PSAT 8/9 Prep with 2 practice tests - Princeton Review 2020

Chapter 11 Math techniques
Part III PSAT 8/9 Prep

In the previous chapter, we mentioned that one of the keys to doing well on the PSAT 8/9 is to have a set of test-specific problem-solving skills. This chapter discusses some powerful strategies, which—though you may not use them in school—are specifically designed to get you points on the PSAT 8/9. Learn them well!

PLUGGING IN

One of the most powerful problem-solving skills on the PSAT 8/9 is a technique we call Plugging In. Plugging In will turn nasty algebra problems into simple arithmetic and help you through the particularly twisted problems that you’ll often see on the PSAT 8/9. There are several varieties of Plugging In, each suited to a different kind of question.

Plugging In Your Own Numbers

The problem with doing algebra is that it’s just too easy to make a mistake.

Whenever you see a problem with variables in the answer choices, use Plugging In.

Start by picking a number for the variable in the problem (or for more than one variable, if necessary), solve the problem using your number, and then see which answer choice gives you the correct answer.

Take a look at the following problem.

20.If a quantity c is decreased by 35%, which of the following is equivalent to the resulting value?

A)65c

B)35c

C)0.65c

D)0.35c

When to Plug In

· phrases like “in terms of” or “equivalent form” in the question

· variables in the question and/or answers choices

Here’s How to Crack It

The question asks for the result of decreasing an unknown value by 35%. There is a variable in the answer choices, so Plugging In can help you make more sense of the calculations here. A great number to plug in for percent questions is 100, as any percent taken of 100 will just be the number of the percent. This means that 35% of 100 is 35. If 100 is decreased by 35, the result is 100 — 35 = 65. This is the target number, so circle it. Next plug in 100 for c in the answer choices to see which one matches the target. Choice (A) becomes 65(100) = 6,500, which doesn’t match the target. Eliminate (A). Choice (B) becomes 35(100) = 3,500, which also can be eliminated. Choice (C) becomes 0.65(100) = 65. This matches the target, so keep it, but check (D) just in case. Choice (D) becomes 0.35(100) = 35, which is the amount of the decrease, not the resulting value after the decrease. Using Plugging In helped you avoid all the traps in (A), (B), and (D). Now you know that the correct answer is (C).

As you can see, Plugging In can turn messy algebra questions into more straightforward arithmetic questions. This technique is especially powerful when the PSAT 8/9 asks you to find the equivalent form of an expression.

5.Which of the following is an equivalent form of the expression x² + 15x — 54?

A)(x + 3)(x — 18)

B)(x — 3)(x + 18)

C)(x — 9)(x + 6)

D)(x + 9)(x — 6)

Here’s How to Crack It

The question asks for an equivalent form of an expression. We looked at factoring in a previous question, but looking at the answer choices shows that there is room for mistakes with the signs on this one. Another approach to try is Plugging In. Pick a small and simple value to work with, such as x = 2. The original expression becomes 2² + 15(2) — 54 = 4 + 30 — 54 = 34 — 54 = —20. Circle this target value, then you are ready to move on to the answer choices. Replace all those x’s with 2’s to determine which answer equals the target of —20. Choice (A) becomes (2 + 3)(2 — 18) = 5(—16) = —80. This doesn’t match your target, so eliminate (A). Choice (B) becomes (2 — 3)(2 + 18) = (—1) (20) = —20. This works, but don’t stop there! Remember to check all four answer choices when plugging in, in case more than one of them works. Choice (C) becomes (2 — 9)(2 + 6) = (—7)(8) = —56, and (D) becomes (2 + 9)(2 — 6) = (11)(—4) = —44. Neither matches the target, so you know that the correct answer is (B).

Plugging In: Quick Reference

· When you see in terms of or equivalent form and there are variables in the answer choices, you can use Plugging In.

· Pick your own number for an unknown in the problem.

· Do the necessary math to find the answer you’re shooting for, which is the target number. Circle the target number.

· Use POE to eliminate every answer that doesn’t match the target number.

Plugging In is such a great technique because it turns hard algebra problems into medium and sometimes even easy arithmetic questions. Remember this when you’re thinking of your POOD and looking for questions to do among the hard ones; if you see variables in the answers, there’s a good chance it’s one to try.

Don’t worry too much about what numbers you choose to plug in; just plug in easy numbers (small numbers like 2, 5, or 10 or numbers that make the arithmetic easy, like 100 if you’re looking for a percent). Also, be sure your numbers fit the conditions of the questions (for example, if they say x ≤ 11, don’t plug in 12).

What If There’s No Variable?

