Cracking the SSAT & ISEE - The Princeton Review 2019

Answer key to fundamental math drills
Part i the basics of both tests

The Building Blocks

Practice Drill 1—Math Vocabulary

1.6

0, 1, 2, 3, 4, 5

2.1, 3, 5

Many sets of integers would answer this question correctly.

3.3

3, 5, and 7

4.8

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

5.That number

The smallest positive integer is 1, and any number times 1 is equal to itself.

6.90

5 × 6 × 3 = 90

7.30

3 + 11 + 16 = 30

8.60

90 — 30 = 60

9.—2, —4, —6

2, 4, and 6 are consecutive integers and the question wants negative. Other sets of consecutive integers would also answer the question correctly.

10.Yes

11 is only divisible by 1 and itself.

11.22

5 + 6 + 4 + 7 = 22

12.6

13 goes into 58, 4 times. 4 × 13 = 52 and 58 — 52 = 6.

13.1, 5, 11, 55

1, 5, 11, and 55 will all divide into 55 evenly.

14.12

5 + 8 + 9 = 22 and 1 + 2 + 0 + 7 = 10.

22 — 10 = 12

15.No

The remainder of 19 ÷ 5 is 4. And 21 is not divisible by 4.

16.2, 2, 3, 13

Draw a factor tree.

17.16

3 + 13 = 16

18.9

12 × 3 = 36 and 9 × 4 = 36.

19.1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Remember that factors are the numbers that can be multiplied together to get 72.

20.There are 9 even factors and 3 odd factors.

The even factors are 2, 4, 6, 8, 12, 18, 24, 36, and 72. The odd factors are 1, 3, and 9.

Practice Drill 2—Adding and Subtracting Negative Numbers

1.—8

2.—14

3.— 4

4.27

5.21

6.4

7.—22

8.—29

9.—6

10.90

11.0

12.29

13.24

14.—30

15.—14

Practice Drill 3—Multiplying and Dividing Negative Numbers

1.— 4

2.—36

3.65

4.11

5.63

6.—13

7.84

8.—5

9.9

10.—8

11.—75

12.72

13.— 4

14.34

15.—11

Practice Drill 4—Order of Operations

1.9

2.16

3.7

4.50

5.6

6.30

7.24

8.60

9.101

10.—200

Practice Drill 5—Factors and Multiples

1.2, 4, 6, 8, 10

4, 8, 12, 16, 20

5, 10, 15, 20, 25

11, 22, 33, 44, 55

2.Yes

3 goes into 15 evenly 5 times.

3.Yes

Use the divisibility rule for 3. The sum of the digits is 9, which is divisible by 3.

4.No

The sum of the digits is 14, which is not divisible by 3.

5.Yes

The only factors of 23 are 1 and 23.

6.Yes

The sum of the digits is 6, which is divisible by 3.

7.No

The sum of the digits is 6, which is not divisible by 9.

8.Yes

250 ends in a 0, which is an even number and is divisible by 2.

9.Yes

250 ends in a 0, which is divisible by 5.

10.Yes

250 ends in a 0, which is divisible by 10.

11.Yes

2 × 5 = 10

12.No

There is no integer that can be multiplied by 3 to equal 11.

13.No

2 is a factor of 8.

14.Yes

4 × 6 = 24

15.No

There is no integer that can be multiplied by 6 to equal 27.

16.Yes

3 × 9 = 27

17.8

6, 12, 18, 24, 30, 36, 42, 48

18.8

Even multiples of 3 are really just multiples of 6.

19.8

Multiples of both 3 and 4 are also multiples of 12.

12, 24, 36, 48, 60, 72, 84, 96

20.48

3 × 16 = 48

Practice Drill 6—Reducing Fractions

1.

2.

3.

4.

5.

6.

7.1

8.

9.If the number on top is larger than the number on the bottom, the fraction is greater than 1.

Practice Drill 7—Changing Improper Fractions to Mixed Numbers

1.5

2.1

3.5

4.2

5.2

6.6

7.1

8.2

9.11

10.10

Practice Drill 8—Changing Mixed Numbers to Improper Fractions

1.

=

2.

=

3.

=

4.

=

5.

=

6.

=

7.

=

8.

=

9.

=

10.

=

Practice Drill 9—Adding and Subtracting Fractions

1.1 or

Multiply using Bowtie to get + = = 1.

2.

Multiply using Bowtie to get + = .

3.

Did you use the Bowtie? You didn’t need to because there was already a common denominator there!

4.

Multiply using Bowtie to get — = .

