SSAT & ISEE Prep 2023 - Princeton Review 2023

Answers and explanations for ISEE practice drills
The ISEE

ISEE MATH

Primary Basics Drill

1.A

2.D

3.A

4.D

5.B

6.C

7.B

8.C

9.B

10.C

11.C

12.B

13.A

14.C

15.C

*Find detailed explanations for these questions in your online Student Tools.

Practice Drill 1—Lower Level

1.C

List all the factors of 16: 1 and 16, 2 and 8, 4 and 4. Since 4 and 4 are not distinct from each other, count 4 only once. There are 5 distinct factors of 16. The correct answer is (C).

2.D

List the factors of 12: 1 and 12, 2 and 6, 3 and 4. All of these are factors of 48. The correct answer is (D).

3.C

A multiple of a number is that number multiplied by another number. Since 7 × 13 = 91, 91 is a multiple of 7. The correct answer is (C).

4.A

A multiple of a number is that number multiplied by another number. 4 is a factor of 8, not a multiple of 8. The correct answer is (A).

5.C

Use the answer choices to help here. 18 is a multiple of 6, but not of 4. Eliminate (A). 32 is a multiple of 4, but not of 6. Eliminate (B). 48 is a multiple of both 4 and 6, so keep this choice. 54 is a multiple of 6, but not of 4. Eliminate (D). The correct answer is (C).

6.B

The question asks what is NOT a divisor of 1,026. Since 1,026 is even, eliminate (A). Do long division to test the other answer choices, and the choice with a remainder will be the answer. The correct answer is (B).

7.B

Use the answer choices to help here. Since the three integers are odd, the square of the smallest integer will also be odd. Eliminate (C) because 36 is even. If the sum of the three is 21, the smallest integer can’t be 7 because 7 + 7 + 7 = 21. Eliminate (D) because 49 is the square of 7. Try one of the remaining two. Let’s look at (B): The square root of 25 is 5, and 5 + 7 + 9 = 21, so that works.

8.B

If the question asks how many times as great, that translates to division. Divide the profit in 2018 by the profit in 2008. = 6 The correct answer is (B).

9.C

Multiply 45 by . 45 × = 18. Another way to think about this is to divide by 5 and multiply by 2. The correct answer is (C).

10.A

If the maximum capacity is 112 gallons and the tank is one-fourth full, multiply (112) - 28. The correct answer is (A).

11.C

To find the total amount of time Jake stands over the course of all four days, add for every day he works. , This reduces to , , or 2.5. The correct answer is (C).

12.A

Test each of the answer choices to see which is the greatest. To add or subtract fractions, use a common denominator. In (A), . Leave this choice and test the others. For (B), find a common denominator, which is 20. and . , which is slightly more than . Eliminate (B) since it is less than (A). When dividing by a fraction, multiply by the reciprocal. To multiply fractions, multiply the numerators and denominators by each other. In (C), . Since is less than , eliminate (C) and try (D). In (D), , which is less than . The correct answer is (A).

13.A

Remember your order of operations and work left to right. Since the equation contains only addition and subtraction, find a common denominator. One possible common denominator of 2, 3, and 4 is 12, so . Now, add all the numerators: 6 + 4 + 3 — 6 + 8 + 9 = 24, and . The correct answer is (A).

14.C

When multiplying by a multiple of 10, move the decimal to the right by however many 0s there are. Since this question contains multiplying by 10,000, move the decimal to the right four places. The correct answer is (C).

15.C

1.576 means 1 whole unit plus a decimal, so eliminate (A) since all that multiplication will result in a very small number and (D) because it is missing the 1. Now, 0.576 breaks down into 0.5 + 0.07 + 0.006, or . The correct answer is (C).

Practice Drill 2—Multiple Choice—Middle and Upper Levels Only

1.A

Since 4 is not a prime number and no multiple of 4 will be prime either, there will not be any numbers in common. Therefore, the correct answer is (A).

2.B

First, list all the factors of 24: 1 and 24, 2 and 12, 3 and 8, 4 and 6. Next, list all the factors of 81: 1 and 81, 3 and 27, 9 and 9. The only factors that 24 and 81 have in common are 1 and 3, so there are two factors in common. The correct answer is (B).

3.D

Guesstimate and let the answer choices help here. If the final bill after 25% tip is $50, $12.50 and $25 are way too small. The tip would be either the same amount as or more than the cost! Eliminate (A) and (B). Try one of the two remaining answer choices. $40 looks easiest to work with, so let’s start there. If the cost is $40, 25% of 40 is calculated by multiplying (40) = 10. $40 + $10 = $50, so that’s our answer. The correct answer is (D).

4.B

List all the numbers from 10 to 150, inclusive, that are multiples of 3 and 5: 15, 30, 45, 60, 75, 90, 105, 120, 135, and 150. Inclusive means to include the ends of the range, so 150 should be included in this list. There are 10 numbers listed, so the correct answer is (B).

5.B

When multiplying by the same base, add the exponents together. 4 + 4 + 4 = 12, so the answer should be 3¹². The correct answer is (B).

6.B

Let the answer choices help here. Remember that exponents mean multiplying a number by itself, so the numbers on the right side of the equation are going to have the same factors as x. Since 4 and 8 are not multiples of 3, eliminate (C). 1 is too small, since 1 to any power is 1 and the right side of the equation has to be at least 8. Eliminate (A). Try one of the remaining two answer choices. Smaller numbers are easier to use with exponents, so try (B). 24 = 2 x 2 x 2 x 2 = 16 and 4(2) + 8 = 16, so that works. The correct answer is (B).

7.C

First, distribute 3 on the right-hand side of the equation. Since 3(3x — 6) = 9x — 18, 18 = 9x — 18. Add 18 to both sides to isolate x: 36 = 9x. Divide both sides by 9 to find that x = 4. Not so fast! Remember the question asks for x + 6, so 4 + 6 = 10. The correct answer is (C).

8.B

Let the answer choices help here. Since 36, 48, and 54 are multiples of 6, eliminate (C), which is not a multiple. The question asks for the largest multiple less than 53, so of these choices, 48 fits this description. The correct answer is (B).

9.D

The questions states that of the cars are silver and that there are 24 silver cars. Therefore, 24 × 3 equals the total number of cars, or 72. The correct answer is (D).

10.C

Add all the roses up to find that 30 yellow roses + 50 red roses + 40 white roses = 120 total roses. The roses that are not yellow are the 50 red and 40 white, which makes 90. = or 75%. The correct answer is (C).

11.C

To find the percent difference, find the difference and divide by the original value. The difference in price is $15, and the original value is $40, since $55 is a markup, which is approximately 38%. The correct answer is (C).

12.C

Guesstimate here. 75% is more than half, and 49.95 is essentially 50. Eliminate (A) and (B) since those are both less than half. is , which is larger than , so eliminate (D). The correct answer is (C).

13.B

To find the percent change, find the difference in points and divide that by the original value, 25. 25 — 15 = 10, which is the difference, or 0.4, which equals 40%. The correct answer is (B).

14.B

If the team adds a fourth person and splits the dues evenly, = $6 per person. The correct answer is (B).

15.C

A regular figure means that all the sides are equal. A pentagon has 5 sides, so to find each individual side, divide the perimeter by the number of sides: 65 ÷ 5 = 13. The correct answer is (C).

16.A

To find the perimeter of a shape, add all the sides together. First, find the hypotenuse by recognizing this is a special right triangle (3-4-5) or using the Pythagorean Theorem: 3² + 4² = c², which simplifies to 9 + 16 = 25 = c², and = 5. Add 3 + 4 + 5 to find the perimeter of the triangle is 12. The correct answer is (A).

More Practice: Upper Level Only

17.A

If bº = 60º, the other angle must be 30 since 180º — 90º — 60º = 30º, which is a special right triangle. In a 30-60-90 triangle, the sides measure x, , and 2x, respectively. The side across from the 30º angle is x, which we know is 4. Therefore, v, the hypotenuse, is twice that value. 4(2) = 8, so v = 8. The correct answer is (A).

18.A

Translate this question very carefully into math. What translates to the unknown x, is to =, of to multiplication, and difference to subtraction. There are 360 degrees in a rectangle and 180 degrees in a triangle. Therefore, the equation should read . Remember PEMDAS and work inside the parentheses first. , so x = 45. The correct answer is (A).

