SAT Math

SAT Math Practice Test: Sat Practice Test 2

Sat Practice Test 2 for SAT Math: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

A runner completes 5 kilometers in 24 minutes at a constant pace. At this pace, how many minutes will it take the runner to complete 8 kilometers?

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Question 1

A runner completes 5 kilometers in 24 minutes at a constant pace. At this pace, how many minutes will it take the runner to complete 8 kilometers?

30.0 min
38.4 min (correct answer)
40.0 min
96.0 min

Explanation: We need to find how many minutes it takes to run 8 kilometers at a constant pace. First, find the rate: 24 minutes ÷ 5 km = 4.8 minutes per kilometer. For 8 kilometers: 8 km × 4.8 min/km = 38.4 minutes. The key is using minutes per kilometer, not kilometers per minute. A common error is using 24/5 as km/min instead of min/km. When setting up rates, make sure the units in your rate match what you're solving for.

Question 2

A square has a diagonal that measures $10\sqrt{2}$ cm. What is the length of each side of the square?

10 cm (correct answer)
20 cm
5 cm
5 $\sqrt{2}$ cm

Explanation: This problem involves finding the side length of a square given its diagonal measurement. A square's diagonal creates two congruent 45-45-90 right triangles, where the sides are in the ratio 1:1: $\sqrt{2}$ (side:side:diagonal). If the diagonal is $10\sqrt{2}$ cm and the diagonal equals side × $\sqrt{2}$ , then side × $\sqrt{2}$ = $10\sqrt{2}$ , so the side length is 10 cm.

Question 3

A composite floor plan consists of a $12\text{ ft} \times 9\text{ ft}$ rectangle with a $6\text{ ft} \times 4\text{ ft}$ rectangular alcove removed from one corner. What is the area of the remaining floor?

$24\text{ ft}^2$
$84\text{ ft}^2$ (correct answer)
$108\text{ ft}^2$
$132\text{ ft}^2$

Explanation: This problem asks for the area of a floor plan where a 6 ft × 4 ft alcove is removed from a 12 ft × 9 ft rectangle. Area of large rectangle = 12 × 9 = 108 ft². Area of removed alcove = 6 × 4 = 24 ft². Remaining area = 108 - 24 = 84 ft². A common error is adding instead of subtracting the alcove area, or misunderstanding which piece is being removed. For composite figures with removed sections, subtract the removed area from the original.

Question 4

In a $30^\circ$ - $60^\circ$ - $90^\circ$ right triangle, the hypotenuse has length $18$ . What is the length of the longer leg?

$9$
$9\sqrt{3}$ (correct answer)
$18\sqrt{3}$
$6\sqrt{3}$

Explanation: This problem asks for the longer leg in a 30°-60°-90° triangle with hypotenuse 18. In a 30°-60°-90° triangle, the sides are in ratio 1 : √3 : 2, where the hypotenuse is 2x, the shorter leg is x, and the longer leg is x√3. Since hypotenuse = 18 = 2x, we get x = 9, so the longer leg = 9√3. Common errors include using 9 as the answer (the shorter leg) or multiplying incorrectly to get 18√3. Memorizing the 30°-60°-90° ratio pattern helps solve these problems quickly.

Question 5

A jar contains 6 red marbles, 5 blue marbles, and 4 green marbles. Two marbles are drawn at random without replacement. What is the probability that both marbles drawn are blue?

$\frac{1}{9}$
$\frac{5}{15}$
$\frac{2}{7}$
$\frac{2}{21}$ (correct answer)

Explanation: This problem asks for the probability of drawing two blue marbles without replacement from a jar containing 6 red, 5 blue, and 4 green marbles (15 total). For the first draw, P(blue) = 5/15. After removing one blue marble, 14 marbles remain with only 4 blue, so P(second blue|first blue) = 4/14. The probability of both events is P(both blue) = (5/15) × (4/14) = 20/210 = 2/21. A key error is forgetting to adjust the counts for the second draw or treating it as with replacement. When drawing without replacement, always update both the favorable outcomes and total count.