5.2 or

Multiply using Bowtie to get + = = 2.

6.—

Multiply using Bowtie to get — = — .

7.4 or

Multiply using Bowtie to get + = = = 4.

8.

Did you use the Bowtie? You didn’t need to because there was already a common denominator there! Subtract to get = .

9.

Multiply using Bowtie to get + = .

10.x

Multiply using Bowtie to get + = = x.

11.

Multiply using Bowtie to get + = = .

12.

Multiply using Bowtie to get — = = .

Practice Drill 10—Multiplying and Dividing Fractions

1.

= =

2.1 or

× = = =

3.

= =

4.1

= = 1

5.

× = = =

Practice Drill 11—Decimals

1.18.7

Don’t forget to line up the decimals. Then add.

2.4.19

After lining up the decimals, remember to add a 0 at the end of 1.7. Then add the two numbers.

3.4.962

Change 7 to 7.000, line up the decimals, and then subtract.

4.10.625

Don’t forget there are a total of 3 digits to the right of the decimal.

5.0.018

There are a total of 4 digits to the right of the decimal, but you do not have to write the final 0 in 0.0180.

6.6,000

Remember to move both decimals right 2 places: and don’t put the decimals back after dividing!

7.5

Remember to move both decimals right 2 places: and don’t put the decimal back after dividing!

Practice Drill 12—Fractions as Decimals

Practice Drill 13—Percents

1.a) 30%

b) 40%

c) 10%

d) 20%

2.18%

100% = 75% + 7% + percentage of questions answered incorrectly

3.a) 20%

b) 30%

c) 40%

d) 10%

sneakers + sandals + boots + high heels = 100%

20% + 30% + 40% + h = 100%

h = 10%

e) 4

sneakers + sandals + boots + high heels = 40

8 + 12 + 16 + h = 40

h = 4

4.90%

5.a) 48%

b) 52%

100% = girls + boys

100 = 48 + b

b = 52

Practice Drill 14—More Percents

1.21

2.9

3.15

4.51

5.8

6.The sale price is $102.

15% of $120 = × 120 = = 18.

$120 − $18 = $102. The sale price is 85% of the regular price. 100% − 15% = 85%.

7.292

80% =

8.27

If she got 25% wrong, then she got 75% correct.

75% of 36 = × 36 = = 27.

9.$72

If 20% = , then × 100 = = 20. The original price ($100) is reduced by $20, so the new price is $80. After an additional 10% mark down , the discounted price is reduced by $8, so the final sale price is 80 — 8 = 72.

Practice Drill 15—Exponents and Square Roots

1.8

2 × 2 × 2 = 8

2.16

2 × 2 × 2 × 2 = 16

3.27

3 × 3 × 3 = 27

4.64

4 × 4 × 4 = 64

5.9

9² = 9 × 9 or 81, so = 9.

6.10

10² = 10 × 10 or 100, so = 10.

7.7

7² = 7 × 7 or 49, so = 7.

8.8

8² = 8 × 8 or 64, so = 8.

9.3

3² = 3 × 3 or 9, so = 3.

Practice Drill 16—More Exponents

1.3⁸

3⁵ × 3³ = 3⁵⁺³ = 3⁸

2.7⁹

7² × 7⁷ = 7²⁺⁷ = 7⁹

3.5⁷

5³ × 5⁴ = 5³⁺⁴ = 5⁷

4.15³

15²³ ÷ 15²⁰ = 15²³⁻²⁰ = 15³

5.4⁹

4¹³ ÷ 4⁴ = 4¹³⁻⁴ = 4⁹

6.10⁴

10¹⁰ ÷ 10⁶ = 10¹⁰⁻⁶ = 10⁴

7.5¹⁸

(5³)⁶ = 5^3×6= 5¹⁸

8.8³⁶

(8¹²)³ = 8^12×3 = 8³⁶

9.9²⁵

(9⁵)⁵ = 9^5×5 = 9²⁵

10.2²⁸

(2²)¹⁴ = 2^2×14 = 2²⁸

Review Drill 1—The Building Blocks

1.No

Remember, 2 is the smallest (and only even) prime number. 1 is NOT prime.

2.9

1, 2, 4, 5, 10, 20, 25, 50, 100

3.—30

4.140

5.

Multiply using Bowtie to get

= .

6.

= =

7.4.08

Don’t forget there are a total of 2 digits to the right of the decimal.

8.20

Multiply to get = 6. Multiply both sides by 100 to get 30x = 600, and then divide both sides by 30 to get x = 20.

