SAT Math - Full-Length Practice Test 3

Calculator is allowed on all questions. Figures are not necessarily drawn to scale.

Module 1

Question 1

If \(5x+8=33\), what is the value of \(x\)?

A) \(3\)
B) \(4\)
C) \(5\)
D) \(8\)

Show solution

Subtract 8 to get \(5x=25\), so \(x=5\).

Answer: C

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Question 2

If \(h(x)=2x^2+3\), what is \(h(2)\)?

A) \(7\)
B) \(11\)
C) \(14\)
D) \(19\)

Show solution

\(h(2)=2(2^2)+3=11\).

Answer: B

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Question 3

A rectangle has length 14 and width 9. What is its area, in square units?

A) \(23\)
B) \(46\)
C) \(112\)
D) \(126\)

Show solution

Use the rectangle area formula \(A=lw\).

Substitute \(l=14\) and \(w=9\): \(A=14(9)=126\).

Answer: D

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Question 4

If \(3(x-2)=21\), enter the value of \(x\).

Enter your answer:

Show solution

Divide by 3: \(x-2=21/3=7\).

Add 2: \(x=7+2=9\).

Answer: 9

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Question 5

A cyclist travels 36 miles in 3 hours at a constant rate. At this rate, how many miles does the cyclist travel in 5 hours?

A) \(15\)
B) \(45\)
C) \(54\)
D) \(60\)

Show solution

The unit rate is \(36/3=12\) miles per hour.

In 5 hours, the cyclist travels \(12(5)=60\) miles.

Answer: D

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Question 6

Which expression is equivalent to \(x^2+7x+12\)?

A) \((x+2)(x+6)\)
B) \((x-3)(x-4)\)
C) \((x+3)(x+4)\)
D) \((x-3)(x+4)\)

Show solution

The factors of 12 that add to 7 are 3 and 4: \(x^2+7x+12=(x+3)(x+4)\).

Answer: C

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Question 7

Which inequality is equivalent to \(2x-7>9\)?

A) \(x>1\)
B) \(x>8\)
C) \(x<8\)
D) \(x>16\)

Show solution

Add 7 to get \(2x>16\), then divide by 2: \(x>8\).

Answer: B

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Question 8

The right triangle shown has side lengths 6, 8, and 10. Which side is the hypotenuse?

A) \(6\)
B) \(8\)
C) \(10\)
D) \(14\)

Show solution

The hypotenuse is the side opposite the right angle and is the longest side.

Its length is 10.

Answer: C

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Question 9

What is the slope of the line through \((1,4)\) and \((5,12)\)?

A) \(\frac12\)
B) \(1\)
C) \(2\)
D) \(4\)

Show solution

Use the slope formula \(m=\frac{y_2-y_1}{x_2-x_1}\).

Substitute the two points: \(m=\frac{12-4}{5-1}=\frac84=2\).

Answer: C

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Question 10

The function \(f(x)=(x-7)(x+2)\) has two zeros. Enter the positive zero.

Enter your answer:

Show solution

A product is zero when a factor is zero.

The zeros are 7 and \(-2\).

The positive zero is 7.

Answer: 7

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Question 11

A jacket originally costs \(80\) and is discounted by 25%. What is the sale price?

A) \(20\)
B) \(60\)
C) \(65\)
D) \(75\)

Show solution

A 25% discount is \(0.25(80)=20\).

The sale price is \(80-20=60\).

Answer: B

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Question 12

What is the \(x\)-coordinate of the solution to \(x+y=11\) and \(x-y=3\)?

A) \(3\)
B) \(4\)
C) \(7\)
D) \(8\)

Show solution

Adding the equations gives \(2x=14\), so \(x=7\).

Answer: C

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Question 13

For \(x\ne -5\), which expression is equivalent to \(\frac{x^2+7x+10}{x+5}\)?

A) \(x+2\)
B) \(x+5\)
C) \(x+7\)
D) \(x^2+2\)

Show solution

Factor the numerator: \((x+5)(x+2)\).

Cancel \(x+5\) to get \(x+2\).

Answer: A

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Question 14

The table shows students by club membership. What fraction of the students who are in Band are juniors?

	Band	Not Band
Juniors	12	18
Seniors	8	12

A) \(\frac{12}{50}\)
B) \(\frac{18}{50}\)
C) \(\frac{12}{20}\)
D) \(\frac35\)

Show solution

Among Band students, there are \(12+8=20\) total, and 12 are juniors.

Thus the fraction is \(12/20=3/5\).

Answer: D

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Question 15

A taxi ride costs a \(4\) starting fee plus \(2.50\) per mile. Which equation gives the total cost \(C\) for \(m\) miles?

A) \(C=2.5m+4\)
B) \(C=4m+2.5\)
C) \(C=6.5m\)
D) \(C=2.5(m+4)\)

Show solution

The fixed fee is the intercept 4 and the per-mile charge is the slope 2.5: \(C=2.5m+4\).

Answer: A

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Question 16

The graph of a quadratic function has \(x\)-intercepts at \(-2\) and 6. Which equation gives its axis of symmetry?