19.B

The perimeter of a square is = 4s and the area of a square is = s². Since one-half the perimeter is equal to the area, (4s) = s². Now solve for s. 2s = s². Divide both sides by s to get s = 2. The correct answer is (B).

20.D

Use your formulas. Area of a circle = πr² and Circumference of a circle = 2πr = πd. So the area of a circle with radius 3 is π(3²) = 9π. That means the circumference of the other circle is 9π. Since 9π = πd, the diameter is equal to 9. The correct answer is (D).

21.A

Use the formula given to find the volume of the first cylinder: V = πr²h = π(3)²(4) = 36π. If the other cylinder is equal in volume to the first cylinder, then plug in the volume and radius of the other cylinder to find its height: 36π = π(6)²h simplifies to 36π = π(36)h. Divide both sides by 36 to get 1 = h. The correct answer is (A).

22.D

Use your formula. Since the perimeter of a square is 4 times the length of one side, you can divide the perimeter by 4 to get the length of one side. The correct answer is (D).

23.D

The length of AB is the same as all the different heights added together on the right-hand side of the figure. Therefore, the perimeter will contain two lengths of 12. Similarly, the length of AC is the same as all the different lengths added together that are across the figure (in this case, above AC), so there will be two lengths of 20. To find the perimeter, add all the sides: P = 12 + 12 + 20 + 20 = 64. The correct answer is (D).

24.B

Notice the three triangles that have been created within the rectangle. Look at the two right angles that surround the larger triangle in the middle. For the right triangle on the left, the value of x is given (70); therefore, to find w, simply subtract 90 and 70 from 180 to get w = 20. For the right triangle on the right, y and z must add up to 90 since a triangle has 180º. Finally, plug in these values into the equation given: yº + zº — wº = 90 — 20 = 70. The correct answer is (B).

25.D

Notice that the part that juts out on the left side of the shape would fit into the indented part on the right side of the shape. Filling in the hole would make a rectangle with a length of 8 and a width of 4 + 3 + 4 = 11. To find the area of a rectangle, use the formula: A = l × w. Therefore, A = 8 × 11 = 88. The correct answer is (D).

26.B

The length of fencing needed to surround a yard is the same as the perimeter. Draw a rectangle and label the length as 28 and the width as 32. Remember, in a rectangle opposite sides are equal to each other. Calculate the perimeter by adding all the sides: 28 + 28 + 32 + 32 = 120. The correct answer is (B).

27.B

The slope of a line is found using the formula or . Therefore, the slope of line segment AB is . To find the slope of the line perpendicular to this line segment, take the negative reciprocal. The reciprocal of is . Change the sign to get —. The correct answer is (B).

28.B

Even though the length of the radius is unknown, it is still possible to find the angle measurements. There is a 90º angle in the center of the circle, and both OQ and OP are radii of the circle, which means they are the same length. Therefore, this is an isosceles right triangle, meaning the two smaller angles are congruent. All triangles have 180º, so 180º — 90º = 90º. The two smaller angles add up to 90º, so . The correct answer is (B).

29.D

Notice that the four intersecting lines form a quadrilateral. All quadrilaterals contain 360º, so keep a tally of the vertices and find the missing angle x. Since 85 is already provided, 360º — 85º = 275º. All straight lines add up to 180º, so use the exterior angles to find the interior angles. For example, if one of the exterior angles is 63º, the supplementary angle must be 117º. Subtract this from 275º to find that 275º — 117º = 158º. The other exterior angle, 112º, is opposite the interior vertex. Since opposite angles are equal, the interior vertex must also be 112º. Subtract this from the current total to find that 158º — 112º = 46º. The missing angle x is 46º. The correct answer is (D).

30.C

Since ABC is an equilateral triangle, all 3 sides are equal to 6. Label AC as 6 and BC as 6. The question asks for the perimeter of the figure. There are two sides of the triangle that are part of the figure’s perimeter, so add them together: AB + AC = 6 + 6 = 12. Eliminate (A) and (B) since the answer must have a 12 in it. Now find the rounded portion, which is half of the circumference (i.e., a semicircle). Since BC is 6, note that the diameter of the semicircle is also 6. If C = πd, then half of the circumference is πd. Plug in the value for the diameter and simplify: π(6) = 3π. The full expression for the perimeter will then read 12 + 3π. The correct answer is (C).

31.D

Break the trapezoid into two triangles and a square. Next, figure out the missing segment lengths. If MN = 8 and MNQP is a square, then NP, QP, and MQ also equal 8. Next find LQ and PO. You may notice that these are 6-8-10 right triangles. Otherwise, use the Pythagorean Theorem to find the base of the triangles: a² + 8² = 10². Solve for a: a² + 64 = 100. Subtract 64 from both sides to get a² + 36. Then take the square root of both sides, and a = 6. To find the areas of the triangles, plug the base and the height into the formula A = bh = (6)(8) = 24. There are two triangles, so 24 + 24 = 48. To find the area of square MNQP, plug the side length into the formula A = s² = 8² = 64. Finally, add the areas: 2 triangles + 1 square = 48 + 64 = 112. The correct answer is (D).

Practice Drill 3—Ratios

1.D

When the question asks about ratios, make a Ratio Box. Place the given information into the Ratio Box:

Add the three ratios to find the ratio total, which is 9. Next determine what times 9 equals 45, which is 5. Finally, multiply 5 by 5 to find the actual number of double scoops, which is 25. The correct answer is (D).

2.C

If 14 of the 26 students have brown eyes, then 12 students have another eye color. Thus, the ratio of students with brown eyes to students with another eye color is 14 to 12, which, when reduced, is 7 to 6. The correct answer is (C).

3.B

When the question asks about ratios, make a Ratio Box. Place the given information into a Ratio Box:

Add the two ratios to find the ratio total, which is 5. Next determine what times 5 equals 25, which is 5. Finally, multiply 2 by 5 to find the actual number of miles he swam, which is 10. The correct answer is (B).

Practice Drill 4—Plugging In

1.B

This is a Plugging In question because there are variables in the choices and the question stem contains the phrase in terms of. Plug in a value, work through the problem to find a target answer, and then check each of the choices to see which yields the target answer. For instance, plug in x = $3. The question asks for the total amount of money donated, so 3 × 200 = 600. $600 is the target answer. Now, plug 3 into the choices for x to see which choice matches your target answer (600). Eliminate (A) because is way too small. Choice (B) works because 200(3) = 600. Eliminate (C) because is too small. Eliminate (D) because 200 + 3 or 203 ≠ 600. The correct answer is (B).

2.D

This is a Plugging In question because there are variables in the choices. Plug in a value, work through the problem to find a target answer, and then check each of the choices to see which yields the target answer. For instance, plug in 6 for d dollars. If 10 magazines cost $6, then $3 would buy 5 magazines—you spend half as much money, so you can get only half as many magazines. So 5 is the target answer. Now, plug 6 into the choices to see which answer yields 5, the target answer. Eliminate (A) because does not equal 5. Eliminate (B) because 30(6) is way too large. Choice (C) is a fraction, , so it will not equal 5. Choice (D) works, as , so keep this choice. The correct answer is (D).

3.C

This is a Plugging In question because there are variables in the choices and the question stem contains the phrase in terms of. Plug in a value, work through the problem to find a target answer, and then check each of the choices to see which yields the target answer. The zoo has four times as many monkeys as lions, so, for instance, plug in 40 for the monkeys, which translates to 4 × lions = 40, so there are 10 lions. There are four more lions than zebras, which means that 10 — 4 = 6 zebras, so z = 6. The question asks how many monkeys are there in the zoo, so the target answer is 40. Now, plug 6 into the choices for z to see which choice matches your target answer (40). Eliminate (A) because 4 × 6 = 24 is not equal to 40. Eliminate (B) because 6 + 4 = 10 is still too small. Since 4(6) + 16 = 40, keep (C). Remember to try all four choices when plugging in, so check (D) as well. 4(6) + 4 = 28, which is too small, so eliminate (D). The correct answer is (C).

More Practice: Lower Level

4.B

This is a Plugging In question because there are variables in the choices and the question stem contains the phrase in terms of. Plug in a value, work through the problem to find a target answer, and then check each of the choices to see which yields the target answer. For instance, say that y = 10, the price of one pair of earrings. The total amount of money for 6 pairs of earrings would be 6 × 10 = 60, which is the target answer. Now, plug 10 into the choices for y to see which choice matches your target answer (60). Eliminate (A) because 6 + 10 = 16, which does not equal 60. Keep (B) because 6(10) = 60. Remember to check the remaining choices when plugging in. 6¹⁰ is a very large number, much greater than 60, so eliminate (C). Eliminate (D) as well because 6 + 6(10) = 66, which does not equal 60. The correct answer is (B).