9.1

1⁵ = 1 × 1 × 1 × 1 × 1. Note: 1 to any power will always equal 1.

10.4

4² = 4 × 4 or 16, so = 4.

11.1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Algebra

Practice Drill 17—Solving Simple Equations

1.x = 12

35 — 12 = 23

2.y = 15

15 + 12 = 27

3.z = 28

28 — 7 = 21

4.x = 5

5 × 5 = 25

5.x = 3

18 ÷ 3 = 6

6.x = 11

3 × 11 = 33

7.y = 5

65 ÷ 5 = 13

8.z = 3

14 = 17 — 3

9.y = 48

× 48 = 24

10.z = 71

136 + 71 = 207

11.x = 12

7 × 12 = 84

12.y = 12

12 ÷ 2 = 6

13.z = 45

45 ÷ 3 = 15

14.x = 18

14 + 18 = 32

15.y = 29

53 — 29 = 24

Practice Drill 18—Manipulating an Equation

1.3

To isolate x, add x to both sides. Then subtract both sides by 8. Check your work by plugging in 3 for x: 8 = 11 — 3.

2.5

To isolate x, divide both sides by 4. Check your work by plugging in 5 for x: 4 × 5 = 20.

3.6

To isolate x, add 20 to both sides. Then divide both sides by 5. Check your work by plugging in 6 for x: 5(6) — 20 = 10.

4.7

To isolate x, subtract 3 from both sides. Then divide both sides by 4. Check your work by plugging in 7 for x: 4 × 7 + 3 = 31.

5.4

To isolate m, add 3 to both sides. Subtract m from both sides. Then divide both sides by 2. Check your work by plugging in 4 for m: 4 + 5 = 3(4) — 3.

6.8

To isolate x, divide both sides by 2.5. Check your work by plugging in 8 for x: 2.5 × 8 = 20.

7.8

To isolate x, subtract 2 from both sides. Then divide both sides by 0.2. Check your work by plugging in 8 for x: 0.2 × 8 + 2 = 3.6.

8.

To isolate x, subtract 4 from both sides. Then divide both sides by 8. Check your work by plugging in for x: 6 = 8 × + 4.

9.7

To isolate x + y, divide both sides by 3. Check your work by plugging in 7 for x + y: 3(7) = 21.

10.7

To isolate x + y, factor out a 3 from both terms on the left side: 3(x + y) = 21. Then divide both sides by 3. Check your work by plugging in 7 for x + y : 3(7) = 21. Note that this question and the previous question are really the same equation. Did you see it?

11.7

To isolate y, subtract 100 from both sides. Then divide both sides by —5. Check your work by plugging in 7 for x: 100 — 5 × 7 = 65.

Practice Drill 19—Manipulating an Inequality

1.x > 4

To isolate x, divide both sides by 4. The sign doesn’t change!

2. x < —2

To isolate x, subtract 13 from both sides. Then divide both sides by —1. Since you divided by a negative number, flip the sign.

3. x > —5

First, combine like terms to get —5x < 25. Then divide both sides by —5. Since you divided by a negative number, flip the sign.

4.x > 4

To isolate x, add x to both sides. Subtract 12 from both sides. Then divide both sides by 3. The sign doesn’t change!

5.x < —7

To isolate x, add 3x to both sides. Subtract 7 from both sides. Then divide both sides by 3. The sign doesn’t change!

Practice Drill 20—Translating and Solving Percent Questions

1.12

Translation: 30 = × 250. To solve, simplify the right side: = , which reduces to . Multiply both sides by 10, and then divide both sides by 25. Check your work by plugging in 12 for x.

2.24

Translation: x = × 200. To solve: = = 24.

3.5

Translation: x = × × 200. To solve, reduce the right side: × × 200. Then simplify: = = 5.

4.80

Translation: × × n = 12. To solve, reduce the left side: × × n. Then simplify: = . Multiply both sides by 20, and divide both sides by 3. Check your work by plugging in 80 in for n.

5.125

Translation: × n = × 80. To solve, reduce both sides to get × n = × 80. Then, multiply to get = . Next, cross-multiply to get 16n = 2,000. Finally, divide both sides by 16. Check your work by plugging in 125 for n.

6.60

Translation: = . To solve, cross-multiply to get 5x = 300, and then divide both sides by 5. Check your work by plugging in 60 for x.

7.40

Translation: 30 = × 75. To solve, simplify the right side: = , which reduces to . Multiply both sides by 4, and divide both sides by 3. Check your work by plugging in 40 for x.

8.2.64 or 2 or

Translation: x = × 24. To solve, × 24 = = = 2.64.