Question 6

A water tank holds 120 liters at noon and drains at a constant rate of 3 liters per minute. The volume after $t$ minutes is modeled by $V(t)=120-3t$ . How much water remains after 15 minutes?

45
40
165
75 (correct answer)

Explanation: Substitute $t=15$ : $V(15)=120-3(15)=120-45=75$ . The values 45 and 165 come from subtracting or adding incorrectly, and 40 results from dividing instead of subtracting per minute.

Question 7

In a circle with center $O$ , chord $$$AB$$ $has length$ 24 $. The perpendicular distance from$ O $to chord$ AB $is$ 5$. What is the radius of the circle?

$11$
$13$ (correct answer)
$12$
$17$

Explanation: The question involves finding the radius of a circle given a chord length and the perpendicular distance from the center to the chord. When a perpendicular from the center meets a chord, it bisects the chord, creating a right triangle with the radius as hypotenuse. Half the chord length is $\frac{24}{2} = 12$ , and the perpendicular distance is 5. Using the Pythagorean theorem: $r^2 = 12^2 + 5^2 = 144 + 25 = 169$ , so $r = 13$ . A common error is using the full chord length instead of half in the Pythagorean theorem. Remember that the perpendicular from the center always bisects the chord.

Question 8

A student investigated whether the number of hours of sleep (hours) is related to reaction time (milliseconds) for 9 trials. The scatterplot shows sleep on the x-axis and reaction time on the y-axis. Which statement best describes the pattern?

The scatterplot proves sleeping more causes faster reactions for everyone.
Reaction time is constant regardless of sleep (no association).
Reaction time decreases as sleep increases (negative association). (correct answer)
Reaction time increases as sleep increases (positive association).

Explanation: The question asks which statement describes the pattern in nine paired measurements of sleep and reaction time. Reading the plot from left to right, the points drift downward: trials with more sleep go with lower reaction times, so the description of reaction time decreasing as sleep increases, a negative association, fits the data. The statement that reaction time increases as sleep increases reverses the direction of the trend. The statement that reaction time is constant regardless of sleep would require a roughly flat, patternless cloud of points, which is not what a clear downward drift looks like. And the statement that the plot proves more sleep causes faster reactions for everyone overreaches twice: nine trials in one student's investigation show an association rather than causation, and no scatterplot can establish that a pattern holds for every person.

Question 9

A school tracked, for 10 students, the number of absences (days) and their final course grade (percent). The scatterplot shows the data and a dashed line of best fit. For the student with $x=8$ absences, approximately how much greater is the actual grade than the grade predicted by the line of best fit?

About 2 percentage points
About 8 percentage points (correct answer)
About 15 percentage points
About 25 percentage points

Explanation: This question asks how much greater the actual grade is compared to the predicted grade for a student with 8 absences. First, we locate x = 8 on the horizontal axis and find both the actual data point and the predicted value on the line. The actual data point at x = 8 appears to be at approximately 68%, while the line of best fit predicts about 60% for 8 absences. The difference is 68% - 60% = 8 percentage points, making the actual grade about 8 percentage points greater than predicted. This represents a positive residual, where the student performed better than the model predicted. When calculating residuals, always subtract predicted from actual: residual = actual - predicted.

Question 10

A square-root function is defined by $f(x)=\sqrt{5-2x}$ . What is the domain of $f$ in real numbers? Choose the option that correctly applies the restriction on the expression under the radical.

$x\le \tfrac{5}{2}$ (correct answer)
$x<\tfrac{5}{2}$
$x\ge \tfrac{5}{2}$
$(-\infty,\infty)$

Explanation: This question asks for the domain of f(x) = √(5 - 2x). For square root functions, the expression under the radical must be non-negative (≥ 0) for real number outputs. We need 5 - 2x ≥ 0, which gives us 5 ≥ 2x, or x ≤ 5/2. The domain is all real numbers x such that x ≤ 5/2, written as (-∞, 5/2] or x ≤ 5/2. A common error is solving the inequality incorrectly or using the wrong inequality direction. When finding domains of radical functions, set up the inequality for the radicand ≥ 0 and solve carefully.

