answer: Remember, the author must explain unfamiliar terms through context in the passage. In the cited line, the author discusses classifying. When you check your Passage Map, you’ll find the word “taxonomists” in the first paragraph in the discussion about classifying pandas. Choice (B) is, therefore, the correct answer.

PERFORM

· Suggested Passage Map notes:

· ¶1: influence of autos in U.S. (purpose)

· ¶2: manufacturing: U.S. v. Europe, pros v. cons

· ¶3: pop. dist. impact: examples

· ¶4: personal impact: examples

· C/Difficulty: Hard

Category: Rhetoric

Getting to the Answer: This is a Rhetoric question in which you are asked to analyze the purpose of part of the passage. Lines 18—22 are part of the author’s discussion, in the second paragraph, of changes to the American workplace brought about by the car. The author mentions the fact that American factories used assembly lines (as opposed to European factories, which did not) in order to emphasize the value that was placed on efficiency, as opposed to workers’ pride in the product, in American factories. That’s choice (C).

· C/Difficulty: Easy

Category: Inference

Getting to the Answer: Determine the author’s reasoning using context. The author uses the key word “unlike” to show contrast. The passage states that railroads brought people together, while cars brought people apart. Choice (C) correctly expresses this idea.

· C/Difficulty: Medium

Category: Vocab-in-Context

Getting to the Answer: On Vocab-in-Context questions, it’s often helpful to use a one-word paraphrase as your prediction. The author’s purpose is to explain how cars have “greatly” affected American life, so look for a similar word among the answer choices. Choice (C) matches. Choice A is extreme; an “absolute” effect would mean that there are no other influences. Choice B doesn’t make sense in context. Choice D is related to a common meaning of “profound,” as when an idea is described as profound. It doesn’t fit here, however, as “effects” cannot be “thoughtful.”

ON YOUR OWN

· Suggested Passage Map notes:

· ¶1: BSW alone; on other’s land

· ¶2: Who am I? must prove membership for land

· ¶3: In. tradition = not say name; asked by gov official for parents; orphan

· ¶4: tradition = not speak of dead; unknown family


· ¶5: proud of name; puzzled by changes/laws; (theme): ? identity

· B/Difficulty: Easy

Category: Global

Getting to the Answer: Predict the main theme and narrative direction of the passage. The author’s focus is on Blue-Star Woman and the effect of a changing world on her identity. Only (B) accurately describes the theme of this story.

· A/Difficulty: Medium

Category: Rhetoric

Getting to the Answer: Predict a reason why the author would use the term identified in the question stem. In the lines surrounding the cited phrase, Blue-Star Woman is compared to the ground squirrel, a creature living in nature. The author has chosen this simile to reinforce the connection between Blue-Star Woman and the natural world. Choice (A) is therefore correct.

· C/Difficulty: Medium

Category: Vocab-in-Context

Getting to the Answer: Predict a synonym for the word in question while making sure your selection also makes sense in the context of the passage. Because the landowner’s attitude toward Blue-Star Woman’s occupation of his land is described as “easy tolerance” (line 7), you can assume that Blue-Star Woman is allowed to stay there. Choice (C), “permission,” conveys this idea.

· D/Difficulty: Hard

Category: Vocab-in-Context

Getting to the Answer: Predict a synonym for the word presented in the question stem within the context of the passage; then, select the answer choice that most closely matches that synonym. A good prediction would be a word like thought. The text of the passage following the word in the question stem (“abstraction”) is about Blue-Star Woman’s careful consideration of the nature of her identity and proper place in the world. Choice (D) most accurately describes her thoughts. Although she has questions about her identity, C is incorrect because she has certainty about the natural promptings of the “law of heart” (line 17); she says, “I am being. I am Blue-Star Woman. A piece of earth is my birthright” (lines 18—19).

· B/Difficulty: Medium

Category: Inference

Getting to the Answer: Predict the central idea of the selection quoted in the question stem. Analyze the quotation along with its context in the passage to arrive at the correct answer. Throughout the passage, Blue-Star Woman’s name is a symbol of her identity and culture. For example, “Blue-Star Woman was her individual name” (lines 46—47). By stating that her name seems meaningless, the author implies that Blue-Star Woman’s culture and customs are also losing their meaning. This is reflected in choice (B).

· B/Difficulty: Medium

Category: Command of Evidence

Getting to the Answer: Answer choices to Command of Evidence questions are listed in the order they appear within the passage. The correct answer to the previous question establishes that Blue-Star Woman’s traditional culture and customs are losing their relevance. Choice (B) most clearly supports the idea that Native American customs are losing relevance by confirming an old custom has since been disregarded.

· A/Difficulty: Medium

Category: Inference

Getting to the Answer: Read the context of the quoted selections to predict the difference between the two types of “laws.” In the second paragraph, the formal nature of “the white man’s law” (line 15) is contrasted with Blue-Star Woman’s more emotional view of what is right. While the government’s law requires proof of tribal membership, the “unwritten law of heart” (line 17) grants a person a piece of earth by virtue of her existence, as a “birthright” (line 19). This is reflected in choice (A).

· B/Difficulty: Medium

Category: Inference

Getting to the Answer: Use your Passage Map to find the parts of the passage that mention the government official to develop a full understanding of his relationship with Blue-Star Woman. The author writes that the official persistently asks Blue-Star Woman who her parents are, even though she doesn’t know and would have a cultural aversion to telling him if she did. His behavior is best described by choice (B).

· C/Difficulty: Medium

Category: Command of Evidence

Getting to the Answer: The correct answer to a Command of Evidence question is the excerpt that best supports the answer to a previous question, not the excerpt that describes the correct answer or itself answers the previous question. The correct answer to the previous question states that the government official “lacks cultural empathy” and is “unable to understand Blue-Star Woman’s difficulties.” The correct answer will provide evidence for both of these claims. Choice (C) explains why Blue-Star Woman’s culture would make her unwilling to tell the government official her parents’ names. The government official is unaware of this custom, so he cannot comprehend why Blue-Star Woman would withhold the information.

Chapter 9

PRACTICE

· Suggested Passage Map notes:

· Passage 1

·


· ¶1: argue U.S. reform campaigns; why: costly, too long, ads mean

Passage 2

· ¶1: argue U.S. elections more democratic if: (1) require debate & (2) law = all vote

· B/Difficulty: Easy

Category: Synthesis

Getting to the Answer: Read this question carefully—it’s asking what element the passages have in common. The answer is going to be more about form than content. This question asks about similarities, and though the authors aren’t disputing each other, they each have very different purposes for writing the passages. Making a specific prediction may be difficult, so evaluate answer choices to find an element that both passages utilize. Both mention other countries in support of their arguments. Choice (B) is correct.

· B/Difficulty: Medium

Category: Synthesis

Getting to the Answer: Just because you pick “Yes” or “No” in questions like these doesn’t mean you are done. You have to read both remaining answer choices carefully to decide which one supports your answer. Author 1 would likely agree with something that could improve American election campaigns. Eliminate C and D. Author 2 proposes requiring candidates to debate, enabling the voters to consider “candidates’ positions on issues” (line 26). Author 1 argues that campaigns should be cheaper, shorter, and friendlier. Predict that a reason for requiring debates within these parameters would be agreeable to author 1. Choice (B) is correct.

