conquering sat math practice test 1 answers

266 CONQUERING SAT MATH

ANSWERS AND EXPLANATIONSSECTION 2

1. Answer: C

Solve the equation.

2. Answer: C

See the pattern below. The sequence 4,9,6,11,8,13, . . . is formed by adding 5 and then subtracting 3. We can seethat the next term in the sequence will be 10.

+5 −3 +5 −3 +5 −3

4 9 6 11 8 13 [10]

3. Answer: A

Multiply the number of styles, 5, by the number of colors, 7, to get the number of bicycles that can be made by thecompany, 5 � 7 = 35. Therefore, there are a total of 35 different types of bicycles that can be made by the company.

4. Answer: C

Look at the graph of the function f(x) = 2(x − 3)2. Notice that the graph does not touch the x-axis when x = −3 butwhen x = 3.

5. Answer: E

Solve the proportion.

346 15

3

4

1

6 15

1

8 15

8 1515

81

7

8

=

= ⇒ =

= ⇒ = =

x

x x

x x

i

cups..

2

4

–2

–4

0 2 4 6–2–4–6

y

x

2 6 4 1

7 2 3 5

x x

x x

+ = −

= ⇒ =.

CHAPTER 19 / SAT MATHEMATICS PRACTICE TEST 1 267

6. Answer: E

Because AB____

≅ BC____

then AB = BC. Therefore, B is the midpoint of AC____

, so statement I is true.

Because B is the midpoint of AC____

, and statement II is true. As stated above, AB = BC; therefore,

so statement III is false. Statements I and II are true. That’s choice E.

7. Answer: A

A = the amount produced by Plant A.

B = the amount produced by Plant B.

T = the total amount produced by the company.

Three times the Plant A production is one-half the total.

Because Plant A and Plant B are the only two production plants and Plant A production is of total production,

8. Answer: D

Multiply the average number of unsold tickets by the number of viewings per week. Total number of unsold tickets = tv.

9. Answer: B

• x + 120 = 180 because a � b, and because same-side interior angles are supplementary. That means x = 60

• Since z and x are vertical angles and vertical angles are congruent, z = 60.

a

b

120°

z°x°

A T B T= =1

6

5

6so .

1

6

31

2

1

6A T A T= ⇒ = .

A B C

1

2AB BC≠ ,

AB AC=1

2,


10. Answer: C

For any number less than −1, x3 < x2

For any number between −1 and 1 x3 < x2

However for any number greater than 1 x 3 > x 2

11. Answer: C

The question mentions that the base area of Cylinder A is four times the base area of Cylinder B. The base of acylinder is a circle. Let rA be radius of Cylinder A and rB be the radius of Cylinder B.

12. Answer: D

The ratio of nonred marbles to total number of marbles is

13. Answer: D

Write an equation for Jim’s monthly earnings. c(4 + x) = $2,000

Jim sold six cars. Substitute 6 for x. c(4 + 6) = $2,000

Solve for c. 10c = $2,000 so c = $200

Substitute $200 for c and find f(x) to be sure. f(x) = $200 (4 + x) = $800 + $200x

The expression tells us that Jim earned a base salary of $800 and a commission of $200 per car.

14. Answer: C

If ay + ax = 0, that means a(x + y) = 0, which means x + y = 0. Therefore, x and y must be opposites. In other words x = −y.

7:12 =7

12.

π πr r r r

r r r

A B A B

A B A

( ) = ( ) ⇒ ( ) = ( )

( ) = ( ) ⇒ =

2 2 2 2

2 2

4 4

4 22rB .

2 83 2( ) = ( )> 4 = 2 .

1

2

1

8

1

2

3 2⎛⎝⎜

⎞⎠⎟ = =

⎛⎝⎜

⎞⎠⎟<

14

.

−⎛⎝⎜

⎞⎠⎟ = − = −

⎛⎝⎜

⎞⎠⎟

1

2

1

8

1

2

3 2

<14

.