5.B

This is a Plugging In question because there are variables in the choices. Plug in a value, work through the problem to find a target answer, and then check each of the choices to see which yields the target answer. For instance, say that p pieces of candy is equal to 5 pieces, and c cents is 10 cents. Therefore, 10 pieces of candy will cost 20 cents—you have twice as many pieces, so it will cost twice as much money. So, the target answer is 20. Now, plug in your values for p and c into the choices to find the choice that equals your target answer (20). Eliminate (A) because , which is too small. , so keep (B). Remember to check the remaining choices when plugging in. , so eliminate (C) as well. Cross off (D) because 10(5)(10) = 500, which is way too large. The correct answer is (B).

More Practice: Middle Level

6.C

In this question, J is an odd integer, so plug in an odd integer for J. Since this is a must be question, see if there is a number that would make the answer untrue. Plug in 1 for J to make (A) untrue, since is not greater than 1. This number for J will also eliminate (B) since 1 — 2 = −1, which is not a positive integer. Choice (C) is true since 2 × 1 = 2, which is an even integer. Eliminate (D) since J could be negative. For example, if J = −3, −3 is not greater than 0. Check that value for (C) to be sure it always works. Again, if J = −3, then 2 × −3 = −6, which is still an even integer. Since it always works, the correct answer is (C).

7.C

When there are percents or fractions without a starting or ending value in the question stem, feel free to use Plugging In. What number would make the math easy? 8 is a common denominator for and , so draw a picture of a fruit tart and divide it into 8 equal parts. Shade in the number of pieces she has eaten. On Monday, she ate of the pie, so of 8 is or 4 slices, leaving 4 slices for later. The next day, she ate of what was left. of 4 slices is 1, so she ate 1 slice. There are now 3 out of 8 slices left. Beware of choosing (A), however! The question asks how much she ate, so add up the slices she consumed. There should be 5 slices shaded (4 + 1 = 5), so the correct answer is (C).

More Practice: Middle and Upper Levels

8.C

When there are percents or fractions without a starting or ending value in the question stem, feel free to use Plugging In. For instance, plug in $100 for the starting price of the suit. It is reduced by 20%, so 20% of $100 is equal to . That is the amount the suit is reduced. Subtract that from $100 to find the resulting price: $100 — $20 = $80. The suit is then reduced by 10%, so 10% of $80 is . Subtract this from $80 to find the final price of the suit: $80 — $8 = $72. The final price is $72. The final price is what percent off of the original is another way of asking the final price is what percent less than the original. So, use the percent change formula: . The difference is $100 — $72 = 28. The original price was $100. Therefore, . The correct answer is (C).

9.C

Try plugging in values that satisfy the question stem, and eliminate choices. It may be necessary to plug in twice on must be true or always true questions. If m is an even number, let m = 2, and let n = 3 since it must be an odd integer. If p is the product of m and n, then p = (2)(3) = 6. Now check the choices. Eliminate (A) because p is not a fraction. Eliminate (B) as well since p is not an odd integer. Keep (C) because 6 is divisible by 2. Finally, keep (D) because 6 is greater than zero. Plug in again to compare the remaining choices. Perhaps keep one number the same, so n = 3, but make m = −2 instead of 2. Now p = (−2)(3) = −6. Choice (C) still works since −6 is divisible by 2, but (D) no longer works since p is less than zero. Since it is always true, the correct answer is (C).

More Practice: Upper Level

10.B

Since there are variables in the choices, plug in a value for p, paying attention to the restrictions in the question. If p is an odd integer, make sure to plug in an odd integer, for instance p = 3. Now, test the choices to see which one can be eliminated. Cross off (A) because (3)² + 3 = 9 + 3 = 12, which is not odd. Choice (B) works since 2(3) + 1 = 6 + 1 = 7, which is odd. Choice (C) works since . Choice (D) does not work since 3 — 3 = 0. Remember 0 is even, not odd. Plug in a second time for the remaining choices. Try p = 5. Choice (B) still works because 2(5) + 1 = 10 + 1 = 11, but eliminate (C) because is no longer an integer. The correct answer is (B).

11.B

The wording on this problem is tricky: it asks for which CANNOT be true, so try to find examples that COULD be true to eliminate choices. Pay attention to the restrictions in the problem, and plug in two positive even integers: say 4 and 6. Thus, 4 + 6 = 10 = m. Next, eliminate choices that WORK. Choice (A) does not work since 10 is greater than 5. Keep it. Choice (B) does not work because 3(10) = 30, which is even, not odd. Keep it. Eliminate (C) because m = 10, which is even, so it works. Eliminate (D) as well because 10³ ends in a zero, which is also even, so this statement works. Now, plug in a second time for the remaining choices. Try new numbers, and remember that the numbers do not have to be distinct from one another. Try plugging in 2 for both positive even integers. Thus, 2 + 2 = 4 = m. Check the remaining answers and eliminate the choices that WORK. For (A), 4 is less than 5. That works, so eliminate (A). For (B), 3(4) = 12, which does not work since it’s even, so keep it. The only choice left is (B), which is the correct answer.

12.D

This is a Plugging In question because there are variables in the choices. Plug in a value, work through the problem to find a target answer, and then check each of the choices to see which yields the target answer. For instance, plug in 20 for Antonia. Since Antonia has twice as many baseball cards as Krissi, Krissi has the number of cards that Antonia has. Therefore, Krissi must have 10 cards, and k = 10. Krissi has one-third as many baseball cards as Ian, so Ian has 3 times as many as Krissi has: 10 × 3 = 30, or 30 cards. Together, Antonia and Ian have 20 + 30 = 50, so 50 is the target answer. Now, plug in 10 for k to find which choice yields 50, your target answer. Eliminate (A) because , not 50. Eliminate (B) because is still too small. Choice (C) is still too small, as . Choice (D) works because . The correct answer is (D).

13.D

This is a Plugging In question because there are variables in the choices. Plug in a value, work through the problem to find a target answer, and then check each of the choices to see which yields the target answer. Let b = 4 and a = 3. Finding the product means multiply, so . The target answer is 18. Now, plug in your values for b and a into the choices to find the choice that equals your target answer (18). Eliminate (A) since (3 × 4)² = (12)² = 144, which is too big. Eliminate (B) since and is not equal to 18. Also eliminate (C) since , which does not equal 18. Choice (D) works: . Keep it. The correct answer is (D).

14.D

Use MADSPM to simplify the exponents in the equations first. When raising a power to a power, multiply the exponents together. For the first equation, , so a = 9. When dividing by the same base, subtract the exponents. For the second equation, , so b = 8. The question asks to find a × b, so 9 × 8 = 72. The correct answer is (D).

15.B

Since the question involves averages, use the Average Pie. However, save yourself some time by reading carefully! Notice that the classes have an equal number of students donating money. Because of this, simply plug in values for the averages since it doesn’t matter how many actual students are donating money from each class. You only need values for the average of each class, so start there. Those two averages will become the numbers for your total in the next part of the problem. For instance, plug in $3 as the average, y, for Mr. Greenwood’s class, and $5 for z, the average for Ms. Norris’s class. Add these two numbers to find the total amount of money donated (3 + 5 = 8), and put 8 in the total spot. There are 2 classes donating money, so the # of items is 2. Find the average by dividing: . The target answer is 4. Now, plug in 3 for y and 5 for z to find which choice yields 4, the target answer. Eliminate (A) because ≠ 4. Choice (B) works because . Eliminate (C) because 3 + 5 ≠ 4. Finally, eliminate (D) because 2(3 + 5) = 2(8) = 16, which is way too large. The correct answer is (B).

16.C

First, simplify the first expression: (3xy)³ = 3³x³y³ = 27x³y³. While comparing it to the other expression, 3x²y⁵, you can work with one aspect of the expression at a time. Start with the coefficients: the greatest common factor of 3 and 27 is 3. Eliminate (A) and (D) since neither contains 3. Both of the remaining answers contain x², so compare y in the two expressions. One has y³ = y × y × y and the other has y⁵ = y × y × y × y × y. The greatest common factor is y³ since both expressions have at least 3 ys. Eliminate (B). The correct answer is (C).