Sometimes you’ll see a problem that doesn’t contain an x, y, or z, but which contains a hidden variable. If your answers are percents or fractional parts of some unknown quantity (total number of marbles in a jar, total miles to travel in a trip), try Plugging In.

Take a look at this problem.

9.Sam is making conical party hats for her daughter’s birthday party. She makes one size for adults and one size for children, both of which have the same height. If the radius of the child-sized hats is one-third that of the adult-sized hats, then the volume of an adult-sized hat is how many times the volume of a child-sized hat?

A)16

B) 9

C) 2

D)

Here’s How to Crack It

The question asks for a relationship between the volumes of the two hats. There is not much geometry on the PSAT 8/9, but there is a handy reference box at the start of each Math section. The formula for the volume of a cone is listed there as V = πr²h. There are no variables in the question, but there is some missing information. If you knew the height and radius of the adult-sized hat, you could determine the height and radius of the child-sized hat and easily compare the volumes. Whenever you need some missing information to solve a problem, see if you can plug it in! Both hats have the same height, so plug in h = 6 (using a value divisible by 3 will help with the fraction later). The radius of the child-sized hat is one-third that of the adult-sized hat, so make the adult r = 9 and the child r = 3. Plug these numbers into the formula to find that the volume of an adult-sized hat is V = π(9)²6 = π(81) = 162π. The volume of the child-sized hat is V = π(3)²6 = π(9) = 18π. To find how many times the adult hat is compared to the child-sized one, divide the volume of the adult one by the volume of the child one to get = 9. The correct answer is (B).

MEANING IN CONTEXT

Some questions, instead of asking you to come up with an equation, just want you to recognize what a part of the equation stands for. It sounds like a simple enough task, but when you look at the equation, they have made it really hard to see what is going on. For this reason, Meaning in Context questions are a great opportunity to plug in real numbers and start to see how the equation really works!

First things first, though, you want to think about your POOD. Does this question fit into your pacing goals? It might take a bit of legwork to get an answer, and you may need that time to go collect points on easier, quicker questions.

If this question does fit into your pacing plan, you should read carefully, label everything you can in the equation, and POE to get rid of any answer choices that are clearly on the wrong track. Then, it’s time to plug some of your own numbers in to see what is going on in there.

Here’s one that is part of a set with a question we already worked on.

Déjà vu?

You may remember Elijah, Betty, and Zeb from the previous chapter. This is the final question that belongs to this set.

Two construction workers are digging holes using shovels and measuring their dig rates to determine who is faster. Both construction workers start in holes below ground level at their assigned dig sites and dig vertically down through the dirt.

The depth below ground level d, in meters (m), that construction worker Elijah has reached in terms of time t, in minutes, after he started digging is described by the function d(t) = 1.8t + 4.5.

Construction worker Betty begins in a hole below ground level. Betty’s depth below ground level, in m, for the first 5 minutes of digging is described in the table below. Betty’s dig rate is constant throughout the competition.

10.Which of the following is the best description of the number 4.5 in the function d(t) = 1.8t + 4.5?

A)The depth below ground level, in meters, of Elijah’s hole at the beginning of the competition

B)The depth below ground level, in meters, of Elijah’s hole 4.5 minutes after digging begins

C)Elijah’s dig rate, in meters per minute, at the beginning of the competition

D)Elijah’s dig rate, in meters per minute, 4.5 minutes after digging begins

Here’s How to Crack It

The question asks for the meaning of a number in a function. As with all sets, the first place to start is to determine where to find the information that you need. The function is in the introductory text, and it refers to Elijah’s digging speed. Start by underlining the number in the equation that the question refers to, which is 4.5. Then label as much of the equation as possible to determine what everything represents. The depth is represented by d, so label the left side of the equation “depth.” Time is represented by t, so label the t as “time.” The number 4.5 is neither of these, so plugging in some numbers may help to determine what is happening here. Choice (A) refers to the depth at the beginning of the competition, or when t = 0. Plug this value into the equation to get d(0) = 1.8(0) + 4.5 = 0 + 4.5 = 4.5. This matches the value you underlined in the equations, so (A) is true. If you are not sure about this, you can always check the other answers to see if you can eliminate them. Choice (B) refers to the depth after 4.5 minutes, which cannot also be 4.5, since that’s the depth at 0 minutes. Eliminate (B). Choices (C) and (D) both refer to rate, in meters per second. Slope relates to rate of change. Looking at the equation, you may have noticed that it is in slope-intercept form, y = mx + b. In this form, the slope is m, so the slope (and the rate of change) is 1.8, not 4.5. Therefore, you can eliminate (C) and (D). The correct answer is (A).