−( ) = − ( )2 83 2

< 4 = −2 .


15. Answer: A

We know that if the sides are equal, the angles across from the sides have equal measure. We also know that the sumof the measures of the angles of a triangle is 180°. Therefore, the given triangle must be an equilateral triangle. Be-cause it is equilateral, we can break the triangle up to find the height of the triangle.

The diagram shows how the equilateral triangle can be partitioned into two 30°-60°-90° right triangles.

The length of each side of the new triangles across from the 30° angle is 5, half of 10.

That means the side across from the 60° angle is

That means that the base of the original triangle is 10, and the height is

Now find the area.

16. Answer: C

Factor the quadratic equation.

Notice that it is a perfect square.

⇒ + =x y 9.

x y+( ) =2

81

x xy y2 22 81+ + =

A bh= = =1

2

1

210 5 3 25 3i i .

5 3.

5 3.

60° 60°

60°

10

5 5

10

10

5√3

90°90°

30°30°


17. Answer: B

Find the slope of line r.

Line r passes through the points (3,2) and (0,−2) [y-intercept of −2]. The slope of line r is

Find the equation of line q.

Line r and line q are parallel, so the slope line q is also .

Line q passes through the point (3, 7).

The equation of line q is .

is the equation of the line with the same y-intercept as line q and perpendicular to line q.

18. Answer: B

Find the sum of A and C to find the average of A and C.

19. Answer: D

Draw a figure to show the height (2x) is twice the base (x).

Use the Pythagorean Theorem to solve for x in the right triangle.

Therefore, the base of the rectangle is 10, and the height of the rectangle is 20, so the area of the rectangle is A = b � h = 10 � 20 = 200.

20. Answer: C

Think of the positive integers less than 17.

The only two positive integers less than 17 that divide into 17 with a remainder of 3 are 7 and 14. Therefore, thesum of the possible value of k is 21. When 21 is divided by 17, the remainder is 4.

x x

x x

x

2 2 2

2 2

2

2 10 5

4 500

5 500

+ ( ) = ( )

+ =

=

That means

,, so x x2 100 10= ⇒ = .

x

2x10√5

Because then Thec a b a c a a b a b= + + = + +( ) = +2 2 2 2 . rrefore, the average of and isa ca b

a2 2

2

+= + bb.

y x= − +3

43

y x= +4

33

y mx b y x b b b b= + ⇒ = + ⇒ = ( ) + ⇒ = + ⇒ =4

37

4

33 7 4 3

4

3

2 2

3 0

4

3

− −( )

−= .


SECTION 5

1. Answer: D

2. Answer: A

Draw dimensions on the diagram.

Use the Pythagorean Theorem to find EG.

x = 10. Therefore, EG = 10. You might have noticed that EG = 10 because �EFG is a 3-4-5 right triangle.

3. Answer: A

The table shows that the number of students in a grade playing a varsity sport is twice the previous grade. Thatmeans the sophomore wedge should be twice the size as the freshmen wedge, the junior wedge should be twice thesize as the sophomore wedge, and the senior wedge should twice the size as the junior wedge.

FreshmanSophomoresJuniorsSeniors

Grade Frequency

Freshmen 10

Sophomores 20

Juniors 40

Seniors 80

x x2 2 2 26 8 36 64 100= + ⇒ = + =

x

8

6

F

G

G

FE

CB

A

E

D

H

x

8

6

x y x y+ = +( ) = ( ) =4 2 2 4 8, .then


4. Answer: D

Let s = number of shovels. Therefore, 10 + s = number of rakes. Write and solve a proportion to find the number ofshovels.

There are 15 shovels, which means that there are 25 rakes.

5. Answer: B

Write the dimensions and draw dotted segments as shown below.

Use formula for the area of triangle AMB to find the width of the rectangle (h = w, b = 3w).

Area of triangle

The base of the rectangle is three times the width, shown as

The length of CM____

is half the length of the base of the rectangle because M is the midpoint of the base CD____

. Therefore,

the length of CM____

is

6. Answer: C

Solve for a.