Practice Drill 5—Plugging In the Answers

1.B

The question is asking for a specific value and there are real numbers in the choices, so use PITA to solve. Tina can read 60 pages per hour, which is 60 pages in 60 minutes, and Nick can read 45 pages in 60 minutes. Combined, they can read 105 pages (60 + 45) in 60 minutes. Now, start with one of the middle choices to see which answer will yield a total of 210 pages. Try (B): if they read for 120 minutes, they will read double the amount they did in 60 minutes. That will make the math easy. In 60 minutes they read 105 pages, so 105 × 2 = 210. This satisfies the question, so (B) is correct.

2.B

The question is asking for a specific value and there are real numbers in the choices, so use PITA to solve, starting with (C). The choices represent how much Emiko pays. If Emiko pays $30 and she pays twice as much as Diego, then Diego would have paid $15 since . Abigail paid three times as much as Emiko, so he would have paid 3 × 30 = 90. This added together is more than $90, so eliminate (C) and (D), as these will amount to a total that is too much. Try (B): if Emiko paid $20, Diego would have paid $10 since . Abigail paid 3 × 20 = 60. Add these amounts together to find that $20 + $10 + $60 = $90, which satisfies the question. The correct answer is (B).

3.D

First, translate the English into math and then use PITA to test the choices. Four less than a certain number translates to n — 4, and two-thirds of a number translates to . So the equation is . Now, use PITA to find the one that satisfies the equation, starting with (C). If n = 8, then the equation will read . Since , eliminate (C) and try another choice. Try (D): if n = 12, then , which is or 8 = 8. Since 12 works, stop here. The correct answer is (D).

More Practice: Lower Level

4.D

The question is asking for a specific value and there are real numbers in the choices, so use PITA to solve. The question asks how many cat-lovers are in the club, so label the choices “cat-lovers” and create another label next to it marked “dog-lovers.” Test the choices, starting with (C). If there are 20 cat-lovers in the club, then subtract 12 to find the number of dog-lovers: 20 — 12 = 8. 20 + 8 = 28, which means that (C) is too small. Eliminate (A), (B), and (C). Choice (D) must be the correct answer, so be aggressive if you’re worried about the time. If you have time to check, you will see that it works because if there are 21 cat-lovers, then 21 — 12 = 9, so there are 9 dog-lovers. 21 + 9 = 30, which matches the total given in the problem. The correct answer is (D).

5.B

The question is asking for a specific value and there are real numbers in the choices, so use PITA to solve. The question asks for how much Jonathan paid, so label the choices as such and create two additional columns next to it, one labeled as Victoria and the other as Russell. Now, test the choices and follow instructions from the question stem, starting with (C). Choice (C) means that Jonathan paid $100. The question states that Victoria paid twice as much as Jonathan, so Victoria paid 100 × 2 = 200. The question then states that Victoria paid half as much as Russell, so Russell paid twice as much as Victoria: 200 × 2 = 400. Find the total: $100 + $200 + $400 = $700, which is too much, so eliminate (C) and (D). Try (B): if Jonathan paid $80, then Victoria paid 80 × 2 = 160, and Russell paid 160 × 2 = 320. Find the total: $80 + $160 + $320 = $560, which is the amount stated in the question. The correct answer is (B).

More Practice: Middle and Upper Levels

6.D

The question is asking for a specific value and there are real numbers in the choices, so use PITA to solve. The question asks for Brinda’s age, so label the choices “B” and create 2 additional columns next to it, one labeled as A and the other as C. Now, follow the steps of the question and test the choices, starting with (C). If Brinda is 18, then Anshuman must be 9 because Anshuman is half as old as Brinda. It is also stated that [Anshuman] is three times as old as Cindy (remember, Anshuman is the subject of the sentence), so Cindy must be 3 since . Find the total of their ages: 18 + 9 + 3 = 30, which is too small. Eliminate (A), (B), and (C). Choice (D) is the only answer left, so it must be correct. Stop work and move on! If you have time later to check, you’ll see that if Brinda is 24, then Anshuman must be 12 and Cindy is 4, which makes the total 24 + 12 + 4 = 40. The correct answer is (D).

7.D

The question is asking for a specific value and there are real numbers in the choices, so use PITA to solve. The question asks for a possible value of x. So, plug in for x and see if there is an integer that would work to make the rest of the equation balance. Start with one of the middle choices. Try (C): if x = 55, then 70(55) = 3,850. Solve the rest of the equation: 3,850 + 33y = 4,233 to see if y is an integer. Subtract 3,850 from both sides to find that 33y = 383. You can try dividing, but you could also guesstimate: 33 × 11 = 363 and 33 × 12 = 396, so 383 is not divisible by 33. Therefore, eliminate (C) and try another choice. Try (D): 70(60) + 33y = 4,233 simplifies to 4,200 + 33y = 4,233. Subtract 4,200 from both sides to find that 33y = 33. Divide each side by 33 to find that y = 1. Since 1 is an integer, this satisfies the question. The correct answer is (D).

8.A

This question gives a fair amount of information in the question, that the smallest of the three integers is 2, the sum of 2 + x + y = 9, and the product of 2xy = 24. Simplify these equations first and then use PITA to solve. Subtract 2 from either side of the sum to find that x + y = 7, and divide by 2 in the second equation to find that xy = 12. Now Plug In, starting with the largest number to find the largest number as efficiently as possible. Eliminate (C) and (D) right away since 8 or 9 added to another positive integer cannot equal 7. Try (B): plug in 6 to get 6 + y = 7, so y would have to equal 1. This cannot be true, however, since the smallest integer has to be 2. Eliminate (B). Choice (A) is the only answer left, so it must be correct. Stop work and move on! If you have time later to check, you’ll see that if you plug in 4 that 4 + y = 7 means that y = 3. These are both larger than 2, the smallest number. Do these numbers work in the second equation? Yes! 3 × 4 = 12. The correct answer is (A).

9.C

Be sure to label the choices very carefully to stay visually organized. The question asks for Lori’s age now, so label the choices “L,” and create another column to the right and label it “C” for Carol’s age. Next, create two more columns and label them as “L + 10” and “C + 10” for their respective ages in 10 years. Now, plug in, starting with (C): if Lori is 20 years now, Carol must be 5 because Lori is 15 years older than Carol. This means that in 10 years, Lori will be 30 (20 + 10 = 30), and Carol will be 15 (5 + 10 = 15). The question states that in 10 years, Lori will be twice as old as Carol. 30 = 2 × 15, so this satisfies the statement. The correct answer is (C).

10.A

Let the choices help here. The choices represent how many people are currently in the group. Eliminate (D) immediately since 30 cannot be divided 12 ways (hopefully there aren’t partial people in the group!). Try (B), which is the middle answer of the remaining choices: if there are 6 people buying the game now, the cost would be $5 each since . If a seventh person joined, the cost per person would be , which is not an integer. Since the problem stated that adding 1 person to the group would result in each person owing $1 less, this does not satisfy the question, so eliminate (B) and try another choice. Try (A): if there are 5 people buying the game, they will each pay $6 since . If a sixth person joins, they will each pay $5 because . Since $5 is exactly one dollar less than $6, this satisfies the question stem. Choice (A) is correct.

Extra Practice

1.B

Guesstimating is one way to work through this problem. The shaded region looks to be about half the square, and half of 144 is 72, (B). To be more precise, the side of the square must be 12 since A = s² and 12² = 144, so the height of each shaded triangle is 12. The base for one triangle ends at P and the base for the other triangle ends at Q, and PQ = 12 since it’s a side of the square. So make the base for each triangle 6 since the two bases must add up to 12. Plug those values into the formula: A = bh = (6)(12) = 36. Since both areas equal 36, the total area of the shaded region is 36 + 36 = 72. The correct answer is (B).

2.B

Break this question down one piece of information at a time. The question asks for the shaded region, so you want the part inside the square but outside the circle. In other words, if you find the area of the square and the area of the circle, you can find the shaded region by removing what you do not need (the area of the circle). First, find the area of the square. The side of the square is equal to 4, so A = s² = 4² = 16. Eliminate (A), (C), and (D), since these do not contain 16. For added security, find the area of the circle. The radius is 2, so A = πr² = π(2)² = 4π. Remember to subtract that from the area of the square, so the full answer is 16 — 4π. The correct answer is (B).