Here are the steps for using Plugging In to solve Meaning in Context questions:

Meaning In Context

1. Read the question carefully. Make sure you know which part of the equation you are being asked to identify.

2. Use your pencil to label the parts of the equation you can identify.

3. Eliminate any answer choices that clearly describe the wrong part of the equation, or go against what you have labeled.

4. Plug in! Use your own numbers to start seeing what is happening in the equation.

5. Use POE again, using the information you learned from plugging in real numbers, until you can get it down to one answer choice. Or, get it down to as few choices as you can, and guess.

Math Techniques Drill 1

Answers can be found in Part IV.

V = πr²h

8.The formula for the volume, V, of a cylinder is given above, where r is the radius of the cylinder and h is the height of the cylinder. Which of the following gives the height of the cylinder in terms of its volume and radius?

A)h = πVr²

B)

C)

D)

9.Jean calculates how much money she spends on her dog Patches, and the results are shown below.

Expenses for Patches

On top of these costs, Jean also pays an annual municipal licensing fee of $55. The cost, P, of caring for Patches for a given number of weeks, w, in the year can be calculated using the equation P = 55 + (20 + 30)w. When this function is graphed in the xy-plane, what does the slope represent?

A)The cost of food and dog-walking per week

B)The cost of licensing and food for a year

C)The total cost of care per week

D)The cost of licensing for a year

4.James joins a local billiards club. The club charges a monthly membership fee of $15 and each hour of play costs $1.50. Which of the expressions below shows James’s monthly cost at the billiards club when he plays for h hours?

A)(15 + h)1.50

B)(15 + h)0.50

C)0.50h + 15

D)1.50h + 15

Albert: t = 9h

Buster: t = 3h

10.The equations above show the number of trees, t, chopped per hour, h, by two lumberjacks, Albert and Buster. Based on these equations, which of the following statements is true?

A)The rate at which Buster chopped trees per hour decreased at a slower rate than the rate at which Albert chopped trees per hour.

B)The rate at which Albert chopped trees per hour decreased at a slower rate than the rate at which Buster chopped trees per hour.

C)For every hour of chopping, the number of trees Albert chopped was one-third as many as the number of trees Buster chopped.

D)For every hour of chopping, the number of trees Buster chopped was one-third as many as the number of trees Albert chopped.

17.The power P, in watts, of an electrical circuit can be calculated using the formula , where V is the voltage of the circuit, in volts, and R is the resistance of the circuit, in ohms. Assuming that the voltage does not change, what is the effect on the power when the resistance is multiplied by a factor of 2?

A)It is divided by 2.

B)It is divided by 4.

C)It is multiplied by 2.

D)It is multiplied by 4.

19.Research scientists are testing the effect of a new antibiotic on bacterial growth. They start with a colony of 1,500 bacteria and take counts every 10 minutes for an hour, as shown in the table below.

Bacterial Count Over Time Following Antibiotic Treatment

Which of the following represents the best interpretation of the y-intercept of the line created when these data points are plotted in the xy-plane if minutes, m, are plotted on the x-axis and number of bacteria, b, are plotted on the y-axis?

A)The number of bacteria at the start of the experiment

B)The rate at which bacteria is decreasing every 10 minutes

C)The number of intervals for which the number of bacteria is decreasing

D)The number of bacteria 60 minutes after the experiment begins

20.In the equation F = 5(g + 2) — 3, which of the following gives the value of g in terms of F?

A)

B)

C)

D)

PLUGGING IN THE ANSWERS (PITA)

You can also plug in when the answer provided to a problem is an actual value, such as 2, 4, 10, or 20. Why would you want to do a lot of complicated algebra to solve a problem, when the answer is right there on the page? All you have to do is figure out which choice it is.

How can you tell which is the correct answer? Try every choice until you find the one that works. Even if this means you have to try all four choices, PITA is still a fast and reliable means of getting the right answer.

If you work strategically, however, you almost never need to try all four answers. If the question asks for either the greatest or the least answer, start there. Otherwise, start with one of the middle answer choices. If that answer works, you’re done. If the answer you started with was too big, try a smaller answer. If the answer you started with was too small, try a bigger answer. You can almost always find the answer in two or three tries this way. Let’s try PITA on the following problem.

3.Lauren and Jolean are gardeners building plant containers in their local community garden. For every plant container, there are at most 110 seeds but no less than 70 seeds. If they build 3 plant containers, which of the following could be the total number of seeds in the 3 containers?

A)465

B)345

C)225

D)105

PITA = Plugging In the Answers

Don’t try to solve problems like this by writing equations and solving for variables. Plugging In the Answers lets you use arithmetic instead of algebra, so you’re less likely to make errors.