· C/Difficulty: Hard

Category: Synthesis

Getting to the Answer: Read the question closely. You’re looking for an answer choice that would undermine the argument. When you examine the infographic, you will notice that there is no information about where the voter turnout is being counted. When you look at the trend, you see that wherever this is, the turnout is trending down. Remember the relevant part of author 2’s argument: U.S. national elections would be more democratic if there were mandatory voting. Think about situations that could be true of the graph that would weaken this argument. A prediction that the graph shows voter turnout after voting became mandatory would weaken the argument, because the trend on the graph is a decline in turnout. Choice (C) matches this prediction.

PERFORM

· Suggested Passage Map notes:

Passage 1

· ¶1: invasive definition; problems; snakehead ex.

· ¶2: actions to contain; U.S. too slow

· ¶3: (central idea): must work together to fix

Passage 2

· ¶1: attention on invasive; inv. = problem; exs.; other issues; media role

· ¶2: lack of evidence; ex.; need data; (central idea): balance

· B/Difficulty: Medium

Category: Synthesis

Getting to the Answer: Think about the central idea of each passage and how they differ. The author of Passage 1 wants groups to work together to make invasive species a “priority” (line 39). While the author of Passage 2 agrees that “containing invasive species is a worthwhile goal (lines 70—71),” the passage concludes that available resources need to be balanced among various problems. Predict that author 2 considers invasive species just one issue to worry about. Choice (B) matches that prediction.

· D/Difficulty: Medium

Category: Synthesis

Getting to the Answer: Sometimes, you won’t be able to make a specific prediction. Just keep in mind the central idea of each passage, and consider what they have in common. Be careful to avoid answers that reflect the view of only one author. Although the authors differ in their views of which environmental issues are most important, they agree that such issues should be addressed. Therefore, choice (D) is correct.

· B/Difficulty: Medium

Category: Synthesis

Getting to the Answer: Pay attention to which passage a question stem cites. This will help you avoid incorrect answer choices. The question stem points you to the part of the infographic you need: the National Invasive Species Council. Checking for trends shows that they spend a good deal of money fighting the problem. Your notes should mention that the author sees some hope for the future if different groups work together (lines 37-44). Predict that the author thinks the organization is one of those groups. Choice (B) matches that prediction.

ON YOUR OWN

· Suggested Passage Map notes:

Passage 1

· ¶1: (central idea): Bio. 2 mission - followed? interest down

· ¶2: problems; criticisms; result: doubt data

Passage 2

· ¶1: Bio. 2 failed; (central idea): media relations changed

· ¶2: lots of publicity; high expectations; turned negative

· B/Difficulty: Easy

Category: Vocab-in-Context

Getting to the Answer: Read around the cited word, looking for the context clues to predict a word that could substitute for “exploded.” The word “exploded” is used in the context of unwanted numbers of bugs. Predict that the populations grew a lot. Choice (B) matches that prediction.

· C/Difficulty: Easy

Category: Command of Evidence

Getting to the Answer: Use your Passage Map to find the author’s claim and how the author supports it. In the first paragraph, the author states that serious questions were raised about the project. In the second paragraph, the author details the problems and actions that raised those questions. Choice (C) is correct.

· A/Difficulty: Medium

Category: Rhetoric

Getting to the Answer: Make sure to summarize the purpose of the paragraph to identify the connection that builds to the central idea. Review your Passage Map for the second paragraph. It should move from problems to results. This reflects a cause-and-effect relationship, choice (A).

· B/Difficulty: Hard

Category: Inference

Getting to the Answer: Use your Passage Map to review author 1’s point of view. Author 1 would likely agree that how the project was conducted cast doubt on its results. Only choice (B) fits these parameters.

· C/Difficulty: Medium

Category: Vocab-in-Context

Getting to the Answer: Read around the cited word to figure out its meaning in context and make a prediction. Because the press called it “the most exciting scientific project . . . ” (lines 64-65), the scientists must have wanted to slow things down and make the project’s goals more realistic. Predict curb or lower. Choice (C) is correct.

· C/Difficulty: Medium

Category: Synthesis

Getting to the Answer: Use your Passage Map to find the same topic in Passage 1. Passage 1 specifically states that food and oxygen were two things that were introduced from the outside. The quote from lines 39—41 makes it sound as though the project was self-sufficient. Therefore, the author of Passage 1 would consider the statement misleading. Choice (C) matches.

· C/Difficulty: Medium

Category: Command of Evidence

Getting to the Answer: Check the lines cited in each answer choice, and match them against the evidence you found for your prediction for the previous question. Where does the author of Passage 1 mention food and oxygen? Predict: lines 26-30. Choice (C) matches.

· D/Difficulty: Hard

Category: Synthesis

Getting to the Answer: Review your Passage Map, looking for the central idea of each passage. Consider how they differ; this will be your prediction. Passage 1 describes the problems with the project itself, resulting in the questioning of its findings. Passage 2 focuses on press relations. Choice (D) reflects this difference in focus.

· C/Difficulty: Medium

Category: Synthesis

Getting to the Answer: Make sure to follow the Kaplan Method for Infographics: Analyze the question, examine the parts and trends of the infographic, and make a prediction. Pay close attention to what this question is asking for: the most consistently published experiment. If you misread this question as asking for the research published most often, you will choose B, which is a trap. The most consistently published type of experiment is manipulative experiments; not only is it the only type that was published every year listed on the chart, but also the numbers across years are more similar to each other than those of other categories. Choice (C) is correct.

· A/Difficulty: Hard

Category: Synthesis

Getting to the Answer: Identify the purpose of each passage, and then combine those purposes. The purpose of Passage 2 is to show the changing perceptions about the project. Passage 1 provides specific reasons for those changing perceptions. Only choice (A) correctly combines these purposes.

Chapter 10

PRACTICE

· Suggested Passage Map notes:

· ¶1: description of sedimentary = water made


· ¶2: types of sedimentary rocks


· ¶3: how fossils are created in most common sed. rocks


· ¶4: use of fossils in science

· D/Difficulty: Easy

Category: Global

Getting to the Answer: Summarize the central idea of the passage even when you can’t make a specific prediction. It’s tough to make a specific prediction before looking at the answer choices, but you do know that the author thinks fossils are great. Choice (D) makes sense; fossils are important because they provide “windows into the past” (lines 9—10) and are a “vital source of information about animals, insects, and plants from long ago” (lines 50-51).

· D/Difficulty: Hard

Category: Detail

Getting to the Answer: Beware of details that are true but not relevant to the question at hand. After making this statement, the author asserts that limestone can be formed when lime settles to the bottom of the ocean or when the shells of sea creatures collect. Lines 22—29 support the assertion that limestone is complex because it can be created in a variety of ways. Choice (D) matches this and is, therefore, correct.

· B/Difficulty: Easy

Category: Command of Evidence

Getting to the Answer: Whether or not you used specific lines of text to answer the question preceding a Command of Evidence question, start looking for the answer in the general portion of the passage that led you to your previous answer. In answering the previous question, you established that limestone can be created in different ways by reviewing lines 22-29. Eliminate C because, although it was a part of the general support for the previous question, it isn’t the best evidence to support the various ways limestone can be formed. Choice (B) details these different ways and is correct.