Practice Drill 6—Functions

1.D

In this equation, 5 will replace b. Thus, 2(5) + 7 = 10 + 7 = 17. The correct answer is (D).

2.B

Since the function is equal to 20, it can be rewritten as 20 = 4n — 4. Solve for n. Add 4 to both sides of the equation to get 24 = 4n. Divide both sides by 4 to get n = 6. The correct answer is (B).

3.D

The value in front of Δ is a, and the value after Δ is b. Therefore, 3Δb can be rewritten as 4(3) + 3b or 12 + 3b. The correct answer is (D).

4.D

Use PITA to evaluate the expression and then determine if the resulting 3-digit number is divisible by 3. The numbers 328, 338, and 358 are not divisible by 3. The number 378 is divisible by 3. The correct answer is (D).

5.C

In this equation, the numerical value inside the will replace c where it appears in the equation. Thus, = 2(10) + 10(10+3) = 20 + 10(13) = 150. Next find , which is 2(3) + 3(3 + 3) = 6 + 3(6) = 24. Finally, find — , which is 150 — 24 = 126. The correct answer is (C). Note: — is not equivalent to , which is (B).

Practice Drill 7—Quant Comp

1.C

Look at column B first. 17 × 2 + 17 is the same as 17 + 17 + 17, or 17 × 3. Thus, the two columns are equal. The correct answer is (C).

2.A

Draw a picture. For instance, draw a pie and divide it into eight parts since eight is a common denominator for and . Since , four of the eight parts of the pie would be colored in. Only three out of eight would be filled in for . Therefore, column A is greater, and the correct answer is (A).

3.B

There are variables in the columns, so plug in twice. For instance, try b = 10. 10 + 80 = 90, and 10 + 82 = 92. In this case, column B is greater, so eliminate (A) and (C). Try another number, perhaps a negative number: —10. Perform the necessary calculations: —10 + 80 = 70 and —10 + 82 = 72. Column B is still greater, so the correct answer is (B).

4.B

Plug in a value here. Say that Maitri is 60 inches tall, making Robbie 58 inches tall since Robbie is two inches shorter than Maitri. The question stem also states that Jasper is four inches taller than Maitri, so Jasper must be 64 inches tall. This makes Jasper taller than Robbie, so column B is greater. The correct answer is (B).

5.C

When dealing with exponents, write it out! Column A can be rewritten as 16 × 16 × 16. Column B can be rewritten as 4 × 4 × 4 × 4 × 4 × 4. Notice that 16 is the same as 4 × 4. Therefore, column A can also be written as (4 × 4) × (4 × 4) × (4 × 4). Since each column contains 6 fours, the two columns are equal. The correct answer is (C).

6.D

The information given does not indicate the direction or orientation of either girl’s house. However, the information does state that Karina lives two miles from school, so column A is 2. However, Jennifer could live another two miles past Karina’s house, four miles in the other direction from the school (making the two houses 6 miles apart), or she could even live 4 miles north or south of the school (making the distance between their houses yet another value). Since there is no way to determine the distance between their houses without more information, the solution cannot be determined. The correct answer is (D).

Practice Drill 8—Quant Comp

1.B

Since there are variables in the columns, plug in a number. Pay attention to the restriction given: plug in a number greater than 1 for x. Let x = 4. Column A is equal to 4, and column B is equal to 4², or 16. Since column B is greater, eliminate (A) and (C). Try a different number to see if column A could be greater or if the quantities could be equal. Since x > 1, x cannot be negative, zero, or one. Try a very large number. 1,000² is much larger than 1,000, so column B is still greater. You could also try a decimal, like 2.5. In this case, column B is still greater since 2.5² = 6.25, which is greater than 2.5. Therefore, since column B is always greater, the correct answer is (B).

2.C

Read the question carefully: b is an integer and —1 < b < 1. There is only one integer between —1 and 1. Therefore, b must be 0. Plug 0 in for b into each of the columns. Column A is . Column B is . The quantities are equal, so (C) is the correct answer.

3.D

Since there are variables in the columns, plug in values for p and m. For instance, let p = 16 and m = 3. Since it takes 4 quarts to make one gallon, column B is less than 1 gallon while column A is 16 gallons. This makes column A greater. However, the question does not state anything about requirements for these numbers, and the values could easily be reversed, that p = 3 and m = 16. The 16 quarts in column B is equal to 4 gallons, which is greater than the 3 gallons in column A. Since this could be true as well, it cannot be determined which quantity is larger. The correct answer is (D).

4.A

Since there are variables in the columns, plug in a number. Pay attention to the restriction given: if x must be a positive integer, plug in a positive integer for x. For example, let x = 3. Column A is while column B is . If you’re not sure which value is greater, draw a picture. You can also use Bowtie to compare fractions. Column A becomes , and column B becomes . Thus, column A is greater. Eliminate (B) and (C). Try plugging in another value for x to see if another outcome is possible. Remember the restriction given, so x cannot be negative or zero, so try a large integer. Make x = 100. Column A is , and column B is . Column A is still greater. You could also try x = 1, but you will get the same result. Column A will be greater since is greater than . The correct answer is (A).

5.D

Since there are variables in the columns, plug in values for p and w, according to the information given: w is an integer less than 4, so let w = 3. You are also given that p is an integer greater than 10, so let p = 11. Therefore, column A is (3)(11) = 33, while column B is equal to 3. In this case, column A is greater. Eliminate (B) and (C). Now, try plugging in different numbers to see if another outcome is possible. Let w = 0 and p = 12. In column A, (12)(0) = 0. This is equal to column B since w = 0. Since column A isn’t always greater nor are the two columns always equal, the correct answer is (D).

6.D

Since there are variables in the columns, plug in a value for c. Let c = 2. In column A, 4(2) + 6 = 8 + 6 = 14. Do the same for column B: 3(2) + 12 = 6 + 12 = 18. In this case, column B is greater, so eliminate (A) and (C). Now, try a different number, perhaps a negative number. Let c = —10. Now, column A will read 4(—10) + 6 = —40 + 6 = —34. Do the same to column B: 3(—10) + 12 = —30 + 12 = —18. In this case, —18 > —34, so column A is now greater. Since neither column is always greater, the correct answer is (D).

Practice Drill 9—Quant Comp

1.C

Find the total cost in each of the columns. Column A contains the statement The total cost of 3 plants that cost $4 each, so 3($4) = $12. Column B contains the statement The total cost of 4 plants that cost $3 each, so 4($3) = $12. The columns are equal, so the correct answer is (C).

2.D

First, simplify the expression in column A. Distribute the 30 in the expression 30(1 — 2n) to get 30 — 60n. Now, plug in a value for n to solve each column. Let n = 2. In column A, 30 — 60(2) = —90. In column B, 30 — 2(2) = 26. Column B is greater, so eliminate (A) and (C). Now, plug in a second time, trying a different number (remember, try 1, 0, fractions/decimals, negatives, or large or small numbers to find different outcomes). Try a negative number here. Let n = —3. Column A will now read 30 — 60(—3) = 210, and column B will read 30 — 2(—3) = 36. Column A is greater in this case. Since neither column is always greater, the correct answer is (D).

3.D

You are given the statement The product of 3 integers is 48. There are many ways to reach a product of 48. For instance, 2 × 3 × 8 = 48. Of the three integers, the smallest is 2, so column A is 2. Compared to column B, column A is greater. Eliminate (B) and (C). However, this is not the only way to multiply integers to get a product of 48. For example, 1 × 2 × 24 = 48. In this case, the smallest of the three integers is 1, which means the two columns are equal. Since column A is not always greater nor are the two columns always equal, the correct answer is (D).

4.C

First, simplify column A: (x + y)(x — y) can be FOILed out to be x² + xy — xy — y². The two middle values cancel each other out, so the expression reads x² — y², which is the same as the expression in column B. Since the two columns are equal, the correct answer is (C). Note that you can also plug in values for x and y and solve the problem this way. You should try more than one set of numbers to check for other possible outcomes.

5.A

Use correct PEMDAS to evaluate the expression in column A. First, work within the parentheses: (7 − 4) × 3 − 3 = (3) × 3 − 3. There are no exponents, so the next step is to do the multiplication and division from left to right: (3) × 3 − 3 = 9 − 3. Finally, add and subtract from left to right: 9 — 3 = 6. Since 6 is greater than 0, column A is greater. The correct answer is (A).