Find the inverse of a. ab

− =13

1

3.

436 4 36

4 36 9

9

2

24 2 4 2

2 6 2 6

6

a

bb a b b

a b a b

a b

= ⇒ =

= ⇒ =

=

i

.

⇒⇒ =a b3 3

6 3

23 3= .

3 2 3 6 3( ) = .w w2 12 12 4 3 2 3= ⇒ = = =i .

AMB w w w= = ⇒ =1

23 18 3 362i i

A 3w B

DC M

w

3

5 1030 3 5 30 2 15=

+⇒ + = ⇒ = ⇒ =

s

ss s s s.


7. Answer: A

Airplane A is flying at 200 miles per hour. So, from 10:00 AM to 10:30 AM, it will travel 100 miles. Airplane A isnow 700 − 100 = 600 miles from the airport.

Airplane B is flying at 250 miles per hour. So, from 10:00 AM to 10:30 AM, it will travel 125 miles. Airplane A isnow 925 − 125 = 800 miles from the airport.

The diagram shows the position of the two airplanes at 10:30 AM. The dotted line represents the distance the two airplanes are from each other.

Use the Pythagorean Theorem. d 2 = 6002 + 8002 ⇒ d 2 = 360,000 + 640,000 ⇒ d 2 = 1,000,000 ⇒ d = 1,000. The twoairplanes are 1,000 miles apart.

You could have also used the fact that this is a 3-4-5 right triangle.

8. Answer: D

Based on the graph we can see that f(2) = −4. So we are looking for a value of a where f(a) = 4.

Inspect answer choices.

Notice that only means f(a) = 4.

5

–5

0 2 4 6–2–4–6

y

x

(2,–4)

( ,4)12

a =1

2

d

A

BAirport 800 miles

600 miles


9. Answer: 4

Each of the three couples invites four guests, none of whom are the same; there are 3 � 4 = 12 guests. Then, we mustconsider the three couples who will attend the party, giving us 6 more people. There are a total of 18 people at theparty. Each table can seat five people. Three tables seat a maximum of 15 people, so it will take a minimum of fourtables to seat 18 people.

10. Answer: 6

Solve the absolute value inequality.

Notice that 6 is a solution to the second inequality but not to the first inequality.

11. Answer: 60

Use the diagram. Write 90 degrees for the right angle.

12. Answer: 10.5

You can use your calculator to guess and check to find that the ten consecutive integers are 6,7,8,9,10,11,12,13,14,and 15. These are ten numbers, the median is the average of the middle two numbers: 10 and 11. Therefore the me-dian of the two numbers is 10.5. The mean and median of consecutive integers are always equal.

xx

x

+ + =+ =

=

90 30 180120 180

60

30°

x° 90°

2 3 7 2 10 52 3 7

2 3 7 2 4 2

x x xx

x x x

x

− ≤ ⇒ ≤ ⇒ ≤− ≤

− ≥ − ⇒ ≥ − ⇒ ≥ −

.

.

−− ≤ ⇒ ≤− ≤

− ≥ − ⇒ ≥ −

2 4 62 4

2 4 2

xx

x x

.

.


13. Answer: 11

14. Answer: 10

Because AB____

≅ BC____

�ABC is an isosceles triangle. That means the measure of ∠A and the measure of ∠C are equal.A triangle has 180°, therefore, ∠A and ∠C each have a measure 25°. Therefore, x + 15 = 25 ⇒ x = 10.

15. Answer: or 0.4

16. Answer: 1.4 or

Write the equation

Divide to findy

xx y

y

x, . .

.

.1 19 0 85

1 19

0 851= ⇒ = ⇒ .. .4 =

y

x

x x y y+ = −. .19 15

75

25

x°

130° 35° 25°

15°

145°

BA

C

f x x

c x

f c c

( ) = +

+( )

+( ) = +

2 3

2

2 2 2

Substitute for .

