SAT Math

SAT Math Practice Test: Sat Practice Test 6

Sat Practice Test 6 for SAT Math: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

In right triangle $\triangle ABC$ with $\angle C=90^\circ$ and $\angle A=52^\circ$ , the leg opposite $\angle A$ is $BC=19$ . What is the hypotenuse $AB$ to the nearest tenth?

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Question 1

In right triangle $\triangle ABC$ with $\angle C=90^\circ$ and $\angle A=52^\circ$ , the leg opposite $\angle A$ is $BC=19$ . What is the hypotenuse $AB$ to the nearest tenth?

$15.0$
$24.1$ (correct answer)
$30.8$
$23.4$

Explanation: We need the hypotenuse AB given angle A = 52° and the opposite side BC = 19. Using sine: sin(52°) = opposite/hypotenuse = 19/AB. Solving for AB: AB = 19/sin(52°) = 19/0.788 ≈ 24.1. A common error is using the wrong trig ratio - since we know the opposite side and need the hypotenuse, sine is the correct choice. When you have an angle and its opposite side, and need the hypotenuse, always use sine.

Question 2

A shipping cost is modeled by $\frac{x}{4}+\frac{x}{6}=10$ , where $x$ is the weight in pounds. What is the value of $x$ ?

$24$ (correct answer)
$60$
$40$
$12$

Explanation: We need to solve $x/4 + x/6 = 10$ by finding a common denominator for the fractions. The LCD of 4 and 6 is 12, so we rewrite: $(3x/12) + (2x/12) = 10$ , which gives us $5x/12 = 10$ . Multiplying both sides by 12 yields $5x = 120$ , so $x = 24$ . A common error is to add denominators directly (getting $x/10 = 10$ , thus $x = 100$ ) or to use 24 as the LCD instead of 12. When adding fractions, always find the least common denominator first.

Question 3

A rectangular poster measures $24$ inches by $18$ inches. What is the area of the poster in square feet?

3 ft $^2$ (correct answer)
36 ft $^2$
2.5 ft $^2$
432 ft $^2$

Explanation: We need to find the area of a 24 inch by 18 inch poster in square feet. First, convert each dimension to feet: 24 in × (1 ft/12 in) = 2 ft and 18 in × (1 ft/12 in) = 1.5 ft. Then calculate the area: 2 ft × 1.5 ft = 3 ft². The key insight is that when converting area units, you must square the linear conversion factor: (1 ft/12 in)² = 1 ft²/144 in². A common mistake is converting only one dimension or forgetting to square the conversion factor. For area conversions, always convert linear dimensions first, then multiply.

Question 4

Let $f(x) = \dfrac{2x - 1}{x + 3}$ and $g(x) = \dfrac{x}{x - 2}$ . Which of the following is $(f \circ g)(x)$ ?

$\dfrac{2x+1}{4x-6}$
$\dfrac{x+2}{2x-3}$
$\dfrac{x+2}{x-2}$
$\dfrac{x+2}{4x-6}$ (correct answer)

Explanation: Compute $f(g(x)) = \dfrac{2\cdot \frac{x}{x-2} - 1}{\frac{x}{x-2} + 3} = \dfrac{\frac{x+2}{x-2}}{\frac{4x-6}{x-2}} = \dfrac{x+2}{4x-6}$ . Other choices drop the common factor, miss the factor 2, or miscompute the numerator.

Question 5

A fitness coach recorded the number of minutes each of 12 clients exercised in a week and the number of calories each client burned during those workouts. The scatterplot shows minutes exercised (in minutes) on the x-axis and calories burned (in calories) on the y-axis, with a dashed line of best fit. Based on the scatterplot, which statement best describes the relationship between minutes exercised and calories burned?

There is a strong negative association; more minutes generally means fewer calories burned.
There is little to no association; calories burned does not change with minutes exercised.
There is a positive association; more minutes generally corresponds to more calories burned. (correct answer)
The scatterplot proves exercising longer causes a specific calorie increase for every person.

Explanation: The question asks which statement best describes the relationship between minutes exercised and calories burned based on the scatterplot. The scatterplot displays points that generally increase from left to right, showing a positive trend with moderate spread around the dashed line of best fit, and no obvious nonlinear patterns. This upward trend indicates that as minutes exercised increase, calories burned tend to increase, aligning with a positive association as in choice C. To confirm, observe that the line of best fit has a positive slope, meaning higher x-values correspond to higher predicted y-values. A key error in choice D is confusing association with causation, as scatterplots describe patterns but do not prove cause-and-effect; choice A incorrectly identifies a negative trend, and B overlooks the evident association. Remember, data literacy involves describing observed trends accurately without inferring unproven causality. A useful test-taking strategy is to mentally trace the overall direction of the points before reading choices to avoid bias.