6.A

Since line m is equal to y = x + 4, the slope of line m is equal to 1. Remember, the slope is the coefficient of x in linear equations. Therefore, column A is 1. Perpendicular lines will have slopes that are negative reciprocals of one another. Therefore, line l, which is perpendicular to line m, will have a slope of —1 since the negative reciprocal of is . Thus, column B is —1. Since 1 is greater than —1, column A is greater. The correct answer is (A).

7.B

Work through the information provided to find the value of column A. If the shoes are $100 and the price is increased by 20%, find 20% of $100 and add that result to the total. reduces to . The price increased $20, so $100 + $20 = $120. The shoes are now $120. However, the price was reduced by 20%. Find 20% of 120 and subtract that result from the total. reduces to . The price decreased $24, so $120 — $24 = $96. The final price of the shoes is $96, so column A is $96. Since column B is $100, it is greater. The correct answer is (B).

8.B

Remember, a negative sign will stay negative with an odd exponent. Both columns contain negative numbers and odd exponents, so both columns will remain negative. With negative numbers, the value that is closer to zero will be the greater value (e.g., −1 > −4). When working with fractions, remember that as the denominator gets larger, the fraction will get smaller (e.g., ). However, with negative fractions, the one that is “less negative” will be greater (e.g., ). In column B, the exponent is greater, so the denominator in column B will be larger and thus the value of the fraction will be smaller. Since the fraction in column B will be less negative than the fraction in column A, the value in column B is greater. The correct answer is (B).

9.A

Both fractions are positive and both are being raised to a positive, even power. However, be careful when working with fractions less than 1. When those numbers are raised to a positive power, they become smaller since the denominator increases. Remember, as the denominator gets larger, the fraction will get smaller (e.g., ). Thus, the denominator in column A (6⁴) will be smaller than the denominator in column B (6⁶), which means the value in column A will be greater. The correct answer is (A).

10.A

The negative sign in both columns will become positive since both columns are being raised to an even power. With fractions less than 1, the larger the denominator, the smaller the fraction. Since the denominator in column A (6²) will be smaller than the denominator in column B (6⁴), the value in column A will be greater. The correct answer is (A).

11.A

With fractions less than 1, the larger the denominator, the smaller the fraction. Since the denominator in column A (6³) will be smaller than the denominator in column B (6⁵), the value in column A will be greater. The correct answer is (A).

12.D

There are no instructions as to the values of x and y, so use Plugging In. For instance, x and y could be equal. Let both x and y equal 4. Column A would be C = 2(4)π, or 8π, and column B would be A = π(4)² = 16π. In this case, column B is greater, so eliminate (A) and (C). Since x and y do not have to be equal, plug in a second time to see if a different outcome is possible. Say that x = 2 and y = 1. Then column A will be C = 2(2)π = 4π, and column B will be A = π(1)² = π. Now, column A is greater. Since neither column is always greater, the correct answer is (D).

13.C

List the prime numbers on the 6-sided die: 2, 3, and 5 (Note! 1 is not a prime number). Since there are 6 sides on the die, the probability of rolling a prime number is , so column A is . This is the same value listed in column B. Since the columns are equal, the correct answer is (C).

14.D

Since there are variables in the columns, plug in values for a and b. If a and b are integers and a + b = 5, choose numbers to make both statements true. Let a = 3 and b = 2. In this case, column A is greater, so eliminate (B) and (C). However, these numbers could easily be reversed because there are no restrictions listed. If a = 2 and b = 3, then column B would be greater. Since neither column is always greater, the correct answer is (D).

15.A

Evaluate the expressions, using order of operations. Column A is , and column B is . Therefore, column A is greater. The correct answer is (A).

16.C

Start with Set B, since it is a finite set. Set B consists of 5, 10, 15, 20, 25, 30, 35, 40, and 45. Set A contains all prime numbers. The only prime number contained in Set B is 5, so the intersection of these two sets (i.e., Set C) will contain only the number 5. There is only one number in Set C, so the columns are equal. The correct answer is (C).

17.B

When multiplying fractions, multiply the numerators across and the denominators across. Therefore, the value of column A is . When adding fractions with a common denominator, add the numerators. Thus, the value of column B is . Column B is greater. The correct answer is (B).

18.C

Since there are variables in the columns, plug in values for a and b, paying attention to restrictions given. Since a > 0, let a = 2. Since b < 0, let b = —3. Now, plug in these numbers to the expressions. Column A is −(2 × −3) = −(−6) = 6. Column B is −2 × −3 = 6 as well. The two columns are equal. You can try testing different values for a and b, but the columns will always be equal because the negative signs will always cancel out and the same numbers are being multiplied in each column, so the result will not vary. The correct answer is (C).

19.B

This question is testing math vocabulary. The median of a set is the middle number when the numbers are listed in order from least to greatest. In Set A, the numbers are already in order, so find the middle number: 8. Thus, column A is 8. In Set B, the numbers are also already in order. However, there is an even number of items in this set, so the median will be found by taking the average of the two middle numbers. The two middle numbers are 8 and 9, so the median is . The value in column B is 8.5. Since column B is greater, the correct answer is (B).

20.C

Inclusive means to include the outer limits—in this case, 1 and 10. In column A, add up all the integers, including 1 and 10: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55. Simplify column B to get 5 × 11 = 55. Since the columns are equal, the correct answer is (C).

21.B

Remember, with exponents, write it out! Thus, column A can be rewritten as (2 × 2 × 2) + (2 × 2 × 2) + (2 × 2 × 2) = 8 + 8 + 8 = 24. Note that in column A, the quantities are being added, not multiplied. Column B can be rewritten as 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2, which will equal a value larger than 24. Therefore, column B is greater. Another way to solve this problem is to use exponent rules. When exponents with the same base and same exponent are being added, simply treat them like any variable. For example, x + x + x = 3x. Thus, 2³ + 2³ + 2³ = 3(2³) = 3(8) = 24. This is not the same as 2⁹, which is a much larger number. Since column B is greater, the correct answer is (B).

22.D

Since there are variables in the columns, plug in. First, simplify column A: 7(x — 3) = 7x — 21. Now, plug in a value for x. For instance, let x = 3. Plug in 3 to column A to find that 7(3) — 21 = 21 — 21 = 0. Now, do the same for column B: 21 — 7(3) = 21 — 21 = 0. The columns are equal, so eliminate (A) and (B). Remember, plug in a second time to see if a different outcome is possible. This time let x = 2. In column A, the expression will read 7(2) — 21, which simplifies to 14 — 21 = —7. In column B, the expression will read 21 — 7(2), which simplifies to 21 — 14 = 7. Now column B is greater. Since the columns are not always equal nor is column B always greater, the correct answer is (D).

23.B

First, take the instructions in column A one piece of information at a time. The smallest positive factor of 25 is 1, and the biggest positive factor of 16 is 16. Thus, 1 × 16 = 16. Column A is 16. Since column B is greater, the correct answer is (B).

24.A

The probability of getting heads on any single flip is , since there are two sides of a coin and only one of those is heads. Column A is the probability of getting heads on 2 consecutive flips, so you need to multiply the probability of the first flip by the probability of the second flip (i.e., Event 1 × Event 2): . To find column B, include the probability of a getting heads on the third flip (i.e., Event 1 × Event 2 × Event 3): . Since column A is greater, the correct answer is (A). Note: If you know that the probability of getting the same result (in this case, heads) on consecutive flips of a coin diminishes with each subsequent flip, you can find the correct answer quickly!

25.C

Use the formula given to find the areas of each figure. In column A, the height of Cylinder A is 10 and the radius is 6. Plug these values into the volume formula to find that V = π(6)²(10) = π(36)(10) = 360π. This means column A is 360π. For column B, you can find the volumes of Cylinder B and Cylinder C separately and then add the results together. Alternatively, since the two figures have the same dimensions, you can multiply the volume formula by 2 to find the total volume. Thus, column B is V = 2π(6)²(5) = 2π(36)(5) = 2π(180) = 360π. Since the columns are equal, the correct answer is (C).

26.C

Since all the angles are labeled as x, they are all equal. The angles in a triangle must add up to 180º; therefore, each angle is equal to 60º since . Thus, the value of column A is 60, which makes the columns equal. The correct answer is (C).