9.200

Translation: × 24 = 48. To solve, simplify the left side: , which reduces to . Then multiply both sides by 25, and divide both sides by 6. Check your work by plugging in 200 for x.

10.2

Translation: × × 500 = 6. To solve, reduce the fraction to and simply the left side: = = 3n. Then divide both sides by 3. Check your work by plugging in 2 for n.

Geometry

Practice Drill 21—Squares, Rectangles, and Angles

1.115°

65° + x° = 180°

2.100°

45° + x° + 35° = 180°

3.The perimeter of PQRS is 16. 4 + 4 + 4 + 4 = 16. Its area is also 16. 4² = 16.

4.The perimeter of ABCD is 20. 7 + 3 + 7 + 3 = 20. Its area is 21. 7 × 3 = 21

5.The area of STUV is 9. If the perimeter is 12, then one side of the square is 3 (12 ÷ 4 = 3). Therefore, the area is 3² = 9.

6.The perimeter of DEFG is 36. If the area is 81, then one side of the square is 9( = 9). Therefore, the perimeter is 9 + 9 + 9 + 9 = 36.

7.The area of JKLM is 24. If the perimeter is 20, then 4 + l + 4 + l = 20. So the length (the other side) of the rectangle is 6. Therefore, the area is 6 × 4 = 24.

8.The perimeter of WXYZ is 22. If the area is 30, then 6 × w = 30. So the width (the other side) of the rectangle is 5. Therefore, the perimeter is 6 + 5 + 6 + 5 = 22.

9.24

V = lwh = 2 × 4 × 3 = 24.

Practice Drill 22—Triangles

1.45°

180° — 90° = 90°. Since two sides (legs) of the triangle are both 3, the angles that correspond to those sides are also equal to each other. Therefore, each angle is 45°, so x = 45°.

2.70°

180° — 40° = 140°. Since sides PQ and QR are equal, then ∠QPR and x° are also equal to each other. Thus, divide 140° by 2 to find that each remaining angle is 70°. So x = 70°.

3.6

Plug the base and height into the area formula for a triangle:

A = bh = (4)(3) = 6.

4.12

In this case, count the height and base of the triangle by counting off the ticks on the coordinate plane. The height is 6 and the base is 4, which means that

A = bh = (4)(6) = 12.

5.14

Plug the base and height into the area formula for a triangle:

A = bh = (4)(7) = 14.

6.WXZ = 5

A = bh = (2)(5) = 5

ZXY = 15

A = = (6)(5) = 15

WXY = 20

A = = (2+6)(5) = 20

7. 4.8

These are similar triangles since all the angles are the same. Set up a proportion to solve: = , so = . Cross-multiply to get 10(QR) = 6(8). Divide both sides by 10, and QR = 4.8.

8.DE = 8

Since this is a right triangle, use the Pythagorean Theorem to find the missing side length: a² + b² = c², so a² + 6² = 10. Subtract 36 from both sides and a² = 64. Take the square root of both sides, and a (or DE) = 8.

9.9.6

These are similar triangles since all the angles are the same. Set up a proportion to solve: = . Cross-multiply to get 16(12) = 20(x). Divide both sides by 20, and x = 9.6.

10.26

Remember that all angles in a rectangle are right angles. This diagonal (AC) cuts the rectangle into two right triangles, so use the Pythagorean Theorem to find the missing side length: a² + b² = c², so 10² + 24² = c², and c (or AC) = 26.

11.40

First, use the right triangle to find AD, which is one side of the square ABCD. 8² + 6² = c², so c = 10. Since all sides of a square are equal, the perimeter is 10 + 10 + 10 + 10 = 40 (or 10(4) = 40).

12.2.4

These are similar triangles since all the angles are the same. Set up a proportion to solve: = . Cross-multiply to get 5x = 6(2). Divide both sides by 5, and x = 2.4.

Practice Drill 23—Circles

1.Circumference = 10π. Area = 25π.

Plug the radius into the circumference formula for a circle: C = 2πr = 2π(5) = 10π. Plug the radius into the area formula for a circle: A = πr² = π(5)² = 25π.

2.16π

Plug the radius into the area formula for a circle: A = πr² = π(4)² = 16π.

3.16π

Since d = 2r, the radius is 4(8 = 2r). Plug the radius into the area formula for a circle: A = πr² = π(4)² = 16π. Note: this is really the same circle as the previous question.

4.3

Remember, you can find the radius from a circle’s area by getting rid of π and taking the square root of 9.