PERFORM

· · Suggested Passage Map notes:

·


· ¶1: Norse colonies disappeared; reason unclear


· ¶2: theory of decline due to massacre; not proven

· ¶3: new theories due to climate change; study of fly remains to support theory


· ¶4: economic changes part of Norse decline; trade between other countries bypassed the Norse


· ¶5: Norse decline due to culture and social problems; Norse did not adapt to needs of environment

· A/Difficulty: Medium

Category: Global

Getting to the Answer: After reading a passage, you should be able to predict a general purpose for the passage. Be careful not to make your prediction overly specific. The author discusses the disappearance of a group of Norse settlers and offers a number of explanations for this occurrence; he does not seem to advocate any one explanation over another. Predict that the main purpose is to explain possible reasons for the disappearance. Choice (A) matches this prediction.

· A/Difficulty: Medium

Category: Inference

Getting to the Answer: This question requires you to put together details from different parts of the passage. The author writes, “a particularly warm period for Greenland . . . occurred between the years 800 c.e. and 1300 c.e.” (lines 36-38). The beginning of the passage states that the colony was founded around the year 1000 c.e., right in the middle of the warm period. Choice (A) works. The “mild,” warm weather was uncharacteristic of the usually cold, harsh climate.

· B/Difficulty: Medium

Category: Command of Evidence

Getting to the Answer: If you used a line reference to answer the previous question, start by seeing whether that line reference is one of the answer choices. In answering the previous question, you determined that the climate during the initial founding of the Norse settlements was “uncharacteristically mild.” This strange, mild weather is detailed in lines 36-38, choice (B).

ON YOUR OWN

· · Suggested Passage Map notes:

·


· ¶1: R: keep mind active; feeling her age


· ¶2: R reflecting on past; sees V as lazy


· ¶3: V takes school for granted; R wanted more schooling


· ¶4—11: father refused to let R continue school, R regrets

· A/Difficulty: Medium

Category: Rhetoric

Getting to the Answer: Look for the answer choice that best paraphrases Rosemary’s attitude toward aging. The last three sentences of the first paragraph (lines 14-19) explicitly discuss Rosemary’s attitude toward old age: “At 87 years of age, she was glad she could still write at all. She had decided long ago that growing old was like slowly turning to stone; you couldn’t take anything for granted.” This attitude is best summarized by choice (A).

· D/Difficulty: Easy

Category: Detail

Getting to the Answer: When you read the lines around the cited text, you see that the trip, though short, “seemed to take a long while” (line 21) and that Rosemary “often experienced an expanded sense of time” (lines 22-23). Identify the answer choice that describes this explicit meaning. Choice (D) restates this perfectly, because “elastic” means to be able to expand and contract.

· D/Difficulty: Easy

Category: Inference

Getting to the Answer: Reread the section of the passage that describes Rosemary’s first day of high school to develop a firm understanding of her memory of that day. The question refers to Rosemary on that “glorious day" (line 65) when she was going to high school. She gets up, puts on her “best dress . . . her heart racing in anticipation” (lines 66-67). This points to choice (D) as the correct answer.

· C/Difficulty: Medium

Category: Command of Evidence

Getting to the Answer: Identify the answer choice that is the best supporting evidence for the idea expressed in the answer to the previous question. The correct answer to the previous question established that Rosemary was eager to start high school and continue her education using lines 64-67. Choice (C) most clearly demonstrates that by describing how she had begged her father for permission to continue her studies.

· C/Difficulty: Hard

Category: Rhetoric

Getting to the Answer: When you see the words “in order to” in a question stem, rephrase that question as a “why” question. For example: Why did the author include Rosemary’s thoughts regarding her grandson in lines 38-43? Choice (C) details why the author includes Rosemary’s thoughts on her grandson; these musings set the stage for her memories of her own educational frustrations as a child, which is quite different from her perception of Victor’s life as easy. While B is tempting, it is presented as a fact; it may be true that Victor does not appreciate how fortunate he is, but you cannot know this for sure, as the passage is filtered through Rosemary’s perspective. Choice (C) is correct.

· C/Difficulty: Hard

Category: Command of Evidence

Getting to the Answer: Review how you answered the previous question to answer Command of Evidence questions. Choice (C) continues where the passage cited in the previous question left off. In the lines cited by choice (C), the author makes the juxtaposition between Victor’s and Rosemary’s experiences more explicit.

· C/Difficulty: Medium

Category: Vocab-in-Context

Getting to the Answer: Predict a close synonym for the word in question, while making sure your selection also makes sense in context of the passage. The sentence that follows “sturdy” talks about Rosemary’s responsibilities as a child. Combined with the overall theme of the passage—Rosemary reflecting on what could have been if she had more schooling—the word “sturdy” demonstrates Rosemary’s capability. This prediction is an exact match for choice (C).

· C/Difficulty: Medium

Category: Inference

Getting to the Answer: Reread the cited line to develop a firm grasp of the author’s underlying point. Rosemary refers to Victor as a “lazy bones” (line 34) who has been given every advantage by his “doting parents . . . his future appeared bright—if he ever got out of bed, that is” (lines 42-44). Choice (C) summarizes this well.

· D/Difficulty: Medium

Category: Global

Getting to the Answer: Identify the answer choice that most directly addresses the central theme and narrative direction of the passage. Though Victor is important in this passage, his real role is to awaken thoughts in Rosemary about her past. Most of the passage is about Rosemary and her life. Choice (D) addresses this.

Chapter 11

PRACTICE

· · Suggested Passage Map notes:

·


· ¶1: new kind of music

· ¶2: ragtime history


· ¶3: American critics dislike ragtime, but the public and Europeans like it


· ¶3: cont.: ragtime popular all over


· ¶4: ragtime musicians = natural, untrained

· C/Difficulty: Medium

Category: Vocab-in-Context

Getting to the Answer: Remember to avoid common meanings when answering Vocab-in-Context questions. Read the sentence without “intricate,” and ask what the author is trying to communicate. Predict: The rhythm had complicated parts. Choice (C) matches.

· A/Difficulty: Medium

Category: Connections

Getting to the Answer: The phrase “According to the author” and the relationship clue word “effect” indicate that you should ask how items are related. The question stem cites the result and asks for the cause. Read around the cited lines. Just before, the author lists the elements of the music that produced the “curious effect.” This matches choice (A).

· B/Difficulty: Hard

Category: Connections

Getting to the Answer: When only one part of a relationship is presented, look for the other part in the answer choices. The question asks about the basis for the author’s opinion. Ask why the author thinks that the piano player is “a natural musician” (line 63). Go to the cited line, and look for the reasons that the author gives: He used his “ear.” Therefore, choice (B) is correct.

PERFORM

· · Suggested Passage Map notes:

·


· ¶1: infants learn language by differentiating sounds


· ¶2: detect diff. sounds


· ¶3: ex. of detecting diff. sounds

· ¶4: capability based on conditions


· ¶5: cond. 1 = loudness


· ¶5, cont.: cond. 2 = familiarity

· D/Difficulty: Easy

Category: Connections

Getting to the Answer: “Whether . . . whether” indicates that you should look for what needs to happen to produce a specific outcome. Refer to your Passage Map for paragraph 5. This paragraph describes how an infant can tell two voices apart when one of the voices is louder or more familiar. Because loudness is not an answer choice, (D) must be correct.