A) \(x=1\)
B) \(x=2\)
C) \(x=3\)
D) \(x=4\)

Show solution

The axis of symmetry is halfway between the intercepts: \((-2+6)/2=2\), so \(x=2\).

Answer: B

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Question 17

The mean of five numbers is 18. Four of the numbers are 12, 17, 20, and 24. Enter the fifth number.

Enter your answer:

Show solution

The total of five numbers is \(5(18)=90\).

The four known numbers sum to 73.

The fifth is \(90-73=17\).

Answer: 17

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Question 18

A linear function satisfies \(f(2)=9\) and \(f(6)=21\). What is \(f(10)\)?

A) \(25\)
B) \(27\)
C) \(30\)
D) \(33\)

Show solution

Use the slope formula \(m=\frac{y_2-y_1}{x_2-x_1}\).

The slope is \(m=\frac{21-9}{6-2}=\frac{12}{4}=3\).

Increasing \(x\) from 6 to 10 increases \(f(x)\) by \(3(4)=12\), so \(f(10)=21+12=33\).

Answer: D

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Question 19

A quantity is multiplied by \(1.21\) every 2 years. If the same growth is expressed as multiplication by \((1+r)\) each year, enter \(100r\).

Enter your answer:

Show solution

The annual factor is \(\sqrt{1.21}=1.10\): \(r=0.10\) and \(100r=10\).

Answer: 10

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Question 20

A circle has diameter 18, as shown. What is its area?

A) \(18\pi\)
B) \(36\pi\)
C) \(81\pi\)
D) \(324\pi\)

Show solution

Use the circle area formula \(A=\pi r^2\).

Substitute \(r=9\): \(A=\pi(9^2)=81\pi\).

Answer: C

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Question 21

The two right triangles shown are similar. What is the value of \(x\)?

A) \(12\)
B) \(15\)
C) \(16\)
D) \(18\)

Show solution

The scale factor from the smaller triangle to the larger is \(20/8=2.5\).

Thus \(x=6(2.5)=15\).

Answer: B

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Question 22

The equation \(|2x-5|=9\) has two solutions. What is the sum of the solutions?

A) \(5\)
B) \(7\)
C) \(9\)
D) \(14\)

Show solution

The solutions satisfy \(2x-5=9\) or \(2x-5=-9\), giving \(x=7\) and \(x=-2\).

Their sum is 5.

Answer: A

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Module 2

Question 1

Which equation represents a line with slope \(\frac34\) and \(y\)-intercept \(-5\)?

A) \(y=-5x+\frac34\)
B) \(y=-\frac34x-5\)
C) \(y=\frac43x-5\)
D) \(y=\frac34x-5\)

Show solution

In \(y=mx+b\), \(m=3/4\) and \(b=-5\): \(y=\frac34x-5\).

Answer: D

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Question 2

A bag contains 5 red, 7 blue, and 8 green marbles. One marble is chosen at random. What is the probability it is blue?

A) \(\frac14\)
B) \(\frac7{20}\)
C) \(\frac25\)
D) \(\frac7{13}\)

Show solution

Use \(P(\text{event})=\frac{\text{favorable outcomes}}{\text{total outcomes}}\).

There are 20 marbles total and 7 are blue, so \(P(\text{blue})=\frac{7}{20}\).

Answer: B

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Question 3

If \(p(x)=x^3-2x\), what is \(p(-2)\)?

A) \(-4\)
B) \(0\)
C) \(4\)
D) \(12\)

Show solution

\(p(-2)=(-2)^3-2(-2)=-8+4=-4\).

Answer: A

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Question 4

Two angles are supplementary. If one angle measures \(137^\circ\), what is the measure of the other?

A) \(37^\circ\)
B) \(40^\circ\)
C) \(43^\circ\)
D) \(53^\circ\)

Show solution

Supplementary angles sum to \(180^\circ\).

The other angle is \(180-137=43^\circ\).

Answer: C

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Question 5

The system \(2x+y=16\) and \(x-y=2\) has solution \((x,y)\). Enter \(x\).

Enter your answer:

Show solution

Adding the equations gives \(3x=18\), so \(x=6\).

Answer: 6

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Question 6

Which are the solutions to \(x^2-5x-14=0\)?

A) \(-7\) and \(2\)
B) \(-2\) and \(-7\)
C) \(-2\) and \(7\)
D) \(2\) and \(7\)

Show solution

Factor: \(x^2-5x-14=(x-7)(x+2)\): \(x=7\) or \(x=-2\).

Answer: C

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Question 7

A circle has center \((3,-2)\) and radius 5. Which point lies on the circle?

A) \((6,2)\)
B) \((8,2)\)
C) \((3,4)\)
D) \((-1,3)\)

Show solution

For \((6,2)\), the displacement from the center is \((3,4)\), whose length is \(\sqrt{3^2+4^2}=5\).

Answer: A

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Question 8

The scatterplot shown has which type of association?

A) Strong negative
B) No association
C) Weak negative
D) Strong positive

Show solution

As \(x\) increases, \(y\) generally increases, and the points stay fairly close to an upward trend.