Question 11

A science class measured plant height each week. The line graph shows height (in cm) for Weeks 0, 1, 2, 3, and 4. If the trend from Week 3 to Week 4 continues for one more week at the same weekly increase, what height would you predict for Week 5? Use the last observed increase as the rate for one more week.

18 cm
19 cm
20 cm (correct answer)
21 cm

Explanation: The question asks for a Week 5 prediction based on the Week 3 to Week 4 trend. From the graph, Week 3 shows 16 cm and Week 4 shows 18 cm, an increase of 2 cm. If this continues, Week 5 would be 18 + 2 = 20 cm. Students often use the wrong weeks to establish the trend or forget to add the increase to the Week 4 value.

Question 12

A photocopier enlarges images so that every length is multiplied by $1.25$ . A logo's original area is $64\text{ cm}^2$ . What is the area of the enlarged logo in square centimeters?

80
100 (correct answer)
90
64

Explanation: When a photocopier enlarges with scale factor 1.25 for linear dimensions, areas are multiplied by (1.25)². The original area is 64 cm², so the enlarged area is 64 × (1.25)² = 64 × 1.5625 = 100 cm². A critical concept: when linear dimensions are scaled by factor k, areas scale by k². Common errors include multiplying area by 1.25 instead of 1.25² (giving 80), or adding 25% to 64 (giving 80). Remember that area scaling involves squaring the linear scale factor, and volume scaling involves cubing it.

Question 13

If $5\sqrt{\dfrac{x}{2}}=20$ , what is the value of $3x$ ?

$16$
$32$
$48$
$96$ (correct answer)

Explanation: Divide both sides by $5$ to get $\sqrt{\dfrac{x}{2}}=4$ . Squaring gives $\dfrac{x}{2}=16$ , so $x=32$ and $3x=96$ . Answering $16$ stops at $\dfrac{x}{2}$ and answering $48$ triples it instead of tripling $x$ .

Question 14

A store sells notebooks for $n$ dollars each and pens for $p$ dollars each. A customer buys 4 notebooks and 3 pens for a total of $27$ dollars, so $4n+3p=27$ . What is $p$ in terms of $n$ ?

$p=\dfrac{27-4n}{3}$ (correct answer)
$p=\dfrac{27+4n}{3}$
$p=\dfrac{4n-27}{3}$
$p=27-12n$

Explanation: We need to express p in terms of n from the equation $4n + 3p = 27$ . To isolate p, first subtract 4n from both sides: $3p = 27 - 4n$ . Then divide both sides by 3: $p = \frac{27 - 4n}{3}$ . This tells us the price of each pen in terms of the notebook price. A common mistake is forgetting to divide all terms by 3 or incorrectly handling the subtraction. When solving for one variable in terms of another, treat the other variable as a constant and use standard algebraic manipulation.

Question 15

A circle has radius $12$ cm. What is the length of an arc that subtends a central angle of $150^\circ$ , in centimeters? (Use $\pi$ in your answer.)

$25\pi$
$10\pi$ (correct answer)
$5\pi$
$20\pi$

Explanation: This question asks for the arc length given the radius and central angle. The arc length formula is L = (θ/360°) × 2πr, where θ is the central angle in degrees. Substituting θ = 150° and r = 12 cm: L = (150/360) × 2π(12) = (5/12) × 24π = 10π cm. A common mistake is forgetting to use the circumference (2πr) as the base for the calculation. Remember that arc length is a fraction of the circumference, not the area.

Question 16

A class has 10 quiz scores with a mean of 84. After discarding the lowest score of 70, what is the mean of the remaining 9 scores, to the nearest tenth?