Here’s How to Crack It

The question asks for the number of seeds that could be in the 3 containers. Label the answers as “seeds,” and start somewhere in the middle. Try (B), 345 seeds. Divided among 3 containers, this would be 345 ÷ 3 = 115 seeds per container. This is outside the range of 70 to 110 seeds per container, so eliminate (B). The value was too large, so try (C) next. Divided among 3 containers, there would be 225 ÷ 3 = 75 seeds per container. Since this is within the range of seeds per container, you can stop here. The correct answer is (C), and you didn’t even need to set up an inequality to solve!

Neat, huh? Let’s try one more.

9.Given the equations —a + 4b = 8 and 2a + 4b = 20, what is the value of a if (a, b) is the solution to the system of equations?

A)—12

B) 3

C) 4

D) 12

Here’s How to Crack It

The question asks for the value of a in the system of equations. We already looked at how to solve systems, but let’s try this one using PITA. Label the answers as a, and start in the middle with (C), 4. Plug this value of a into both equations and solve for b.

Both equations yield the same value for b, so we’ve found the correct value of a. Always be on the look-out for opportunities to plug in or plug in the answers as you do your PSAT 8/9 practice. Once you get good at it, you can decide on any test question which way is faster—solving or Plugging In.

Math Techniques Drill 2

Answers can be found in Part IV.

3.What is the value of z given the equation 4(z + 3) — 6 = 5(z — 1)?

A)11

B) 7

C) 2

D)—2

6.If and x ≠ —2, what is the value of x?

A) 2

B) 3

C) 9

D)27

9.Which of the following is a solution to the equation p² + 20p + 75 = 0?

A)—75

B)—15

C) 0

D) 20

12.Gail’s homemade iced tea is a combination of green tea and raspberry purée. If the green tea and raspberry purée are in a ratio of 23 : 7, and a pitcher of iced tea can hold 900 mL, what is the amount of green tea, in mL, that Gail needs to make a pitcher of iced tea?

A) 63

B)207

C)210

D)690

15.Which if the following is a solution to the equation

(x + 8) = 56?

A)The equation has no solution.

B)24

C)32

D)40

17.Chris is coating strawberries in chocolate for dessert. The table above displays 6 strawberries and their corresponding amounts of chocolate coating in grams (g). If Chris adds a 7th strawberry coated in chocolate, what is one possible value for the amount of chocolate coating that would decrease the mean of the data but increase the range?

A)15.8 g

B)17.1 g

C)19.2 g

D)23.1 g

For more information on dealing with range and mean, see the next section of this chapter.

18.In the expression x² + cx — 8, c is an integer. If the expression can be rewritten as the equivalent expression (x — c)(x + 4), which of the following is the value of c?

A) 4

B) 2

C)—2

D)—4

DATA ANALYSIS

In the calculator-allowed Math section (Section 4), there will be questions that will ask you to work with concepts such as averages, percentages, and unit conversions. Luckily, The Princeton Review has you covered! The rest of this chapter will give you techniques and strategies to help you tackle these questions.

Averages and T = AN

You probably remember the average formula from math class, which says Average (arithmetic mean) = . However, the PSAT 8/9 will not always ask you to take a simple average. Of the three parts of an average problem—the average, the total, and the number of things—you’re often given the average and the number of things, and finding the total will be the key to solving the question.

If you multiply both sides of the above equation by # of things, you get (Average)(# of things) = total. It’s easier to remember this as

T = AN

or, Total = Average × Number of Things. Once you know the total, you’ll have the key information needed to finish solving the question.

Total

When calculating averages, always find the total. It’s the one piece of information that PSAT 8/9 loves to withhold.

Let’s try this example.

12.0, 12.5, 13.0, 13.5, 14.0

8.The volumes, in milliliters, of five water bottles are shown above. What is the average volume, in milliliters, of the five bottles?

A)14.0

B)13.0

C)12.0

D) 2.0

Here’s How to Crack It

The question asks for the average volume of the five listed water bottle volumes. Use T = AN to calculate the average. The number of things is 5, and the total of the 5 volumes is T = 12.0 + 12.5 + 13.0 + 13.5 + 14 = 65. Plug these values into the formula to get 65 = A(5), then divide both sides by 5 to get 13 = A. You may also have noticed that since the 5 numbers are equally spread out by 0.5, the middle value will also be the average. Either way, the correct answer is (B).

Median

Another concept that is often tested along with average is median.

The median of a group of numbers is the number in the middle, just as the “median” is the large divider in the middle of a road. To find the median, here’s what you do:

· First, put the elements in the group in numerical order from lowest to highest.