· D/Difficulty: Hard

Category: Vocab-in-Context

Getting to the Answer: Be wary of answer choices that sound or look like the word or phrase in question. This question is hard for two reasons. First, the word “predicated” is used infrequently; second, the context you need is in the following paragraph. Your Passage Map notes that, in the fourth paragraph, the author attributes the ability to distinguish sounds to two conditions; in the next paragraph, the author informs you of how each condition affects the results. Predict that the capability is altered by the specific conditions. Choice (D) matches this prediction.

· C/Difficulty: Medium

Category: Connections

Getting to the Answer: When you see the phrase “which choice best describes,” and the question provides both sides of the relationship, look for the answer choice that joins the sides correctly, even if the author doesn’t state it directly. Because describing this relationship is describing why the author wrote the passage, check the first and last paragraphs for clues. In the first paragraph, the author suggests that learning the language is important. In the last paragraph, the author asserts that concentrating on a single stream speeds up the process of language acquisition. Choice (C) matches.

· C/Difficulty: Medium

Category: Vocab-in-Context

Getting to the Answer: Be careful when a Vocab-in-Context question asks about a phrase that includes a preposition. Make sure your answer uses the correct word and preposition. Read the sentence without “attend to,” and ask what the author is trying to communicate. Because the paragraph is about focusing on one specific speaker, predict: pay attention to. Choice (C) is correct.

ON YOUR OWN

· · Suggested Passage Map notes:

·


· ¶1: G happy and surprised


· ¶2: G’s book to be published


· ¶3: pre-pub excitement


· ¶4: book based on G’s home town


· ¶5: excited for article about him


· ¶6: difficult writing process


· ¶7: owns his process


· ¶8: article = mixed feelings

· B/Difficulty: Medium

Category: Global

Getting to the Answer: You need to develop a firm grasp of the passage as a whole to be able to accurately summarize central ideas and characters; your Passage Map is extremely useful for this type of question. Over the course of the passage, the author describes George experiencing a wide range of emotions, including joy over his recent publication, fear over how people in his hometown will feel, and disappointment over how his story in Rodney’s Magazine is received. Choice (B) is correct.

· A/Difficulty: Medium

Category: Global

Getting to the Answer: Remember that your approach to reading U.S. and World Literature passages should include keeping a running tally of a character’s traits. Choice (A) is correct because the main character, George Webber, is hungry for fame and recognition. This is alluded to throughout the passage and is addressed clearly in lines 14-15 (“So . . . knocking at his door”).

· B/Difficulty: Medium

Category: Detail

Getting to the Answer: Look at your notes for the paragraph that introduces and discusses the subtopic mentioned in the question stem. Once you have a firm grasp on the details, select the answer choice that best summarizes the subtopic. In the fourth paragraph, the author explains that George’s new book is a “novel” (line 24) and that it is based on his hometown of Old Catawba (line 26). Choice (B) is correct.

· C/Difficulty: Easy

Category: Command of Evidence

Getting to the Answer: The correct answer choice should explicitly address the subtopic discussed in the previous question. Choice (C) is correct because it is in these lines that the author explicitly addresses that George’s new book is a novel based on his experiences in his hometown of Old Catawba.

· A/Difficulty: Hard

Category: Vocab-in-Context

Getting to the Answer: Pay attention to the author’s use of the word in the sentence and the point being made in that section of the passage. In lines 4-8 (“All he could remember . . . broad grin”), the author describes George’s transition from confusion to remembrance and then excitement for his recent success at getting his book published. George is described as feeling “incredulity” because he still has trouble believing his good fortune; predict skepticism. Choice (A) is correct.

· C/Difficulty: Medium

Category: Inference

Getting to the Answer: Develop a firm grasp of the author’s intent in the section cited in the question stem. Choose the answer that best describes what is implied by the text. The cited section is located within paragraph 4, which discusses the book’s basis in George’s hometown. In this section, George seems nervous that people from his hometown “might recognize themselves” (line 36) in characters from his new book and “be offended” (line 36). The reader can infer that the book is closely based on details of real events and people in Old Catawba. The correct answer is choice (C).

· D/Difficulty: Medium

Category: Detail

Getting to the Answer: Review your notes for the section that discusses George’s writing process: paragraph 6. In lines 61-71 (“Already . . . morning”), the author describes George’s tumultuous creative periods. It is made explicit that the writing process takes a toll on George’s body and mind. Choice (D) is correct.

· B/Difficulty: Medium

Category: Command of Evidence

Getting to the Answer: The answer to the previous question will not be directly stated in the answer choice to this question. Rather, the correct answer to this question will provide the strongest support for your answer to the previous question. Identify lines 61-71 as the lines that helped you describe George’s writing process. These lines are featured in choice (B), which is correct.

· B/Difficulty: Medium

Category: Vocab-in-Context

Getting to the Answer: Reread the sentence that includes “fury” and predict another word that could be substituted. Your Passage Map notes that paragraph 6 is about George’s difficult writing process. Choose a close synonym for “fury” that would describe his “creation” (line 65); energy or intensity works well. Choice (B) is correct, as agitation is the best synonym of “fury” in this particular context. The definitions of “fury” that imply indignation or animosity are not valid to describe George’s creative process.

Chapter 12

PRACTICE

· Suggested Passage Map notes:

· ¶1: changed rel. between U.S. press & readers

· ¶2: some causes = indust. & urban landscape

· ¶3: result = press influences pop culture

· A/Difficulty: Medium

Category: Rhetoric

Getting to the Answer: When faced with an EXCEPT question, locate each answer choice within the passage. The answer choice you cannot find is the answer. This question stem references the author’s thesis, which you know from your Passage Map is in lines 16-21: “Yet, the formative significance of these periods in the constitution of the American press aside, it was the last few decades of the nineteenth century that produced the most profound change in the relationship between the American press and its readership.” If you read this sentence closely, you’ll notice that the thesis has to do with how the American press related to its readership—therefore, increased readership, choice (A), is not evidence, but instead part of the original thesis, or claim, itself.

· D/Difficulty: Hard

Category: Rhetoric

Getting to the Answer: When you see the phrase “in order to,” pay attention to how the element in question fits into the overall structure of the passage. The author spends the majority of the passage discussing developments in the history of the newspaper and the causes and effects of those developments. The last sentence, which is what this question is about, discusses how the “advent of radio and television” (lines 63—64) expanded the “social position” (line 62) established by the American press. This statement points beyond the contents of the passage itself, rendering choice (D) correct.

· B/Difficulty: Medium

Category: Rhetoric

Getting to the Answer: This question stem is rather long and involved. Make sure you take time to figure out what the question is asking before trying to predict an answer. Your Passage Map will indicate that the factors mentioned in the question stem are discussed in the second paragraph. The last sentence of that paragraph reads, “But to look at these two areas of change, one social and the other technological, as separate catalysts is to miss the point; it was their union that created a massive newspaper readership that has only grown over the past century” (lines 40-44). This conclusion supports the idea that the factors worked together, choice (B).