Question 6

A line passes through $(3,8)$ and has slope $\frac{1}{3}$ . Which equation is the line in point-slope form?

$y-8=\frac{1}{3}(x-3)$ (correct answer)
$y-3=\frac{1}{3}(x-8)$
$y-8=3(x-3)$
$y+8=\frac{1}{3}(x+3)$

Explanation: The question asks for the point-slope form of a line through (3,8) with slope 1/3. The form is y - y1 = m(x - x1), so y - 8 = (1/3)(x - 3). This directly uses the given point and slope. It models the linear equation without expanding. A common error is switching x and y coordinates, like y-3=(1/3)(x-8). Another mistake might be using reciprocal slope like 3. For point-slope, plug in the known point and slope carefully.

Question 7

A square has side length $11$ . What is the length of its diagonal?

$242$
$113$
$112$ (correct answer)
$22$

Explanation: A square's diagonal forms a 45°-45°-90° triangle with two sides of the square. For a square with side length s, the diagonal has length s√2. With side length 11, the diagonal = 11√2. Common mistakes include doubling the side (getting 22) or using 11√3. Remember that a square's diagonal always equals side × √2, which follows from the Pythagorean theorem: 11² + 11² = 2(121) = 242, so diagonal = √242 = 11√2.

Question 8

In the coordinate plane, the circle shown has center $(1,1)$ and passes through $(5,1)$ . What is the area of the circle, in terms of $\pi$ ?

$4\pi$
$8\pi$
$16\pi$ (correct answer)
$25\pi$

Explanation: This question asks for the area of a circle with center $(1,1)$ that passes through $(5,1)$ . First, find the radius as the distance between these points: $r=\sqrt{(5-1)^2+(1-1)^2}=\sqrt{16+0}=4$ . The area formula is $A=\pi r^2=\pi(4)^2=16\pi$ . A common error is forgetting to square the radius or confusing radius with diameter. Since both points have the same y-coordinate, this is a horizontal distance calculation, making it easier to avoid errors.

Question 9

A water tank holds 18 quarts. A manual describes capacity in gallons. Using $1\text{ gal}=4\text{ qt}$ , what is the tank's capacity in gallons?

3.6 gal
72 gal
4.5 gal (correct answer)
14 gal

Explanation: We need to convert 18 quarts to gallons. Set up the conversion: 18 qt × (1 gal/4 qt). The quart units cancel: 18/4 = 4.5 gallons. To divide: 18 ÷ 4 = 4 remainder 2, so 18/4 = 4 + 2/4 = 4 + 0.5 = 4.5. A common error is multiplying by 4 instead of dividing. Remember: gallons are larger than quarts, so converting quarts to gallons gives a smaller number.

Question 10

A car's value, $V$ , depreciates by 18% each year. The value after $t$ years is modeled by $V=22000(0.82)^t$ . What does the number $22000$ represent in this situation?

Value after 1 year
Annual depreciation amount
Initial value at $t=0$ (correct answer)
Annual growth factor

Explanation: This question tests understanding of exponential decay model parameters, specifically what each number represents in context. In the equation V = 22000(0.82)^t, the number 22000 is the coefficient that appears when t = 0, making it the initial value of the car. The base 0.82 represents the decay factor (1 - 0.18 = 0.82, since the car loses 18% of its value each year). A common mistake is thinking 22000 represents the value after one year, but that would be 22000(0.82)^1 = 18040. When interpreting exponential models, always evaluate at t = 0 to find the initial value.

Question 11

A coordinate plane shows an exponential curve that passes through $(0,8)$ and decreases to the right, passing near $(1,4)$ and $(2,2)$ . Which equation best matches the graph?

$y=8(2)^x$
$y=8\left(\dfrac12\right)^x$ (correct answer)
$y=8-x$
$y=4\left(\dfrac12\right)^x$

Explanation: We need to identify an exponential curve passing through (0,8), (1,4), and (2,2). This is exponential decay since values are decreasing. From 8 to 4 is ×1/2, and from 4 to 2 is also ×1/2, confirming constant ratio. Testing y = 8(1/2)^x: at x = 0, y = 8(1/2)^0 = 8(1) = 8 ✓; at x = 1, y = 8(1/2)^1 = 8(1/2) = 4 ✓; at x = 2, y = 8(1/2)^2 = 8(1/4) = 2 ✓. The decay factor of 1/2 means the function value halves with each unit increase in x. Exponential decay has base between 0 and 1. When consecutive y-values show constant ratios less than 1, it indicates exponential decay.

Question 12

A taxi fare is shown on a coordinate plane with points $(1,6)$ and $(5,14)$ , where $x$ is miles and $y$ is total cost in dollars. Assuming a linear model, what is the cost for a 9-mile ride?