· If the number of elements in your group is odd, find the number in the middle. That’s the median.

· If you have an even number of elements in the group, find the two numbers in the middle and calculate their average (arithmetic mean).

Finding a Median

To find the median of a set containing an even number of items, take the average of the two middle numbers after putting the numbers in order.

Try this on the following problem.

19.A baseball statistician analyzed 21 different games for the total number of runs scored by a baseball team. The distribution of the runs scored in each game is displayed in the frequency table below.

What is the median number of runs scored of these games?

A)2

B)3

C)4

D)5

Here’s How to Crack It

The question asks for the median number of runs based on the chart. The PSAT 8/9 will often make medians more difficult to calculate by putting the data in a chart or graph rather than in a list. On a frequency chart such as this one, there were three games that the team scored 0 runs, then 4 that the team scored 1 run, and on and on. Rather than listing out all 21 numbers, figure out which number would be in the middle if you did list them out. It would be the 11th number, as there would be 10 numbers before it and 10 after it. The first 7 numbers on the list are 0 or 1, then the 8th and 9th are 2. This means that the 10th through 13th numbers are 3, making the 11th number, and the median, 3. The correct answer is (B).

Mode and Range

Two more statistical concepts you may see on the PSAT 8/9 are mode and range, and they can often appear in the same question.

The mode of a group of numbers is the number that appears the most. (Remember: mode sounds like most.) To find the mode of a group of numbers, simply see which element appears the greatest number of times.

The range of a list of numbers is the difference between the greatest number on the list and the least number on the list. For the list 4, 5, 5, 6, 7, 8, 9, 10, 20, the greatest number is 20 and the least is 4, so the range is 20 — 4 = 16.

Let’s look at a problem:

14.Dot plot A has 15 data points, and one of the data points is considered an outlier. Dot plot B has the 14 data points representing the same data as plot A without the outlier. Which of the following statements accurately describes the two dot plots?

A)The range of dot plot A is equal to the range of dot plot B.

B)The median of dot plot A is equal to the median of dot plot B.

C)The mode of dot plot A is equal to the mode of dot plot B.

D)The mean of dot plot A is equal to the mean of dot plot B.

Here’s How to Crack It

The question asks for a description of the two dot plots, and all the answers contain statistics about the data. When given many options for data analysis, start with the easier ones and use POE. On a dot plot, the easiest thing to see is mode, as that will be the number with the most dots above it. On both dot plots, there are 4 dots at the value of 5, and no other values has 4 or more dots. This means that the mode of both dot plots is equal, making (C) the answer. If you aren’t sure, you can check out the other measures. Range is the next easiest one to calculate. The range of dot plot A is 11 — 0 = 11, but the range of dot plot B is 11 — 5 = 6. Therefore, the ranges are not equal and (A) can be eliminated. The median of dot plot A will be the middle, or 8th, number. This is 7. The median of dot plot B will be the average of the 7th and 8th numbers, since there are an even number of numbers. The 7th number is 7, and the 8th number is 9, so the median of dot plot B is 8. This eliminates (B). The mean or average is the total divided by the number of things. Both lists will have the same total, as only the value of 0 is different. However, the total in A will be divided by 15 to get the average, while the total in B will only be divided by 14. This makes the average of A less than that of B, so eliminate (D). The correct answer is (C).

Rates

Rate is a concept related to averages. Cars travel at an average speed. Work gets done at an average rate. Because the ideas are similar, you can use Distance = Rate × Time (D = RT) or Work = Rate × Time (W = RT) to find the total distance traveled or work done.

Here’s a simple example:

Problem: If a mover can load boxes onto a truck at a rate of 2 every 7 minutes, how long will it take him, in minutes, to load 37 boxes onto the truck?

Solution: Use the formula W = RT to calculate the mover’s time. The W, or work, is 37 boxes, and the rate is , so the formula becomes 37 = T. Multiply both sides by to get (37) = T, or T = 129.5 minutes.

Now let’s look at a rate question.

23.Gabriel and David are assigned to clean a certain number of whiteboards during weekend detention. If Gabriel works alone, he can clean 7 boards per half-hour session. If David works alone, he can clean 5 boards per half-hour session. If the supervising teacher assigns them to work together in half—hour sessions, how many white boards will they clean in 2 hours?

Here’s How to Crack It

The question asks for the number of whiteboard Gabriel and David will clean in two hours. Use the formula W = RT to find the answer. They are working together, so their combined rate is 5 + 7 = 12 boards in one half-hour session. Multiply this by two to get the combined rate: 24 whiteboards per hour. Now that the units of time match, plug the values of R = 24 and T = 2 into the formula to get W = (24)(2) = 48. The correct answer is 48.