PERFORM

· Suggested Passage Map notes:

· ¶1: amt. of ozone ↓

· ¶2: chlorine destroys ozone


· ¶3: 1985 - ozone hole reported

· ¶4: CFCs deplete ozone

· ¶5: choose CFC or ozone

· ¶6: nat. events make prob. worse


· ¶7: stopping CFC won’t completely fix ozone


· ¶8: sci. working on CFC subst.

· D/Difficulty: Hard

Category: Rhetoric

Getting to the Answer: Avoid answer choices that go beyond the scope of the passage. Even stopping all CFC production today wouldn’t solve the problem, you’re told, because CFCs can live for up to 400 years. Choice (D) captures the underlying point here; measures against ozone depletion may take years to have an effect.

· A/Difficulty: Medium

Category: Rhetoric

Getting to the Answer: Sometimes, you will be given evidence and asked to find what it supports instead of the other way around. In this part of the passage, the author is providing a progress report on ozone depletion in the 1990s. He states that the layer is “depleting faster than expected” (line 53), in addition to natural events “making the problem worse” (lines 55-56). Choice (A) summarizes this idea that the situation is worsening.

· B/Difficulty: Medium

Category: Rhetoric

Getting to the Answer: Use your Passage Map to summarize the purpose of any one paragraph within a passage. The last paragraph focuses on what scientists are doing about the problem, and your Passage Map states that scientists are working on a CFC substitute. Even if things are bad, the author says that “nobody can say that the situation will not improve” (lines 74-75) if people lend a hand. Choice (B) captures this positive note.

ON YOUR OWN

· Suggested Passage Map notes:

· ¶1: human brain flexibility = plasticity

· ¶2: continues to grow and develop

· ¶3: can adapt/rewire

· C/Difficulty: Medium

Category: Global

Getting to the Answer: You must have a firm grasp of the overall passage to understand the author’s central idea. If you’re uncertain, look at your Passage Map notes for the first and last paragraphs. Predict that the author is emphasizing the brain’s plasticity, or flexibility. Choice (C) is correct.

· A/Difficulty: Medium

Category: Detail

Getting to the Answer: When presented with a Detail question, use your Passage Map to locate the section that describes the issue raised by the question stem. Paragraph 2 discusses the brain’s growth and development. In lines 26-31 (“The first few . . . synapses”), the author states that three-year-olds have “15,000 synapses per neuron” and that adults have about half of that number. Use this prediction to determine that choice (A) is correct.

· B/Difficulty: Easy

Category: Command of Evidence

Getting to the Answer: The lines cited in the correct answer choice should explicitly address the topic of the previous question. Predict that the best evidence will come from lines 27-31. Choice (B) is correct.

· B/Difficulty: Medium

Category: Vocab-in-Context

Getting to the Answer: The author is describing the brain as “unrivaled” (line 8) and “unique” (line 16). Predict that “sophisticated” most nearly means complicated or elaborate. Choice (B) is correct.

· A/Difficulty: Medium

Category: Vocab-in-Context

Getting to the Answer: Remember to use the Kaplan Strategy for Vocab-in-Context questions. In this section of the passage, the author explains the significant capacity changes that the human brain can experience, even well into adulthood. Predict that “marked” most nearly means significant or extraordinary. Choice (A) is correct.

· D/Difficulty: Medium

Category: Inference

Getting to the Answer: Reread around the lines cited in the question stem to determine what can be concluded. In this section, the author describes an experiment that showed that rats in enriched environments developed “heavier, thicker brains with more neurons and synaptic connections” (lines 69-70). The author goes on to say that these results have also been observed in humans. Predict that the results from the experiments show that enriched environments positively impact both rats and humans. Choice (D) is correct.

· C/Difficulty: Medium

Category: Global

Getting to the Answer: Use your Passage Map to locate the appropriate section of the passage. In the second paragraph, the author explains that the brain’s ongoing flexibility was guessed by William James (lines 42-46). The author then details how this flexibility was proven in late twentieth-century experiments (lines 49-73). Predict that the brain’s flexibility was hypothesized over 100 years ago but only recently proved correct. Choice (C) is correct.

· B/Difficulty: Medium

Category: Command of Evidence

Getting to the Answer: Remember to use Kaplan’s Strategy for Command of Evidence questions. In the previous question, lines 42-46 and lines 49-73 were used to describe the various scientific views of brain flexibility over time. The transition between these two sections, “Then, several provocative experiments dramatically complicated conventional thinking about the human brain” (lines 46-48), is an excellent summary of the change in thinking over time. Predict that the best evidence for the answer to the previous question involves lines 42-48. Choice (B) is correct.

· B/Difficulty: Medium

Category: Vocab-in-Context

Getting to the Answer: Read around the cited line and word for context to determine the correct answer. The author describes the rats’ environment as being “enriched,” or enhanced, by more stimuli than the environment of the control group. Predict that “enriched” most nearly means enhanced. Choice (B) is correct.

· D/Difficulty: Easy

Category: Vocab-in-Context

Getting to the Answer: Make sure to avoid common meanings when answering Vocab-in-Context questions. In this sentence, the author is referring to new stimuli’s ability to “mold the brain’s . . . architecture” (lines 88—89). Predict that “architecture” most nearly means structure. The correct answer is (D).

Chapter 13

PRACTICE

· C/Difficulty: Hard

Category: Sentence Formation

Getting to the Answer: Make sure introductory modifiers are modifying the correct items. The introductory modifier phrase in this sentence is “During their seasonal migration.” However, “there are large numbers migrating” appears after this phrase. As written, this sentence states that the large numbers are migrating, not birds and bats. Even though both (C) and D correct the error, only choice (C) retains the original meaning: that the number of birds and bats migrating is large.

· B/Difficulty: Medium

Category: Sentence Formation

Getting to the Answer: Make sure verbs and verb phrases in a sentence are parallel in structure. The underlined word “assessing” does not match the sentence’s prior verb: “to allow.” “Assessing” should be “to assess,” which is choice (B).

· A/Difficulty: Medium

Category: Usage

Getting to the Answer: Make sure verbs agree with their subjects in person and number. This sentence is correct as written because the word “data” is the plural form of the singular “datum.” Therefore, the plural verb “provide” is necessary. Choice (A) is correct.

PERFORM

· D/Difficulty: Medium

Category: Effective Language Use

Getting to the Answer: When you see an underlined portion with no obvious grammatical errors, check to make sure the style and tone are appropriate for the passage. As written, this segment uses the cliché “Life goes on,” which is inappropriate in style and tone. Choices B and C also use clichés and overly familiar language, making choice (D) the correct answer.

· C/Difficulty: Easy

Category: Usage

Getting to the Answer: Make sure you don’t confuse possessive determiners, contractions, and adverbs. “It’s” is a contraction for “it is,” which makes no sense in the context of the underlined portion, and “their” is plural, but this pronoun refers to “the business,” which is singular. The “it’s” in the passage should be the possessive form: “its,” which matches choice (C).

· B/Difficulty: Hard

Category: Usage

Getting to the Answer: Whenever you see the word “it,” make sure you know exactly what it is referring to. In this sentence, it’s ambiguous what “this” refers to, so replacing the pronoun “this” with the noun it refers to is necessary. It is the “sole proprietorship” that “does not come without its share of disadvantages,” so the answer is choice (B).