(( ) + =

+ + = ⇒ + = ⇒ =

3 15

2 4 3 15 2 7 15 2 8

.

.Solve for c

c c c ⇒⇒ =

( )( ) = ( ) = ( ) + = + =

c

f c

f c f

4

4 2 4 3 8 3 11

.

.

.

Find

Original number of red marbles

in jar

Number of red marbles in jar

after adding three green 6

Probability of choosing a red

marble after three green have been

added, making a total of 15 marbles

615

25

=

12

12 6i =


17. Answer: 4

Write square root of r in exponent form.

Use the formula for volume of a cylinder.

18. Answer: 12

Use distance formula:

Square both sides.

Simplify.

Take square root of each side.

24 12 12= + ⇒ =x x.

576 122

= +( )x

676 100 12 576 122 2

= + +( ) ⇒ = +( )x x.

676 10 122 2

= −( ) + +( )x

26 3 13 122 2 2

2

( ) = −( ) + +( )( )x .

26 3 13 122 2

= −( ) + − −( )[ ]x

d x x y y= −( ) + −( )2 1

2

2 1

2

32 32 32 45

2

5

2

2

5

5

2

2

5π π= ⇒ = ⇒ ( ) = ( ) ⇒ =r r r r.

Substitute forr h

r r r r r

1

2

21

2

4

2

1

2

5

2

.

.π π π= =

V r h= π 2

r r h r= =1

2

1

2, .


SECTION 8

1. Answer: B

The only number that is both even and prime is 2. Therefore, the intersection of the set containing all the even num-bers and the set containing all the prime numbers is {2}.

2. Answer: E

Solve the equation.

Square both sides.

Solve for x.

3. Answer: E

Therefore, there were a total of 190 + 273 = 463 Not in Favor of the law.

4. Answer: E

Use the diagram.

We know the top angle in the bottom triangle is 40° because vertical angles are congruent.

The sum of the angles in a triangle is 180°.

180 − 40 = 140. That means x + y = 140.

y°x°

40°

40°

80°

Number Percentage Number Polled Not in Favor Not in Favor

Men 250 76% 0.76 � 250 = 190

Women 350 78% 0.78 � 350 = 273

x = 30

x x−( ) = ⇒ − =5 5 5 252

2

x x− + = ⇒ − =5 6 11 5 5


5. Answer: B

Inspect to eliminate choices A, C, and E, which are all less than 50%.

The 2000 price for houses B and D is the same, while the 1995 price is less for house B. That means there is agreater percent change for house B.

House B:

House D:

6. Answer: C

Because f(x) = f(−x), the graph of f(x) is symmetric with respect to the y-axis, as shown below. Inspect the graph tosee that f(1.5) = 2.

1.5

450 300

300

150

30050

−= = %

450 275

275

175

27563 63

−= = . %

House Price

0

100

200

300

400

500

50

150

250

350

450

A B C D E

House

Pri

ce

1995 Appraisal Price

2000 Appraisal Price


7. Answer: A

The key is to rewrite 9 as 32. 9a � 3c = (32)a � 3c = 32a � 3c = 32a + c = 177,147.

Use the guess and check method to find the power of 3 that will result is 177,147.

311 = 177,147. 32a + c = 177,147 so 2a + c = 11.

8. Answer: C

The distance between the center and a point on the circle is equal to the radius of the circle.

The radius of the circle is 13. The area of the circle is A = πr 2 = π(13)2 = 169π.

9. Answer: D

The arrows are pointing out. Choose a greater than inequality. The midpoint of −12 and 4 is −4. This is only true forChoice D. Check your solution.

–12 0 4

− − ⇒ − ⇒

− ⇒ − ⇒

x x xx

x x x

4

4

> 8 > 12 < −12.− − 4 > 8

− < −8 < −4 > 4.

r = − −( )[ ] + −( ) = + −( ) = + = =2 3 1 13 5 12 25 144 169 132 2 2 2

..

r

(–3,13)

(2,1)


10. Answer: D

Cylinder A:

Use the volume formula to find the radius of Cylinder A.

h = 4.