PROBABILITY

One topic that is often tested with two-way tables is probability. Probability refers to the chance that an event will happen, and it is given as a percent or a fractional value between 0 and 1, inclusive. A probability of 0 means that the event will never happen; a probability of 1 means that it is certain to happen.

For instance, if you have a die with faces numbered 1 to 6, what is the chance of rolling a 2?

There is one face with the number 2 on it, out of 6 total faces. Therefore, the probability of rolling a 2 is .

What is the chance of rolling an even number on one roll of this die? There are 3 faces of the die with an even number (the sides numbered 2, 4, and 6) out of a total of 6 faces. Therefore, the probability of rolling an even number is , or .

Let’s look at how this concept will be test on the PSAT 8/9.

7.The data table above shows the results of a survey of 300 employees of a car dealership who were asked how many cars they had owned in their lifetimes. What is the probability that a randomly-selected employee had owned 3—4 cars?

A)0.40

B)0.20

C)0.15

D)0.08

Here’s How to Crack It

The question asks for a probability based on a chart. The definition of probability is , and the question states that there were 300 total employees surveyed. To determine the want, check the chart to see how many of those surveyed owned 3—4 cars. This number on the chart is 120, so the probability is . The answers are in decimal form, so divide 120 by 300 on your calculator to find that the probability is 0.40. The correct answer is (A).

PERCENTS

Percent just means “divided by 100.” So 20 percent = , or .2.

Likewise, 400 percent = .

Any percent question can be translated into algebra—just use the following rules:

Take a look at some examples of phrases you might have to translate on the PSAT 8/9:

Try a question.

17.Joanna is a florist, and she estimates that of the 23 bouquets she sells in a day, 8.7% of them will not be gift-wrapped. At this rate, which of the following is the approximate total number of bouquets sold after 14 days that will not be gift-wrapped?

A)2,800

B) 200

C) 28

D) 2

Looking for More Practice?

If you want some more difficult practice (or want a jump start on the PSAT), check out Princeton Review PSAT/NMSQT Prep.

Here’s How to Crack It

The question asks for an approximation of the number of bouquets sold in 14 days that will not be gift-wrapped. Use the given percentage to determine how many are not gift-wrapped each day. Joanna sells 23 bouquets in a day, and 8.7% are not gift-wrapped. “Percent” means to divide by 100, so 8.7% can be written as . This percent is of the bouquets, and “of” means to multiply, so this becomes (23) ≈ 2 bouquets each day that are not gift-wrapped. It is okay to round this, as the question asks for an approximate number and the answers are spread apart. Before you jump in and choose (D), make sure to pay attention to what the question actually asks for—the number of bouquets that are not gift-wrapped over the course of 14 days. If there are 2 per day, the result is (2)(14) = 28. The correct answer is (C).

RATIOS AND PROPORTIONS

Some questions in the calculator-allowed Math section (Section 4) will ask about ratios and proportions. With the strategies that you’ll learn on the next few pages, you’ll be well prepared to tackle these concepts on the PSAT 8/9.

Ratios

Ratios are about relationships between numbers. Whereas a fraction is a relationship between a part and a whole, a ratio is about the relationship between parts. So, for example, if there were 3 boys and 7 girls in a room, the fraction of boys in the room would be . But the ratio of boys to girls would be 3:7. Notice that if you add up the parts, you get the whole: 7 + 3 = 10. That’s important for PSAT 8/9 ratio problems, and you’ll see why in a moment.

Ratio problems usually aren’t difficult to identify. The problem will tell you that there is a “ratio” of one thing to another, such as a 2:3 ratio of boys to girls in a club. On the PSAT 8/9, you’ll often be asked to compare different ratios. Use the definition that looks like a fraction (or, in other words, divide), and be sure to have the correct order (first term in the numerator, second term in the denominator).

Gridding In

A ratio is usually expressed as 2:3 or 2 to 3, but if you need to grid a ratio, grid it as .

Try this one.

5.The preferred pets of a random selection of high school students are shown in the table below. Students were asked to select their preferred pet from the following list: Dog, Cat, Fish, or Lizard.

Preferred Pet Based on High School Year

Based on the results in the table above, the ratio for the least- to most-preferred pet of the senior class is about how many times greater than the ratio for the least- to most-preferred pet of the freshman class?