27.A

To find percent change, use this formula: . In column A, the difference from 1 to 2 is 1. Since you are looking for percent increase, the smaller number will be the original number. Thus, the original number is 1. Column A is , which simplifies to . For column B, the difference from 2 to 1 is still 1. However, since you are looking for percent decrease, the larger number will be the original number. Thus, the original value is now 2. Column B is , which simplifies to . The value in column A is greater, so the correct answer is (A).

28.C

Remember, if you see the word average, you can use an Average Pie. First, find the sum: 4 + 6 + 8 + 10 = 28. That number goes in the total place. There are 4 numbers, so put 4 in the # of items spot. Divide to get . Column A is 7. Remember that the median of a set is the middle number when the numbers are listed in order from least to greatest. Since there is an even number of items in this set, the median will be found by taking the average of the two middle numbers, 6 and 8. The average of 6 and 8 is , so the value in column B is 7. Since the two columns are equal, the correct answer is (C).

29.D

Since there are variables in the columns, plug in values for the variables and evaluate the expressions. For instance, let x = 3 and y = 5 since x and y must be positive numbers. Column A is , whereas column B is , which is a little bit less than 4 since . Since column A is greater, eliminate (B) and (C). Now, plug in a second time to see if a different outcome is possible. Let x = 1 and y = 1. Column A is , and column B is . This time, column B is greater. Since neither column is always greater, the correct answer is (D).

30.C

This is a common trick on the ISEE. Notice that all the numbers are exactly the same and in the exact same order. All that is different is the placement of the decimals. 567.83 and 5.6783 are off by two decimals, or a factor of 100. Similarly, 0.40 and 40.0 are related by a factor of 100, or two decimal places as well. This cancels itself out, and the product will be the same for both columns. Check if you want to see for yourself, but if you know this trick it will save you some time on the test! The correct answer is (C).

31.C

First, find the probability of not picking red shoes, which is the same as picking purple or white shoes, and then compare the result to column B’s value. There are 10 total pairs to pick from, and there are 8 pairs of shoes that are purple or white (i.e., not red), so the probability is . Column A is which is the value of column B. Since the columns are equal, the correct answer is (C).

32.B

Find the total of each column. Column A is 10 × 8 = 80, and column B is 20 × 4.5 = 90. Column B is greater, so the correct answer is (B). Remember, you can estimate column B’s value if you’re running short on time. 20 × 4 = 80; however, since 20 is really being multiplied by 4.5, not 4, you know the product has to be more than 80.

33.C

Remember, if you see exponents, you can always write it out! You could also plug in since there are variables in the columns. However, another way to find the values of the two columns is to use the exponent rules (MADSPM). For column A, . Column B is also x³. Since the two columns are equal, the correct answer is (C).

34.D

The information given is x² = 36. Remember that 6 is not the only solution to this equation. 6² is equal to 36, but (—6)² is also equal to 36 since two negatives multiplied together will equal a positive number. Since column A can be greater than column B but could also be equal to column B, the correct answer is (D).

35.C

Remember that factors are numbers you can multiply together to equal another number. The largest positive factor of 16 is 16, since 1 × 16 = 16. Thus, column A is 16. Multiples are the result of multiplying two numbers together. The smallest positive multiple of 16 is also 16, since 16 × 1 = 16. Therefore, column B is also 16. Since the columns are equal, the correct answer is (C).

36.B

Since there are variables in the figure, plug in a value for x. For instance, let x = 2. The perimeter of square ABCD is equal to 2 + 2 + 2 + 2 = 8. The perimeter of MNOP is equal to 2 + 2(2) + 2 + 2(2), or 2 + 4 + 2 + 4 = 12. In this case, column B is greater, so eliminate (A) and (C). Try plugging in a second time to see if a different outcome is possible. Try x = . The perimeter of square ABCD is equal to , or 1 + 1 = 2. The perimeter of MNOP is equal to , or 1 + 1 + 1 = 3. Column B is still greater. Since there are figures, it is not possible to plug in a negative number or zero; the only possible numbers will be positive. All positive numbers will yield the same outcome since the rectangle has two sides that are double the side length of the square, meaning that the rectangle will always have a perimeter a little bit (or a lot) bigger than the square’s. Therefore, column B will always be greater, and the correct answer is (B).

37.C

Draw the triangle on the figure provided and write down the formula for the area of a triangle: . Next, find the values for the base and height. To find the base, find the value of DE. AD and BC are congruent since the shape is a rectangle. From there, the base must be 4, since E is the midpoint of AD, which splits 8 into two equal parts. The height of the triangle must meet the base at a right angle, so the height of the triangle is the same as the measure of side AB, which has a length of 6. Plug in these values to the formula to find that . Column A is 12, which is equivalent to the value in column B. Since the columns are equal, the correct answer is (C).

38.C

Remember, with exponents, you can always write it out. However, this is a common trick on the ISEE. Another way to evaluate these expressions is to rewrite them with the same base. In column B, 64 is the same as 4³, so the whole expression can be rewritten as (4³)⁴. Using the exponent rules (MADSPM), you can further simplify: . Since the value in column B is the same as the value of column A, the correct answer is (C).

39.A

When the question asks about ratios, use a Ratio Box. The ratio of blue to red tickets is 3:5, so fill in that information into the top row of the Ratio Box. Remember to add the two numbers together to get the total for the ratio row: 3 + 5 = 8. No other information is provided. To make things simple, let’s say that there are a total of 8 actual tickets. That makes the multiplier 1. Column B asks for the fractional part of all the tickets that are blue. Use the numbers from the Ratio Box. There are 3 blue tickets and 8 total tickets: . Thus, column B is . Remember that when the numerators are the same, the fraction with the bigger denominator is actually the smaller number. Therefore, column A is greater, and the correct answer is (A).

40.C

Remember, if you see the word average, you can use an Average Pie. Luke made two trips, so the # of items is 2. Column A represents the average speed for the entire trip, so to find the total speed, add the speeds from both parts of the trip: 50 + 60 = 110. Put that value in the total spot. Finally, divide to find the average: . Thus, column A is 55, which is the same as column B. Since the two columns are equal, the correct answer is (C).

41.A

First, rearrange the equation in column A to y = mx + b form. Subtract 12x from both sides to get — 4y = —12x + 16. Divide by —4 on both sides to get y = 3x — 4. The slope is the coefficient of x, so the slope is 3. Column A is 3. Now, find the slope from the two points given in column B using the formula . Thus, , and column B is 1. The value of column A is greater, so the correct answer is (A).

42.B

Don’t be intimidated by the decimals under the square root! You probably know that 9² = 81, right? To go from 81 to 0.81, you need to move the decimal to the left two places, so move the decimal one place to the left for each 9 and multiply: 0.9 × 0.9 = 0.81. Thus, . Column A is 0.9. For column B, the decimal needs to be moved only one place. That means the two numbers that will be multiplied together will have to be bigger than 0.9. Therefore, column B will be greater. The correct answer is (B). Note, you can also guesstimate: and , so the will be less than 3 but greater than 2, which is still larger than 0.9.

43.D

There are infinitely different ways in which the length and width could multiply together to equal 36. For instance, the length of y could be 4 and the width of z could be 9, or vice versa. The values of y and z could also each be 6, which would make the two columns equal. Since it is impossible to know if y or z is the greater value, or if they are equal. Without more information, the correct answer must be (D).

44.A

First, list the nonnegative even integers less than 10. Don’t forget about zero! The list should contain 0, 2, 4, 6, and 8. Therefore, column A is 5. Since this is greater than the value of column B, the correct answer is (A).

45.D

Don’t make assumptions! As for all geometry questions, use the rules of the shape you are given to help you solve and ignore anything unstated about the illustration. Isosceles triangles have 2 equal sides. There are therefore three possibilities for the two equal sides: 1) both AB and BC equal 2, 2) both BC and AC equal 2, or 3) BC equals 2 and both AB and AC equal some other number. Because the shape is not drawn to scale, it’s possible that AB and AC could be equal to, say, 100, which would make the area of the triangle much greater than 4. In other words, there are so many possibilities, many of which have an area greater than 4, that the correct answer must be (D).

Math Review

1. Yes

2. It is neither positive nor negative.

3. Addition

4. Multiplication

5. The quotient

6. Yes

3 + 1 + 2 = 6, which is divisible by 3;

No

3 + 1 + 2 = 6, which is not divisible by 9.

7. Exponents

8. Yes

3 goes into 12 evenly 4 times;

No

12 cannot go into 3.