84
85.6 (correct answer)
86
86.7

Explanation: Total is 10×84=840; removing 70 leaves 770, and 770/9≈85.6. Other choices reflect not adjusting the total correctly or dividing by the wrong count.

Question 17

A population of deer is shown in the graph of $P$ versus $t$ (years). The curve rises slowly at first and then more rapidly. Which model is more appropriate, and which feature of the graph supports your choice?

Linear; straight-line trend
Exponential; increasing rate of change (correct answer)
Linear; constant ratio each year
Exponential; constant difference each year

Explanation: This question describes a graph that rises slowly at first and then more rapidly, which is characteristic of exponential growth. Exponential functions show an increasing rate of change - the curve gets steeper as time progresses because each increase is a percentage of the current value. Linear functions maintain a constant rate of change and would appear as a straight line. The key feature supporting exponential growth is the accelerating rate of increase. A common error is choosing linear because of confusion about what "constant ratio" means. For visual identification: if the curve bends upward (concave up), it's likely exponential growth.

Question 18

A recipe calls for between 2 and 5 cups of flour, inclusive. If $x$ is the number of cups of flour used, which compound inequality matches the requirement?

$2<x<5$
$2\le x\le 5$ (correct answer)
$x\le 2\text{ or }x\ge 5$
$2\le x<5$

Explanation: This question asks for a compound inequality representing "between 2 and 5 cups of flour, inclusive." The word "inclusive" is crucial—it means both endpoints (2 and 5) are included in the acceptable range. This translates to $2 ≤ x ≤ 5$ , which reads as "x is greater than or equal to 2 AND less than or equal to 5." Without the word "inclusive," we would use strict inequalities ( $2 < x < 5$ ), excluding the endpoints. A common mistake is using "or" instead of "and"—the inequality $x ≤ 2$ or $x ≥ 5$ would mean flour amounts outside the 2-5 range, which is the opposite of what we want. When you see "between" in a problem, it typically means a compound inequality with "and," not "or."

Question 19

A train travels 180 miles in 2.5 hours. At the same speed, how far will it travel in 4 hours? Mistakes include dividing 180 by 4 or using 2.5 as if it were miles per hour.

144 miles
240 miles
288 miles (correct answer)
450 miles

Explanation: We need to find distance traveled in 4 hours at a constant speed. First, calculate speed: 180 miles ÷ 2.5 hours = 72 miles per hour. In 4 hours: 72 mph × 4 hours = 288 miles. The key is finding speed first, then using it to calculate the new distance. Common mistakes include dividing 180 by 4 or using 2.5 as if it were the speed. For distance problems, remember: distance = speed × time.

Question 20

A bag contains 3 black socks and 7 white socks. Two socks are selected at random without replacement. What is the probability that at least one of the socks selected is black?

$\frac{7}{15}$
$\frac{8}{15}$ (correct answer)
$\frac{3}{10}$
$\frac{8}{45}$

Explanation: This problem asks for P(at least one black sock) when drawing 2 socks from 3 black and 7 white (10 total) without replacement. It's easier to use the complement: P(at least one black) = 1 - P(both white). P(both white) = (7/10) × (6/9) = 42/90 = 7/15. Therefore, P(at least one black) = 1 - 7/15 = 8/15. The complement approach is often simpler than calculating P(exactly one black) + P(both black). When you see "at least one," consider using the complement rule with "none."

Question 21

A store recorded the number of items sold each hour: 4, 5, 5, 6, 6, 6, 7, 30. Which measure of center is most affected by the outlier, and what happens to it when the outlier is included?

Median; it increases a lot
Mean; it increases (correct answer)
Mode; it becomes 30
Range; it decreases

Explanation: This question asks which measure of center is most affected by the outlier 30 in the data: 4, 5, 5, 6, 6, 6, 7, 30. The mean without the outlier would be about 5.7, but with it: (4+5+5+6+6+6+7+30)÷8 = 69÷8 = 8.625, showing a large increase. The median would be 6 without the outlier and remains 6 with it (middle values of ordered data). The mode stays at 6 (most frequent value) regardless. The mean is most sensitive to outliers because it uses all values in its calculation. When analyzing data with outliers, consider reporting both mean and median to show the full picture.