A)1.2

B)1.7

C)3.0

D)3.4

Here’s How to Crack It

The question asks for a comparison of the ratios of least- to most-preferred pets for two different classes. It is important to read the question carefully and find the correct numbers for the ratios. Start with the freshman class. The freshmen’s least preferred pet is the one with the smallest number of votes—lizard, with only 45. The freshmen’s most preferred pet is a dog, with 246 votes. Strictly speaking, the ratio is 45:246, but the question says about how many times, indicating that you can estimate. Let’s just call this 50:250, or 1:5. Ratios can be written as fractions, so that’s , or decimals, making it 0.20. This last version will be most useful for comparing the ratios. Now move on to the seniors, whose least-preferred pet was a fish with 62 votes and most-preferred pet was a cat with 113 votes. Again, round this to about 60:120 or 1:2, making the ratio 0.5. The question asks how many times greater the 0.5 ratio is than the 0.2 ratio, so take 0.5 and divide it by 0.2. The result is 2.5, which is closest to (C).

If you are nervous about rounding on one like this, given that the answers are somewhat close together, you can always do the exact calculations on your calculator, as calculator use is allowed on statistics questions. Just make sure to only do one bite-sized piece at a time and to write things down on your paper. That way, you won’t get confused or miscalculate. Either way, the correct answer is (C).

Direct Variation

Direct variation or proportion problems generally ask you to make a conversion (such as from ounces to pounds) or to compare two sets of information and find a missing piece. For example, a proportion problem may ask you to figure out the amount of time it will take to travel 300 miles at a rate of 50 miles per hour.

To solve proportion problems, just set up two equal fractions. One will have all the information you know, and the other will have a missing piece that you’re trying to figure out.

Be sure to label the parts of your proportion so you’ll know you have the right information in the right place; the same units should be in the numerator on both sides of the equals sign and the same units should be in the denominator on both sides of the equals sign. Notice how using a setup like this helps us keep track of the information we have and to find the information we’re looking for, so we can use Bite-Sized Pieces to work through the question.

Now we can cross-multiply and then solve for x: 50x = 300, so x = 6 hours.

Let’s try the following problem.

3.A full water jug in an office contains 3,785 mL of water. What is the water jug’s volume, in liters? (1 liter = 1,000 milliliters)

A) 3.785

B) 378.5

C) 378,500

D)3,785,000

Here’s How to Crack It

The question asks for the measure of a volume of 3,785 mL in liters. On these, try to ballpark if you can. A liter is a bigger unit of measurement than a milliliter, so the volume in liters must be a smaller number than the volume in milliliters. This enables you to eliminate (C) and (D). Sometimes this is not obvious, though, so only eliminate this way if you are sure you will not make a mistake, especially given that all of the answers are 3875 times a different power of 10. To make sure you always get proportions right and to decide between (A) and (B), set up two equal proportions with the known information and matching units in the numerators and denominators. For this question, that becomes . Cross-multiply to get 1,000x = 3,875, then divide both sides by 1,000 to get x = 3.875. The correct answer is (A).

Inverse Variation

Inverse variation is simply the opposite of a direct, or ordinary, proportion. In a direct proportion when one variable increases, the other variable also increases; however, with inverse variation, when one variable increases, the other variable decreases, or vice versa. These types of problems are generally clearly labeled and all you have to do is apply the inverse variation formula:

x₁y₁ = x₂y₂

Once you memorize the formula, applying it will become second nature to you.

17.Vidhi is studying Boyle’s Law, where the pressure of a gas varies inversely to its volume. If the pressure of a certain gas is 45 kilopascals for a volume of 5 liters, then what is the pressure of the gas in kilopascals when the gas is transferred to a container with a volume of 9 liters?

A)16

B)20

C)25

D)81

Here’s How to Crack It

The question asks for the pressure of a gas in a container of a certain volume. According to the question, the pressure and the volume vary inversely, so use the inverse variation formula x₁y₁ = x₂y₂. Plug in the given values and P for the missing pressure to get (45)(5) = (P)(9), which simplifies to 225 = 9P. Divide both sides of the equation by 9 to get 25 = P. The correct answer is (C).

Math Techniques Drill 3

Remember, answers to these drill questions can be found in Part IV!

a. If a student scores 70, 90, 95, and 105, what is the average (arithmetic mean) for these tests?

b. If a student has an average (arithmetic mean) score of 80 on 4 tests, what is the total of the scores received on those tests?

c. If a student has an average of 60 on tests, with a total of 360, how many tests has the student taken?

d. If the average of 2, 8, and x is 6, what is the value of x?