Cylinder B:

The radius of Cylinder A is 8, so

h = 8.

Use the volume formula to find the radius of Cylinder B.

11. Answer: E

Choose two values of k to find two points.

Use these points to find the slope of the line.

Find the y-intercept of the line using the point (−2,1).

Therefore, the equation of the line is

12. Answer: A

A, B, and C represent the populations of the towns.

A = 0.75B

C = 0.20B

Divide A by C.

The population of Town A is 375%. The population of Town C.

A

C

B

B= =

0 75

0 203 75

.

..

y x= −1

2.

y mx b y x b b b b= + ⇒ = − + ⇒ = − −( ) + ⇒ = + ⇒ =1

21

1

22 1 1 0 .

m =−

− − −( )=

−2 1

4 2

1

2

2 2 1 4 2= −( ) = −( ), , .and 4

V r h r

r r

= = ⇒ ( ) =

= ⇒ = =

π π π π2 2

2

256 8 256

32 32 4 2.

V r h r

r r

= = ⇒ ( ) =

= ⇒ =

π π π π2 2

2

256 4 256

64 8.


13. Answer: C

I. One of the numbers is odd

Let x be odd.

Let y and z be even.

y + z is even. An even plus an odd is odd, so x + y + z is odd.

II. Two of the numbers are odd.

Let x be even.

Let y and z be odd.

y + z is even ⇒ x + y + z is even.

III. Three of the numbers are odd.

Let x, y, and z be odd.

x + y is even ⇒ x + y + z is odd.

14. Answer: A

By choosing one value of x to test all three statements we can see that only statement I is true. Let x = −0.5. Remem-ber you are multiplying negative numbers.

I. (−0.5)3 > −(0.5) True: (−0.5)3 = −0.125, which is greater than (−0.5).

II. −0.5 > (−0.5)2 False: (−0.5)2 is positive.

III. (−0.5)3 > (−0.5)2 False: (−0.5)2 is positive; (−0.5)3 is negative.

15. Answer: C

Begin by sketching what appears to be the best fitting line. Notice that the y-intercept is between 65 and 70. Inthe equation of a line, y = mx + b, b represents the y-intercept. Since the y-intercept for choices A, B, and E is 24,these can be eliminated. The line has a positive slope. In the equation of a line y = mx + b, m represents the slope.For choice B, the slope is −24 and for choice C the slope is 24. Choice B can be eliminated leaving, leaving uswith choice C as the correct answer.

Study Hours

Gra

de

on

Tes

t

1.00.80.60.40.20.0

100

90

80

70

60

88

95

9086

9088

8580

85

77

80

70

75

60

Scatterplot of Grade on Test vs Study Hours


16. Answer: D

There are four congruent hexagons and four congruent isosceles triangles. Find the area of one isosceles triangle andone hexagon, add them together, and then multiply by 4.

There are four isosceles triangles in the original figure. Find the area of one of them. Partition the triangle into two30°-60°-90° triangles. Use the properties of 30°-60°-90° triangles to complete the diagram.

Use the formula to find the area.

A bh= = =1

2

1

210 3 5 25 3i i .

10

30° 30°

105

60°60°

5√3 5√3

120°120°

120°10

10


Hexagon:

Find the area of one of the central equilateral triangles and multiply that area by 6. We already know that the base

of the triangle is 10. Use the rules of a 30°-60°-90° right triangle. Find the height of the triangle to be The area

of the triangle is

Therefore, the sum of these two areas is so the area of the region is 700 3.

4 175 3i =25 3 150 3 175 3+ = ,

30°

60°

5√3

5 5

10

A = = ⇒ ( ) =1

210 5 3 25 3 6 25 3 15× × Area of hexagon is 00 3.

5 3.

conquering sat math practice test 1 answers

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