· A/Difficulty: Easy

Category: Usage

Getting to the Answer: When a preposition is underlined, make sure it’s used correctly. The word preceding the preposition is “separate.” When things are separate, they are separate from each other, which means the correct preposition is being used, and the sentence is correct as written, choice (A).

· D/Difficulty: Medium

Category: Usage

Getting to the Answer: Use who when the pronoun acts as the subject of a sentence and whom when the pronoun acts as the object. Always use the objective form when using a pronoun in a prepositional phrase, so “with whom” is grammatically correct. “Who” comes after “with,” a preposition, in this sentence. Therefore, “who” should be “whom” so that it’s correct as the object of a preposition. Choice (D) is correct.

· C/Difficulty: Hard

Category: Sentence Formation

Getting to the Answer: This sentence needs to be separated into two independent clauses, because, as written, it is a run-on sentence. One way to correct a run-on sentence is to separate the two independent clauses with a semicolon. The only answer choice that corrects the run-on is choice (C), which uses a semicolon and a comma after “however.”

· D/Difficulty: Medium

Category: Effective Language Use

Getting to the Answer: Avoid the passive voice whenever possible. The underlined portion is written in the passive voice. Choices B and (D) both correct this issue. However, choice (D) is more succinct and, therefore, the correct answer.

· D/Difficulty: Medium

Category: Quantitative

Getting to the Answer: Read answer choices to Infographics questions carefully so you don’t get caught in a misused-word trap. As written, this sentence claims that partnerships are less lucrative than sole proprietorships. The graph, however, indicates the opposite, as each of the partnership bars soars above the corresponding sole proprietorship bars. You need to find the answer choice that states partnerships are more lucrative than sole proprietorships, which is choice (D).

· D/Difficulty: Medium

Category: Usage

Getting to the Answer: When you see a pronoun underlined, always check to make sure it refers to a clearly identifiable noun. It’s unclear what “this” is referring to. The beginning of the sentence mentions the business being sold; therefore, “this” should be “this sale,” choice (D).

ON YOUR OWN

· C/Difficulty: Medium

Category: Effective Language Use

Getting to the Answer: When a phrase is underlined, look for the choice that best conveys the author’s intended meaning in the context of both the sentence and paragraph. In this sentence, the author is addressing the perceived need that people had after World War II to move away from crowded, dirty urban centers. Choice (C) is correct because the suburbs were a response to this need.

· A/Difficulty: Easy

Category: Punctuation

Getting to the Answer: When an underlined section includes punctuation, make sure that the punctuation enhances the reader’s understanding of the passage. If the punctuation harms understanding, consider changing it or eliminating it altogether. In this case, the comma included in the underlined section clearly separates an independent and dependent clause. No punctuation at all would create a run-on, and words are not being omitted, so there is no reason to use the ellipsis. The semicolon is used to separate two independent clauses and is incorrect in this context. Because the sentence is correct as written, choice (A) must be correct.

· B/Difficulty: Easy

Category: Effective Language Use

Getting to the Answer: Look for the clearest, most straightforward wording of the idea underlined in the passage, and avoid answers that contain unnecessarily complicated syntax. In the underlined section, the author is simply trying to address the self-sufficiency of the suburbs. Choice (B) is correct because it conveys this idea in the clearest manner.

· C/Difficulty: Hard

Category: Development

Getting to the Answer: The last sentence of a paragraph usually ties the different claims in that paragraph together. In this paragraph, the author introduces the tension between the desire to move away from the city and the need to return to the city for work and leisure. Choice (C) is correct because it effectively expresses that dichotomy.

· D/Difficulty: Medium

Category: Effective Language Use

Getting to the Answer: When a single word is underlined, make sure it clearly conveys the author’s intended meaning in the context of both the sentence and paragraph. The author explains how suburbanites do not want to be too far away from things that can be found in the city. Therefore, it makes sense to describe these things using a positive adjective such as “benefits” rather than using a negative adjective. Choice (D) is correct.

· A/Difficulty: Medium

Category: Punctuation

Getting to the Answer: One use of commas, like parentheses, is to set aside unnecessary information. Choice (A) is correct because the clause “while offering a respite from city pollution” needs to be set aside by punctuation. The existing use of commas is the best available option, meaning that no change is necessary.

· A/Difficulty: Easy

Category: Usage

Getting to the Answer: When a single verb is underlined, make sure it agrees with its subject in person and number and that the tense makes sense in the context of the sentence and paragraph. Choice (A) is correct because “have” agrees with its subject (“laws”) and is the correct tense in the context of the sentence and paragraph.

· D/Difficulty: Easy

Category: Effective Language Use

Getting to the Answer: When an underlined portion is grammatically correct, look for words or phrases that differ from the overall tone of the passage. Most of the answer choices for the question depart from the professional tone of the passage. Choice (D) is correct because it maintains the author’s meaning and tone.

· C/Difficulty: Medium

Category: Effective Language Use

Getting to the Answer: Select the answer choice that clearly conveys the author’s position with the most effective syntax. Choice (C) is correct because it clearly and concisely demonstrates the author’s opinion. The other answer choices relate the same thing, but with many unnecessary words.

· B/Difficulty: Hard

Category: Quantitative

Getting to the Answer: Study the graph to develop a sound understanding of the information it is showing. Choose the answer choice that accurately describes what the graph displays. The graph shows that population growth in primary cities outpaced growth in the suburbs from 2010 to 2011, a reversal of the dominant trend of the previous decade. Choice (B) is correct because it accurately describes the graph’s data and draws a reasonable conclusion from it.

· A/Difficulty: Hard

Category: Organization

Getting to the Answer: Make sure sentences are presented in the most logical order that suits the passage’s clarity and narrative. Choice (A) is correct because the information from the graph serves as logical support to this paragraph’s first sentence. The population shift back to primary cities makes sense in the context of growing anti-suburb sentiment.

Chapter 14

PRACTICE

· D/Difficulty: Easy

Category: Usage

Getting to the Answer: The noun closest to an underlined verb may not be the subject of that verb. It is important to read enough of the sentence to identify the subject of the underlined verb.

In this sentence, the noun closest to the underlined verb is “Pennsylvania,” a singular noun that agrees with “was.” Rereading the sentence reveals that “Pennsylvania” is part of a phrase that modifies the actual subject of the sentence, “Canvases.” Choice (D) is correct.

· B/Difficulty: Medium

Category: Usage

Getting to the Answer: The correct preposition can often only be determined in the context of the sentence.

This sentence states that the Hudson River School was suited to the task of competing with Europe. The job of the prepositional phrase is to explain how that was accomplished. Choice (B) conveys this precisely.

· C/Difficulty: Hard

Category: Usage

Getting to the Answer: If you can’t identify who or what a pronoun refers to, the pronoun’s antecedent is unclear.

Because the paragraph mentions only Europe and not Europeans, eliminate D. In the previous sentences, the passage discusses “writers” and their interest in the influence of “wilderness” on their “nationality.” Choice (C) is correct.

· A/Difficulty: Easy

Category: Usage

Getting to the Answer: Look for clues that indicate when some things are being compared. Then, check to make sure that the comparison is logical, parallel, and properly phrased.