2, 3, 3, 4, 6, 8, 10, 12

e. What is the median of the group of numbers above?

f. What is the mode of the group of numbers above?

g. What is the range of the group of numbers above?

h. What percent of 5 is 6?

i. 60 percent of 90 is the same as 50 percent of what number?

j. Jenny’s salary increased from $30,000 to $33,000. By what percent did her salary increase?

k. In 1980, factory X produced 18,600 pieces. In 1981, factory X produced only 16,000 pieces. By approximately what percent did production decrease from 1980 to 1981?

l. In a certain bag of marbles, the ratio of red marbles to green marbles is 7:5. If the bag contains 96 marbles, how many green marbles are in the bag?

m. One hogshead is equal to 64 gallons. How many hogsheads are equal to 96 gallons?

n. The pressure and volume of a gas are inversely related. If the gas is at 10 kPa at 2 liters, then what is the pressure when the gas is at 4 liters?

4.Kendall is baking a cake that requires 0.8 quarts of cream, but her measuring cup only measures liters. There are 1.06 quarts in 1 liter. Which of the following is the approximate number of liters of cream Kendall needs for the recipe?

A)1.55

B)0.85

C)0.75

D)0.26

5.Jeshua is making flashcards. He can make 18% of the flashcards he needs per hour, on average. If Jeshua continues at this rate, which of the following is the approximate number of flashcards he can make in one hour if he needs 398 flashcards?

A)68

B)72

C)76

D)80

Number of Times Jolly Retrieved the Toy Throughout the Day

8.The two-way table above shows the number of times Jolly the puppy will retrieve a ball or a stuffed animal at different times of the day. What was the average number of times Jolly retrieved the stuffed animal throughout the day?

A)10.50

B)10.88

C)11.25

D)11.50

11.Fibi has a fish tank that can hold a given number of cubic centimeters of water. She measures her fish tank and finds that it has a length of 1,219 millimeters, a width of 0.305 meters, and a depth of 406 millimeters. What are the measurements (length, width, and depth, respectively) of her fish tank in centimeters? (Note: 1 centimeter = 10 millimeters, and 1 meter = 100 centimeters)

A)121.9 cm × 30.5 cm × 4.06 cm

B)121.9 cm × 30.5 cm × 40.6 cm

C)12,190 cm × 305 cm × 4,060 cm

D)12,190 cm × 305 cm × 406,000 cm

15.Brent, an accounting firm owner, donates a specific percentage of his firm’s annual profit to charity. If Brent’s firm made $654,000 last year and donated $52,320, how much money will his firm donate if its annual profit is $575,000 this year?

A) $46,000

B) $59,508

C) $79,000

D)$131,320

Questions 24—25 refer to the following information.

One batch of hand sanitizer can be created using the materials and amounts listed in the table above. A single dose of hand sanitizer is 0.5 ounces and one batch of this recipe makes 4 doses.

24.Jack used d% of one batch of the hand sanitizer. If this amount contained 0.25 milliliters of essential oil, what is the value of d?

25.How many liters of rubbing alcohol are in one dose of this hand sanitizer?

Summary

o The test is full of opportunities to use arithmetic instead of algebra—just look for your chances to use Plugging In and Plugging In the Answers (PITA).

o If a question has in terms of or variables in the answer choices, it’s a Plugging In problem. Plug in your own number, do the math, find the target number, and use POE to get down to one correct answer.

o If a question doesn’t have variables but asks for a fraction or a percent of an unknown number, you can also plug in there. Just substitute your own number for the unknown and take the rest of the problem step by step.

o For Meaning in Context questions, start with RTFQ, labeling the equation, and using POE. If you still have more than one answer choice remaining, plug in to the equation to help you narrow down the answers.

o If a question has an unknown and asks for a specific amount, making you feel like you have to write an equation, try PITA instead.

o Average is defined as . Often on the PSAT 8/9, you’ll be given the average and the number of things. Use T = AN on those problems to determine the total.

o The median is the middle value in a list of consecutive numbers. If there are an odd number of elements, the median is the average of the two middle values.

o The mode is the most commonly occurring value in a list of numbers.

o The range is the difference between the greatest and least values in a list of numbers.

o Rates are closely related to averages. Use D = RT or W = RT just like you use T = AN. Remember that the PSAT 8/9 likes to make you find the totals (distance or work in rate questions).

o Probability is a fractional value between 0 and 1 (inclusive), and it is equal to the number of outcomes the question is asking for divided by the total number of possible outcomes. It can also be expressed as a percent.

o Percent simply means “per 100.” Many percent questions can be tackled by translating English to math.

o Set up ratios like fractions. Take care to put the first term of the ratio in the numerator and the second term in the denominator.

o Sometimes you’ll need to treat ratios like fractions or decimals. Use your calculator to turn the numbers into the easiest form to work the problem.

o Direct variation or proportion means as one value goes up, the other goes up. The formula is .

o Inverse variation means as one value goes up, the other goes down. The formula is x₁y₁ = x₂y₂.