Question 22

Researchers recruited 80 volunteers from a college gym and recorded each person's weekly exercise hours and resting heart rate. They found a negative correlation ( $r=-0.62$ ). The study did not randomly assign exercise amounts, and diet and sleep were not measured. Which claim is best supported?

For these gym volunteers, higher reported exercise hours tend to be associated with lower resting heart rate, but the study cannot show exercise alone caused the lower rates. (correct answer)
Exercising more hours per week will lower resting heart rate for every adult, because a correlation of $-0.62$ proves a strong causal relationship.
The negative correlation means that people with lower resting heart rates choose to exercise less, because correlation indicates the direction of cause.
Because $r$ is negative, there is no relationship between exercise and heart rate in the sample, since only positive correlations indicate association.

Explanation: This question tests understanding of correlation versus causation with exercise and heart rate data. The study found r = -0.62 among 80 gym volunteers, showing a moderate negative correlation—as exercise hours increase, resting heart rate tends to decrease. Choice A correctly states that higher exercise is associated with lower heart rate in these volunteers but cannot prove exercise alone caused it, since the study didn't randomly assign exercise amounts or control for diet and sleep. Choice B wrongly claims causation and overgeneralizes, choice C reverses the causal direction without justification, and choice D misunderstands that negative correlations do show relationships (inverse ones). The key insight is that correlation, even strong correlation, doesn't prove causation—healthier people might both exercise more and have lower heart rates due to genetics or overall lifestyle. Always distinguish between observational associations and causal conclusions.

Question 23

A copy center charges $18 for 120 pages of printing. At this rate, how much will it cost to print 350 pages?

$\$ 42.00
$\$ 52.50 (correct answer)
$\$ 54.00
$\$ 63.00

Explanation: We need to find the cost for 350 pages when 120 pages cost $18. First, find the cost per page: $18 ÷ 120 pages = $0.15 per page. For 350 pages: $0.15/page × 350 pages = $52.50. To verify: 350/120 = 2.917, and $18 × 2.917 ≈ $52.50. A common error is rounding the per-page rate too early, leading to accumulated error. Keep extra decimal places in intermediate calculations.

Question 24

A salesperson earns a weekly base pay plus a commission per item sold. In a week with 12 items sold, the pay is $460, and in a week with 20 items sold, the pay is $620. Assuming a linear relationship, what is the commission per item?

$18
$22
$20 (correct answer)
$16

Explanation: Let b be the base pay and c be the commission per item. We have two equations: $b + 12c = 460$ and $b + 20c = 620$ . Subtracting the first from the second: $8c = 160$ , so $c = 20$ . The commission per item is $20$ . To verify: $b = 460 - 12(20) = 460 - 240 = 220$ , and checking: $220 + 20(20) = 220 + 400 = 620$ ✓. A common error is setting up the equations incorrectly or making arithmetic mistakes when solving the system. When dealing with pay structures, clearly define your variables before writing equations.

Question 25

A store discounts an item by $15\%$ , so the sale price is $0.85x$ . If the sale price is $51$ , what is the original price $x$ ? Solve the equation $0.85x=51$ .

$x=60$ (correct answer)
$x=55$
$x=58$
$x=65$

Explanation: This question asks for the original price x where the sale price after 15% discount is 0.85x = 51. Divide both sides by 0.85: x = 51 / 0.85. Calculating, 51 / 0.85 = 5100 / 85 = 60. A common error is using 0.15 instead of 0.85, like dividing 51 by 0.15 to get 340, which is irrelevant. Another mistake might be adding 15% incorrectly without setting up the equation. For test-taking, solve decimal equations by converting to fractions if needed, like 51 / (85/100) = 51 * (100/85) = 5100/85=60, and check: 0.85*60=51.