At the beginning of the sentence, the phrase “Unlike European painters” tells you that the writer is comparing “European painters” to something else. Logically, that something else must also be painters. Choice (A) logically compares painters to painters.

PERFORM

· C/Difficulty: Easy

Category: Usage

Getting to the Answer: Adjectives in the comparative form compare only two items or people. Use the superlative form to compare three or more items or people.

In context, it is clear that many “financial panics” have occurred in U.S. history. As a result, the superlative form must be used when comparing them. Because the superlative of “bad” is “worst,” there is no need to add “most” to indicate the comparison. Choice (C) is correct.

· A/Difficulty: Medium

Category: Usage

Getting to the Answer: When nouns in a sentence relate to each other, they need to agree in number.

The sentence discusses “types of gadgets” and gives two examples. Each of the examples must match the plural nouns “types” and “gadgets.” Choice (A) makes all the related nouns plural. While D uses the plural forms, it uses an improper connection between the two nouns.

· B/Difficulty: Hard

Category: Usage

Getting to the Answer: Remember that a pronoun and its antecedent may not be close to each other.

The possessive pronoun correctly indicates the owner of the loan as “investor,” but does not match in number. Because the gender of the investor is not specified, be careful not to use a gendered pronoun. Referring to the loan in generic terms corrects this issue. Choice (B) is correct.

· D/Difficulty: Easy

Category: Usage

Getting to the Answer: Remember that barely, hardly, and scarcely imply a negative; when paired with explicitly negative words, they create a double negative, which is grammatically incorrect.

Adding the word “hardly” when the predicate contains “could not” creates a double negative and is incorrect. Choice (D) corrects the error by eliminating “hardly.”

· C/Difficulty: Medium

Category: Usage

Getting to the Answer: A pronoun is ambiguous when its antecedent is either missing or unclear.

To find the antecedent for the underlined pronoun, read the previous sentence and think about the focus of the passage. Although the previous sentence has more than one possible antecedent, in the context of the passage, choice (C) is the clearest and most relevant.

· B/Difficulty: Medium

Category: Usage

Getting to the Answer: The pronoun “which” may be used only to refer to things or animals. “Who” and “whom” may be used only to refer to people.

In this sentence, the pronoun “which” is used incorrectly to refer to “investors,” who are people. To determine whether “who” or “whom” is correct, substitute the appropriate third-person pronoun for “who” or “whom,” and read the sentence. Choice (B) is correct.

· D/Difficulty: Hard

Category: Usage

Getting to the Answer: Idiom questions often hinge on correct preposition usage. The preposition must convey the correct meaning.

Read the sentence carefully to understand the meaning that the preposition must convey. The “pressure” was on the “market,” not “on everyone.” Choice (D) conveys the sense that the sellers were the source of the pressure.

· A/Difficulty: Medium

Category: Usage

Getting to the Answer: Subjects and verbs must agree in person and number. Singular subjects are paired with singular verbs, and plural subjects are paired with plural verbs.

Although many things and people make up a market, the noun is singular and requires a singular verb. Because the market is a thing being spoken about, it takes the third-person verb. The verb “to start” is a regular verb, and its third-person singular form is “starts.” Choice (A) provides the third-person singular verb.

ON YOUR OWN

· A/Difficulty: Medium

Category: Effective Language Use

Getting to the Answer: When a single word is underlined, make sure that it is the word that best conveys the author’s intended meaning in the context of both the sentence and paragraph.

No change is necessary. One of the definitions of “propagate” is “to cause to spread out” over a great area or population. This sense of the word is correct in the context of this sentence and paragraph. Choice (A) is correct.

· D/Difficulty: Medium

Category: Punctuation

Getting to the Answer: When an underlined sentence fragment includes a parenthetical remark, look for the answer that shows the correct way to punctuate that clause.

Choice (D) is correct. “Such as murder” is a parenthetical remark because it is not necessary for the sentence to be complete. It is best set aside by two commas.

· B/Difficulty: Easy

Category: Punctuation

Getting to the Answer: An underlined section that includes punctuation requires an assessment of the punctuation. Choose the answer that provides the correct punctuation for the sentence and context.

Choice (B) is correct. The sentence cannot be correctly combined with the following sentence using a semicolon, so a question mark is the only appropriate option to follow the word “child.”

· A/Difficulty: Medium

Category: Organization

Getting to the Answer: Make sure sentences are presented in the most logical order for the clarity of the passage.

Choice (A) is correct, as the current placement is the correct one for this paragraph. Moving sentence 2 before sentence 1 would be confusing for the reader, and moving it later in the paragraph would interfere with the author’s transition to the next paragraph.

· A/Difficulty: Hard

Category: Development

Getting to the Answer: Determine whether the underlined sentence is effective and correct in its current form or whether one of the other options more clearly develops the author’s ideas.

No change is necessary. The current sentence covers all of the points it needs to in order to address the questions posed in the previous paragraph. It also serves as an introduction to the new paragraph. Choice (A) is correct.

· C/Difficulty: Medium

Category: Effective Language Use

Getting to the Answer: Pay attention to the construction of the sentence (its syntax). Choose the option that correctly and effectively conveys the author’s narrative intent, while also preserving additional points the author is trying to make.

Choice (C) is correct because it uses a more effective and clear active voice sentence structure. Choice D is also active voice (unlike choices A and B), but it leaves out the author’s additional point about the exchange being “one of the most terrifying conversations in all literature.”

· B/Difficulty: Medium

Category: Effective Language Use

Getting to the Answer: Determine whether the underlined sentence is effective in its current form. Pay attention to word choice and precision of language.

In this case, the original sentence is not grammatically incorrect, but there are less wordy options that get the point across clearly. Choice (B) is the best option, as its language is precise and helpful to the overall narrative.

· D/Difficulty: Medium

Category: Development

Getting to the Answer: For the first sentence of a paragraph, look for the answer choice that embodies the central idea of the paragraph while also serving as an effective connection to, or transition from, the previous paragraph.

Choice (D) is correct because it serves as a logical and effective transition from the previous paragraph. Here, the author is shifting from describing the unsavory aspects of fairy tales to explaining a theory for why they evolved to be that way.

· A/Difficulty: Medium

Category: Effective Language Use

Getting to the Answer: When a single word is underlined, make sure that it is the word that best conveys the author’s intended meaning in the context of both the sentence and paragraph.

No change is necessary because “fantastic” is the best option for this context; the author is creating a parallel between “realistic” threats in the real world and the unrealistic elements of fairy tales. Choice (A) is correct.

· B/Difficulty: Medium

Category: Sentence Formation

Getting to the Answer: When an underlined section includes the connection between two sentences, consider whether the two sentences should remain separate. If they should be joined, choose the best option for doing so.

Choice (B) is correct because these two sentences need to be combined, and a comma is the correct way to do so in this instance. If kept separate, the second sentence is actually a fragment.

· C/Difficulty: Medium

Category: Development

Getting to the Answer: The final sentence serves the important dual function of concluding the paragraph and the passage. Choose the option that is true to the author’s intent for both.

Choice (C) is correct because it effectively states the author’s main idea that violence in children’s media has been present for centuries in fairy tales. It also connects the main idea of the passage to the main idea of the final paragraph: Fairy tales possibly helped children navigate the dangers of real life.