5.6

Find the radius from a circle’s area by getting rid of π and taking the square root of 9. Then multiply the radius by 2 to find the diameter.

6.10π

Find the radius from a circle’s area by getting rid of π and taking the square root of 25. Then, plug the radius into the circumference formula for a circle: C = 2πr = 2π(5) = 10π.

Practice Drill 24—3-D Shapes

1.128π

Plug the radius and height into the volume formula for a cylinder: V = πr²h = π(4)²(8) = 128π.

2.1,000

Plug the side length into the volume formula for a cube: V = s³ = 10³ = 1,000.

3.216

Plug the length, width, and height into the volume formula for a rectangular box: V = lwh = 12 × 3 × 6 = 216.

4.162

First, find the volume of the cube: V = s³ = 6³ = 216. Next, to find the remaining liquid needed to completely fill the cube, subtract the number of gallons already poured into it: 216 — 54 = 162.

5.12

One way to solve this problem is to divide the length, width, and height into segments of 2. The length is 8, so 4 cubes could fit along the length of the rectangular box since each cube has a side length of 2. The width of the box is 2, so only 1 cube could fit along the width of the box. That means the bottom layer of the box could hold 4 cubes (4 cubes across by 1 cube deep). The height of the box is 6, so you could stack 3 cubes on top of each other to fill the box. If each layer has 4 boxes and 3 layers of cubes can be stacked, then a total of 12 cubes can fit into the box (4 boxes per layer times 3 layers equals 12 boxes).

6.48π

First, find the volume of the cylinder: V = πr² = π(4)²(9) = 144π. Since the grain fills only a third of the cylinder, then find of the volume, or (144π) = 48π. Just treat the π like a variable in questions like these.

Word Problems

Practice Drill 25—Word Problems

1.4 quarts

Set up a proportion: = = . Then cross- multiply to get 32(x) = 128. Divide both sides by 32, and x = 4.

224 ounces

Set up a proportion: = = . Then cross-multiply to get 32(7) = x, and x = 224.

2.6 hours

Set up a proportion: = = . Then cross- multiply to get 50x = 300. Divide both sides by 50, and x = 6.

3.44

Start with the given age: Rufus’s. If Rufus is 11, then find Fiona’s age. Fiona is twice as old as Rufus translates to: Fiona = 2(Rufus) or F = 2(11), so Fiona is 22. Next find Betty’s age. Betty is twice as old as Fiona translates to: Betty = 2(Fiona) or B = 2(22). Therefore, Betty is 44.

4.5

Translate the parts of the question. This year’s sales = 1,250, how many times greater than means to divide, and last year’s sales = 250. Thus, .

5.120

Translate the first part of the problem: of means to multiply and the total students = 500. So, the number of freshman is (500) = = = 200. Now, translate the second part of the problem: of means to multiply and all the freshmen = 200. Therefore, the number of freshmen girls is (200) = = = 120.

Review Drill 2—The Building Blocks

1.45

Translate the problem: (b) = 15. Multiply both sides by 3, and b = 45. Check your work by plugging in 45 for b: (45) = 15.

2.8

To isolate x, add 7 to both sides. Then divide both sides by 7. Check your work by plugging in 8 for x: 7(8) — 7 = 49.

3.10

To isolate y, divide both sides by 4. Then add 5 to both sides. Check your work by plugging in 10 for y: 4(10 — 5) = 20.

4.x< 8

To isolate x, subtract 1 from both sides. Then divide both sides by 8. The sign doesn’t change!

5.160

Translation: 16 = (10). To solve, simplify the right side: (10) = = , which reduces to . Then, multiply both sides by 10. Check your work by plugging in 160 for x.

6.75

Translation: (32) = 24. To solve, simplify the left side of the equation: (32) = = , which reduces to . Then multiply both sides by 25, and divide both sides by 8. Check your work by plugging 75 for x.

7.21

Plug the base and height into the area formula for a triangle: A = bh = (7)(6) = 21.

8.14

Find the radius from a circle’s area by getting rid of π and taking the square root of 49. Then multiply the radius by 2 to find the diameter.

9.6

Find the radius from a circle’s circumference (C = 2πr) by getting rid of π from both sides (they cancel out), which leaves 12 = 2r. Divide both sides by 2. Check your work by plugging in 6 for the radius.

10.25π

Be careful not to just fill in a familiar formula with the given numbers. Here, you aren’t given r. Instead, you’re given the diameter. Since d = 2r, the radius is 5 (10 = 2r). Plug the radius into the area formula for a circle: A = πr² = π(5)² = 25π.