9. No

No integer times 12 is equal to 3;

Yes

3 × 4 = 12

10. 0

The tens digit is two places to the left of the decimal.

11. 2

The tenths digit is one place to the right of the decimal.

12. 8

Write it out! 2 × 2 × 2 = 8

13. Over 100 or

14. Multiplication

15. Total column

16. Average Pie

17. Plug in a number

18. Add; all four

19. Multiply; two (or square one side, since the sides of a square are the same)

20. 180

21. 3; 180

22. 360

23. 2; equilateral

24. Hypotenuse; right angle

25.

REVIEW—THE VERBAL PLAN

Pacing and Verbal Strategy

I will do the verbal questions in this order.

1.Sentence completions

2.Synonyms with words I sort of know

3.Synonyms with words I know

I should spend a little more than 10 minutes on sentence completions.

I will always eliminate wrong (or “worse”) answers.

If possible, I will eliminate choices before I guess, but even if I can’t eliminate any, I will still guess productively.

No, I cannot eliminate choices that contain words I do not know.

SYNONYMS

Practice Drill 1—Write Your Own Definition

Possible Definitions

1.weird

2.introduction

3.giving

4.lesson found in a fable or tale

5.change

6.circle around

7.optimistic

8.stick around

9.help

10.build

11.bend down

12.honest

13.tease

14.rough

15.self-centered

16.calm

17.use

18.full of life

19.stretch out

20.help

Practice Drill 2—Write Another Definition

Look up these seven words in a dictionary to see how many different meanings they can have.

Practice Drill 3—Basic Synonym Techniques

1.C

2.A

3.C

4.D

5.D

6.D

7.B

8.C

9.D

10.A

11.C

12.B

13.D

14.B

15.A

16.C

17.D

18.C

19.B

20.C

Practice Drill 4—Making Your Own Context

Possible Contexts (Answers Will Vary)

1.Common cold; common man

2.Competent to stand trial

3.Abridged dictionary

4.Untimely demise; untimely remark

5.Homogenized milk

6.Juvenile delinquent; delinquent payments

7.Inalienable rights

8.Paltry sum

9.Auspicious beginning; auspicious occasion

10.Prodigal son

Practice Drill 5—Using Your Own Context

1.C

2.D

3.D

4.C

5.A

6.B

7.C

8.A

9.C

10.A

Practice Drill 6—All Synonyms Techniques

1.C

2.A

3.C

4.A

5.D

6.A

7.D

8.B

9.C

10.B

11.B

12.C

13.D

14.C

15.D

16.B

17.C

18.B

19.D

20.B

21.C

22.B

23.B

24.D

25.D

26.D

27.C

28.C

29.A

30.D

31.B

32.B

33.B

34.D

35.D

36.C

37.C

38.D

39.B

40.D

SENTENCE COMPLETIONS

Practice Drill 7—Coming Up with Your Own Word

These words are just to give you an idea of what you could use. Any words that accurately fill the blank, based on the clue and the direction word, will do.

1.good

2.rare

3.remarkable

4.awake

5.lucky

6.thoughtful

7.alike

8.waste time

9.frugal

10.movement

11.simple

12.produce

13.changed

14.strong

15.not necessary

16.repetitive

17.risky

18.outgoing

19.balanced

20.generous

21.steadfast

22.intimidated; shy

23.annoyed

24.on time

25.skill

26.variety

27.creative

28.sharing

29.flexible

30.inborn

31.affable; talkative

32.awestruck

Practice Drill 8—Eliminating Answers Based on Your Word

Below are the correct answers to the problems. You should have eliminated the other choices.

1.D

2.B

3.C

4.D

5.A

6.A

7.D

8.C

9.B

10.A

11.D

12.D

13.B

14.C

15.D

16.C

17.B

18.A

19.C

20.C

21.B

22.D

23.A

24.B

25.A

26.C

27.B

28.B

29.C

30.B

31.A

32.C

Practice Drill 9—Using Positive/Negative

1.+

2.+

3.‒

4.+

5.‒

6.+

7.‒

8.‒

9.‒

10.+

Practice Drill 10—Eliminating Based on Positive/Negative

1.A

2.C

3.B

4.D

5.C

6.A

7.C

8.D

9.A

10.C

Practice Drill 11—Two-Blank Sentence Completions (Upper Level Only)

1.B

2.C

3.A

4.A

5.D

6.A

7.C

8.C

9.A

10.C

Review—The Sentence Completions Plan

One-Blank Sentence Completions

For each and every sentence completion, the first thing I do is ignore the answers.

I look for the clue, and I mark it by underlining or highlighting it.

I look for any direction words, and I circle or highlight them.

Then I come up with my own word for the blank. If I have trouble coming up with a word for the blank, I decide if the blank is positive or negative (or neither).

Then I eliminate choices, and I guess from the remaining choices.

Two-Blank Sentence Completions- Upper Level Only

For each and every sentence completion, the first thing I do is ignore the answers.

I look for the clue, and I mark it by underlining or highlighting it.

I look for any direction words, and I circle or highlight them.

If the sentence completion has two blanks, I do them one at a time.

I do the blank that is easier first—the one that has the better clue.

I come up with a word for one of the blanks, and when I uncover the choices, I uncover only the words for the blank that I am working on, and I eliminate based on those.

Then, I go back to the sentence and come up with a word for the other blank, uncover the choices that are left, and eliminate.

Eliminating Choices and Guessing

No, I cannot eliminate choices that contain words I do not know.

If I can eliminate only one or two choices, then I guess from the remaining choices.

If the sentence or vocabulary looks so difficult that I can’t come up with a word or decide if the blank is positive or negative, then I fill in my “letter-of-the-day.”

I spend my last minute filling in the “letter-of-the-day” for any questions I have not gotten around to answering.

I should never leave a question unanswered because there is no penalty for guessing.

Practice Drill 12—All Sentence Completion Techniques

1.A

2.C

3.B

4.A

5.D

6.B

7.B

8.C

9.C

10.B

11.D

12.A

13.B

14.C

15.A

16.B

17.C

18.D

19.A

20.D

READING COMPREHENSION

Practice Drill 1—Getting Through the Passage

You should have brief labels like the following:

Practice Drill 2—Answering a General Question

1.D

2.D

Practice Drill 3—Answering a Specific Question

1.C

2.A

3.B

4.D

5.C

Review—The Reading Plan

After I read each paragraph, I label it either on the page or on my scratch paper.

After I read an entire passage, I ask myself: What? and So what?

The five main types of general questions, and the questions I can ask myself to answer them, are:

·  Main idea: What was the “What? So what?” for this passage?

·  Tone/attitude: How did the author feel about the subject?

·  General interpretation: Which answer stays closest to what the author said and how he said it?

·  General purpose: Why did the author write this?

·  Prediction: How was the passage arranged? What will come next?

To find the answer to a specific question, I can use three clues.

·  Paragraph labels

·  Line or paragraph reference

·  Lead words

If the question says “In line 22,” then I begin reading at approximately line 17.

On a general question, I eliminate answers that are:

·  Too small

·  Not mentioned in the passage

·  In contradiction to the passage

·  Too big

·  Too extreme

·  Against common sense

On a specific question, I eliminate answers that are:

·  Too extreme

·  Contradicting passage details

·  Not mentioned in the passage

·  Against common sense

When I’ve got it down to two possible answers, I should:

·  Reread the question

·  Look at what makes the two answers different

·  Go back to the passage

·  Eliminate the answer that is worse

Practice Drill 4—All Reading Techniques—Lower Level

What? Tides

So what? Are caused by the Moon

1.A

2.D

3.D

4.C

Practice Drill 5—All Reading Techniques —Lower Level

What? Brooklyn Bridge

So what? There were problems building it.

1.D

2.B

3.C

4.A

5.C

Practice Drill 6—All Reading Techniques—All Levels

What? William Levitt

So what? Built homes efficiently

1.C

2.D

3.C

4.D

5.A

6.C

Practice Drill 7—All Reading Techniques—All Levels

What? Etymology

So what? Has many words to explore

1.B

2.C

3.A

4.C

5.B

Practice Drill 8—All Reading Techniques—Upper Level

What? Bob Dylan

So what? Was destined to be a musician

1.D

2.B

3.D

4.A

5.C

Practice Drill 9—All Reading Techniques—Middle and Upper Levels

What? Science

So what? Doesn’t have all the answers

1.A

2.D

3.D

4.A

5.C