Chapter 15

PRACTICE

· D/Difficulty: Medium

Category: Punctuation

Getting to the Answer: Make sure to use the correct punctuation to connect independent clauses. “However” cannot join two independent clauses with just a comma. It can join them with a semicolon or start a new sentence, as long as “however” is still followed by a comma. Choice (D) is correct.

· D/Difficulty: Medium

Category: Punctuation

Getting to the Answer: Remember to use punctuation to separate nonessential information from the rest of a sentence. In this case, it might be useful information to know what the author says the restorers claimed; however, the sentence still makes logical sense without the information. Only (D) properly sets off the parenthetical (nonrestrictive) phrase.

· B/Difficulty: Easy

Category: Punctuation

Getting to the Answer: Make sure you don’t confuse possessive determiners, contractions, and adverbs. “It’s” is a contraction for “it is,” which makes no sense in the context of the underlined portion. “Their” can refer only to people, not objects. The “it’s” in the passage should be possessive, which is “its,” (B).

· A/Difficulty: Medium

Category: Punctuation

Getting to the Answer: Because the underlined portion contains a dash, find the other dash and check whether the information inside the dashes is essential to the logic of the sentence. The information inside the dashes is nonessential, so the underlined portion is correct as written, (A).

PERFORM

· D/Difficulty: Medium

Category: Punctuation

Getting to the Answer: When you see an underlined segment featuring a noun followed by “the,” make sure the correct punctuation is being used. As written, the sentence to which the underlined segment belongs is a run-on because of the lack of punctuation between the introductory phrase (“Unlike most city museums”) and the independent clause that follows (“the design museum displays . . .”). Choice (D) corrects the error by introducing a comma between “museums” and “the.”

· C/Difficulty: Hard

Category: Punctuation

Getting to the Answer: When a period is underlined, make sure it’s being used correctly. The second clause is dependent and, with a period, creates a fragment; eliminate A. You can eliminate B because a semicolon won’t fix the error. Choices C and D are both grammatically correct but have different meanings. Choice D, without the comma, suggests that the “spotlights” were breaking down barriers. When separated with a comma, the phrase modifies the entire preceding clause, which makes (C) correct.

· D/Difficulty: Medium

Category: Punctuation

Getting to the Answer: Check whether the sentence makes sense without “however.” It does. When “however” is used within a clause, it is nonrestrictive and must be set off by commas. Choice (D) is correct.

· A/Difficulty: Hard

Category: Punctuation

Getting to the Answer: Adjectives are single-word modifiers that modify nouns. When two adjectives modify the same noun, they must be separated by either a comma or the conjunction “and.” The adjectives “impressive” and “innovative” both modify “products.” Choice (A) properly separates the items in the list.

· C/Difficulty: Medium

Category: Punctuation

Getting to the Answer: When you encounter two nouns in a row, check to see whether the possessive form is needed. Who or what has “visitors”? The “museum.” Is “museum” singular or plural? “A design museum” is singular, so choose the singular possessive. Choice (C) is correct.

· B/Difficulty: Medium

Category: Punctuation

Getting to the Answer: When a comma is underlined, check to see whether it can be replaced by a period. The clause before the comma can stand on its own, but the rest of the sentence can’t. A quick read indicates that the verb “show” has the same subject as “illustrate,” forming a compound predicate with “museums” as the subject. The comma is unnecessary. Choice (B) correctly eliminates the punctuation.

· D/Difficulty: Medium

Category: Punctuation

Getting to the Answer: Remember that dashes are used to indicate a hesitation or break in thought. Read the sentence without the verbs. The sentence still makes sense, signaling that the verbs are either nonessential information or an aside. Choice (D) sets the information off with a dash.

· B/Difficulty: Easy

Category: Punctuation

Getting to the Answer: With two nouns in a row, check to see whether the possessive is needed. Who or what has “humorous aspects”? The “societies.” Is “societies” singular or plural? The author never refers to multiple societies, so choose the singular possessive, which is (B).

ON YOUR OWN

· D/Difficulty: Hard

Category: Development

Getting to the Answer: With an underlined “and,” make sure to identify what job the conjunction is doing in the sentence. Because both clauses can stand on their own as sentences, you need to fix the run-on in the best possible way. This question is difficult because all of the answer choices correct the grammatical error. Carefully read each answer choice into the sentence. Choice (D) makes the clause dependent, giving information about the professionals’ complaints.

· C/Difficulty: Medium

Category: Punctuation

Getting to the Answer: When an underlined apostrophe appears with a noun, determine the noun’s number. Look for context clues that tell you how many “professions” possess “status.” Logically, each profession has its own status. In the previous sentence, the author refers to “groups” that professionals are part of. Choice (C) properly uses the plural possessive form.

· A/Difficulty: Hard

Category: Sentence Formation

Getting to the Answer: It might seem like the comma is unnecessarily separating the predicate from the object, but reading around the underlined section shows that the comma closes a nonrestrictive phrase. The underlined portion needs no change, so choice (A) is correct.

· D/Difficulty: Hard

Category: Development

Getting to the Answer: Whenever you have the chance to omit the underlined portion, carefully consider what function the underlined portion is performing. As written, the dash correctly sets off the phrase. However, the phrase itself contains information that shifts the focus of the passage. Choice (D) is correct.

· B/Difficulty: Medium

Category: Punctuation

Getting to the Answer: When you see a phrase set off by commas, always read the sentence without the phrase. In this case, you need the information enclosed by the commas; without the information, the sentence doesn’t make sense. The “aspirants” must be restricted to those whom the author discusses. Choice (B) correctly removes both commas.

· D/Difficulty: Medium

Category: Development

Getting to the Answer: When a transition is underlined, quickly summarize each paragraph and how the ideas are connected. Paragraph 2 talks about how it used to be, while paragraph 3 discusses how it is now. Things have changed. When things are different, look for the answer that provides contrast, which is choice (D).

· A/Difficulty: Easy

Category: Usage

Getting to the Answer: “Who” and “whom” refer only to people; “when” refers to time. Only (A) has the pronoun match.

· B/Difficulty: Hard

Category: Effective Language Use

Getting to the Answer: Check whether a period is the best way to join clauses. The period correctly separates the two sentences, but the change is abrupt. Choice (B) provides a better connection by showing the logical connection between the two clauses.

· C/Difficulty: Easy

Category: Punctuation

Getting to the Answer: If an underlined comma does not connect two independent clauses, ask what else it is doing. Here, the comma is separating items in a list containing only two elements. There is no need for the comma, so choice (C) is correct.

· B/Difficulty: Medium

Category: Effective Language Use

Getting to the Answer: Identify the overall tone of the passage. The tone in this passage is formal and informative. The author is careful to explain each idea introduced. The phrase “jumped on the bandwagon” does not fit because it is casual and, while giving a sense of the responses, does not clarify the author’s meaning. Choice (B) matches the overall tone and provides a detailed explanation.

· D/Difficulty: Easy

Category: Effective Language Use

Getting to the Answer: When you can’t identify a grammatical error, make sure to check for a stylistic error. In the context of the sentence, “sustain” and “preserve” mean the same thing. Choice (D) eliminates the redundancy error.