$18
$20
$22 (correct answer)
$24

Explanation: This problem requires finding a linear equation from two points and then extrapolating. Given points (1,6) and (5,14), first find the slope: m = (14-6)/(5-1) = 8/4 = 2. Using point-slope form with (1,6): y - 6 = 2(x - 1), which simplifies to y = 2x + 4. For a 9-mile ride, y = 2(9) + 4 = 18 + 4 = 22 dollars. The key steps are calculating slope from two points, finding the equation, then evaluating at the desired x-value. A common error is using the slope incorrectly or forgetting the y-intercept when writing the final equation.

Question 13

A line passes through $(5,2)$ and has $y$ -intercept $-3$ . Which equation represents the line in slope-intercept form?

$y=\frac{1}{5}x-3$
$y=\frac{5}{2}x-3$
$y=x-3$ (correct answer)
$y=-x-3$

Explanation: Given a line through $(5,2)$ with y-intercept $-3$ , we can find the equation. Since the y-intercept is $-3$ , we have $y = mx - 3$ . Using point $(5,2)$ : $2 = m(5) - 3$ , which gives us $2 = 5m - 3$ , so $5 = 5m$ and $m = 1$ . Therefore, the equation is $y = x - 3$ . Common errors include arithmetic mistakes when solving for the slope or confusing which value is the y-intercept. When given the y-intercept explicitly, use it directly in the slope-intercept form.

Question 14

A train travels at 72 miles per hour. How long will it take the train to travel 210 miles at this constant speed? (Give the time in hours.)

2.5 hr
2.75 hr
2.92 hr (correct answer)
3.5 hr

Explanation: We need the time to travel 210 miles at 72 mph. Using time = distance ÷ speed: 210 miles ÷ 72 mph = 2.917 hours, which rounds to 2.92 hours. Don't confuse hours with minutes or round too early. For time calculations at constant speed, always use distance divided by rate.

Question 15

A store sells notebooks for $x$ dollars each and pens for $y$ dollars each. A customer buys 3 notebooks and 2 pens for a total of $19$ . Which equation represents this relationship between $x$ and $y$ ?

$2x+3y=19$
$3x+2y=19$ (correct answer)
$3x+2y=5$
$x+y=19$

Explanation: We need to write an equation representing the total cost of 3 notebooks at x dollars each and 2 pens at y dollars each, totaling $19. The cost of 3 notebooks is 3x dollars, and the cost of 2 pens is 2y dollars. Adding these together equals the total: 3x + 2y = 19. A common mistake is reversing the coefficients, writing 2x + 3y = 19, which would mean 2 notebooks and 3 pens. When translating word problems into equations with two variables, carefully match each coefficient to its corresponding variable based on the problem context.

Question 16

The number of pages $p$ remaining in a book after reading $r$ pages is modeled by $p=320-r$ . If $p$ decreases by 12, how does $r$ change?

$r$ decreases by 12
$r$ increases by 12 (correct answer)
$r$ increases by 320
$r$ decreases by 320

Explanation: Given p = 320 - r, we need to find how r changes when p decreases by 12. If p decreases by 12, then p_new = p - 12 = (320 - r) - 12 = 308 - r. Setting this equal to 320 - r_new, we get 308 - r = 320 - r_new, which gives us r_new = r + 12. Therefore, r increases by 12. The key insight is that in the equation p = 320 - r, p and r change in opposite directions. When working with inverse relationships, a decrease in one variable causes an increase in the other.

Question 17

A gym charges a one-time sign-up fee plus a monthly cost. The total cost after $x$ months is modeled by $45+19x=12x+94$ . What is the value of $x$ , the number of months when the two cost plans are equal?

$7$ (correct answer)
$\dfrac{7}{49}$
$49$
$-7$

Explanation: This problem asks us to find when two gym cost plans are equal by solving the equation $45 + 19x = 12x + 94$ . To solve, we first collect like terms by subtracting 12x from both sides: $45 + 19x - 12x = 94$ , which gives us $45 + 7x = 94$ . Next, we subtract 45 from both sides: $7x = 49$ . Finally, we divide both sides by 7: $x = 7$ . A common error is to incorrectly combine the x terms or make arithmetic mistakes when moving constants. When solving equations with variables on both sides, always move all variable terms to one side first.

Question 18

In a study on diet and weight loss, participants who ate less than 1500 calories a day lost weight faster than those who did not. Which limitation affects the conclusion?

The study did not consider exercise habits. (correct answer)
The study proves reducing calories causes weight loss.
All participants followed the same diet plan.
The study included only those who wanted to lose weight.

Explanation: This question asks which limitation affects the conclusion in a diet and weight loss study. The study found that people eating under 1500 calories lost weight faster, but this doesn't account for other factors that might influence weight loss. Answer A correctly identifies that not considering exercise habits is a major limitation, as exercise could be a confounding variable affecting the results. Options B incorrectly claims the study proves causation, option C makes an unsupported assumption about diet consistency, and option D describes participant motivation rather than a methodological limitation. When evaluating study limitations, look for important variables that weren't controlled or measured.