This indicates a strong positive association.

Answer: D

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Question 9

The formula \(A=\frac12bh\) gives the area of a triangle. Which equation expresses \(h\) in terms of \(A\) and \(b\)?

A) \(h=\frac{A}{2b}\)
B) \(h=\frac{2A}{b}\)
C) \(h=\frac{b}{2A}\)
D) \(h=2Ab\)

Show solution

Multiply by 2: \(2A=bh\).

Divide by \(b\): \(h=\frac{2A}{b}\).

Answer: B

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Question 10

If \((x+3)^2=64\) and \(x\) is positive, enter \(x\).

Enter your answer:

Show solution

\(x+3=\pm8\), so \(x=5\) or \(x=-11\).

The positive solution is 5.

Answer: 5

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Question 11

In a right triangle, \(\cos\theta=\frac{12}{13}\). If the side adjacent to \(\theta\) has length 24, what is the hypotenuse length?

A) \(13\)
B) \(24\)
C) \(26\)
D) \(36\)

Show solution

\(24/H=12/13\).

Since 24 is twice 12, the hypotenuse is twice 13, or 26.

Answer: C

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Question 12

Which expression is equivalent to \(3(2x-5)+x\)?

A) \(7x-15\)
B) \(6x-15\)
C) \(7x-5\)
D) \(5x-15\)

Show solution

Distribute and combine like terms: \(3(2x-5)+x=6x-15+x=7x-15\).

Answer: A

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Question 13

A value decreases from 320 to 272. What is the percent decrease? Enter the number only.

Enter your answer:

Show solution

The decrease is 48.

Since \(48/320=0.15\), the percent decrease is 15%.

Answer: 15

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Question 14

For what value of \(k\) does the system \(4x+6y=10\) and \(6x+9y=k\) have infinitely many solutions?

A) \(10\)
B) \(12\)
C) \(15\)
D) \(20\)

Show solution

Multiplying the first equation by \(3/2\) gives \(6x+9y=15\), so \(k=15\).

Answer: C

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Question 15

A quadratic function has zeros at \(-5\) and 3 and satisfies \(f(0)=-30\). Which equation defines \(f\)?

A) \(f(x)=(x+5)(x-3)\)
B) \(f(x)=2(x+5)(x-3)\)
C) \(f(x)=-2(x+5)(x-3)\)
D) \(f(x)=2(x-5)(x+3)\)

Show solution

Write \(f(x)=a(x+5)(x-3)\).

Then \(-15a=-30\), so \(a=2\).

Answer: B

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Question 16

A right circular cylinder has radius 3 and volume \(108\pi\). Enter its height.

Enter your answer:

Show solution

Using \(V=\pi r^2h\), \(108\pi=9\pi h\), so \(h=12\).

Answer: 12

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Question 17

Which equation is equivalent to \(x^2+6x-7=0\)?

A) \((x+3)^2=7\)
B) \((x-3)^2=16\)
C) \((x+6)^2=16\)
D) \((x+3)^2=16\)

Show solution

Move \(-7\) to the other side: \(x^2+6x=7\).

Add 9 to both sides to get \((x+3)^2=16\).

Answer: D

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Question 18

A school wants to estimate the average number of hours its students sleep each night. Which sampling method is least likely to be biased?

A) Randomly select students from the complete enrollment list.
B) Survey students arriving before 7:00 a.m.
C) Survey members of the athletics teams.
D) Invite students to respond voluntarily on social media.

Show solution

A random sample from the complete enrollment list gives all enrolled students a chance to be selected and avoids targeting a subgroup.

Answer: A

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Question 19

For \(x\ne2\), \(\frac{3}{x-2}+1=\frac{x}{x-2}\). Which statement is true?

A) The equation has solution \(x=1\).
B) The equation has solution \(x=3\).
C) The equation has two real solutions.
D) The equation has no solution.

Show solution

Multiply by \(x-2\): \(3+(x-2)=x\), which simplifies to \(x+1=x\), an impossibility.

Therefore there is no solution.

Answer: D

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Question 20

For a constant \(a\), the equation \(a(x-4)=3x-12\) is true for every real value of \(x\). Enter \(a\).

Enter your answer:

Show solution

The right side factors as \(3(x-4)\).

Therefore \(a=3\) makes the two sides identical for every \(x\).

Answer: 3

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Question 21

The roots of \(x^2-14x+c=0\) differ by 8. What is \(c\)?

A) \(21\)
B) \(33\)
C) \(40\)
D) \(49\)

Show solution

The roots sum to 14.

their midpoint is 7.

If they differ by 8, the roots are 3 and 11.

Their product is \(c=33\).

Answer: B

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Question 22

A culture starts with 800 cells and doubles every 6 hours. It is modeled by \(800(2)^{t/6}=800(b)^t\). Enter \(1000(b-1)\) rounded to the nearest whole number.

Enter your answer:

Show solution

\(b=2^{1/6}\approx1.12246\).

Thus \(1000(b-1)\approx122.46\), which rounds to 122.

Answer: 122

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