Question 19

A right triangle has legs 9 and 12. Let $\theta$ be the acute angle adjacent to the 12-unit leg and opposite the 9-unit leg. What is $\theta$ to the nearest degree?

$53^\circ$
$60^\circ$
$45^\circ$
$37^\circ$ (correct answer)

Explanation: We have a right triangle with legs 9 and 12, and need angle θ adjacent to the 12-unit leg and opposite the 9-unit leg. Using tangent: tan(θ) = opposite/adjacent = 9/12 = 0.75. Taking inverse tangent: θ = arctan(0.75) ≈ 36.87°, which rounds to 37°. This is a 3-4-5 triangle scaled by 3 (sides 9-12-15), and the angle opposite the shortest side is always the smallest acute angle. When working with integer-sided right triangles, recognizing Pythagorean triples can help verify your work.

Question 20

A gym membership fee increases from 80 to 92. What is the percent increase?

12
15 (correct answer)
13
1.15

Explanation: Percent increase is (92 - 80)/80 * 100 = 15. 12 is the absolute difference, 13 uses the new price as the base, and 1.15 is the multiplier, not the percent.

Question 21

A parallelogram has base $13$ cm and height $7$ cm. What is the area of the parallelogram in square centimeters?

20 cm $^2$
45.5 cm $^2$
91 cm $^2$ (correct answer)
182 cm $^2$

Explanation: This question asks for the area of a parallelogram with base 13 cm and height 7 cm. The area of a parallelogram is A = base × height. Substituting: A = 13 × 7 = 91 square centimeters. A common mistake is confusing the height with a slanted side length - the height must be perpendicular to the base. Unlike triangles, parallelograms don't have the ½ factor in their area formula.

Question 22

Lines $p$ and $q$ are parallel. A transversal intersects them, forming an angle of $125^\circ$ in an exterior position at the top intersection. What is the measure of the alternate exterior angle at the bottom intersection?

$55^\circ$
$125^\circ$ (correct answer)
$180^\circ$
$235^\circ$

Explanation: This question asks about alternate exterior angles formed by parallel lines and a transversal. The key property is that alternate exterior angles are equal when lines are parallel. If one exterior angle measures 125°, its alternate exterior angle on the opposite side of the transversal also measures 125°. A common mistake is confusing alternate exterior angles with corresponding angles or same-side exterior angles. When identifying alternate exterior angles, look for angles on opposite sides of the transversal and outside the parallel lines.

Question 23

A landscaper spreads mulch at a constant rate of 3.2 cubic yards per hour. How many cubic yards of mulch will the landscaper spread in 7.5 hours?

21.3 yd $^3$
10.7 yd $^3$
25.6 yd $^3$
24.0 yd $^3$ (correct answer)

Explanation: We need to find cubic yards of mulch spread in 7.5 hours at a rate of 3.2 cubic yards per hour. This is a direct multiplication: 3.2 yd³/hour × 7.5 hours = 24.0 cubic yards. To calculate: 3.2 × 7.5 = 3.2 × 7 + 3.2 × 0.5 = 22.4 + 1.6 = 24.0. Common errors include decimal multiplication mistakes or forgetting to include units. When given a rate and time, multiply to find the total quantity.

Question 24

A circle is centered at $(2,-3)$ . Point $(8,1)$ lies on the circle. What is the radius of the circle?

$\sqrt{52}$ (correct answer)
$\sqrt{20}$
$\sqrt{36}$
$\sqrt{16}$

Explanation: This question asks for the radius of a circle centered at $(2,-3)$ that passes through $(8,1)$ . The radius is the distance between these two points, calculated using the distance formula: $r = \sqrt{(8-2)^2 + (1-(-3))^2} = \sqrt{6^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52}$ . Note that $\sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13}$ , but the answer is given as $\sqrt{52}$ . A common error is making arithmetic mistakes with the signs, especially when subtracting negative numbers like $1-(-3) = 1+3 = 4$ .

Question 25

A phone plan costs $30 plus $0.10 per text message. You want to keep the total cost under $45. Let $t$ be the number of texts. What is the greatest whole number of texts you can send?

$t=149$ (correct answer)
$t=150$
$t=151$
$t=200$

Explanation: We need to find the maximum number of texts while keeping the total cost under $45. The cost is $30 + $0.10t, and we need 30 + 0.10t < 45. Solving: 0.10t < 15, so t < 150. Since we want the greatest whole number less than 150, the answer is t = 149. A common error is using ≤ instead of < when the problem says "under," or forgetting that we need a whole number of texts. When finding maximum values with strict inequalities, remember to use the largest integer that's strictly less than the bound.