SAT Math

SAT Math Practice Test: Sat Practice Test 1

Sat Practice Test 1 for SAT Math: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

A scientist randomly assigned 60 identical plants to two groups for 6 weeks: Fertilizer X or no fertilizer. All plants received the same light and water. Final heights were recorded. Based on this randomized experiment, which conclusion is most appropriate?

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Question 1

Fertilizer X will increase the height of all plant species in any environment because the experiment used plants and measured growth.
Fertilizer X caused plants to grow taller on average than no fertilizer in this experiment, since treatment was randomly assigned and conditions were controlled. (correct answer)
All plants given Fertilizer X will be at least 4 cm taller than plants given no fertilizer because the group averages differ by 4 cm.
Fertilizer X is associated with taller plants, but the study cannot determine whether Fertilizer X caused the difference in height.

Explanation: The question asks what conclusion a randomized, controlled experiment supports. Because the 60 identical plants were randomly assigned to Fertilizer X or no fertilizer, and light and water were held constant, the only systematic difference between the groups is the fertilizer, so a difference in average final height can reasonably be attributed to the fertilizer itself. That is exactly what the statement crediting Fertilizer X with causing taller average growth in this experiment says. The statement that the study can show only an association and cannot determine cause understates what random assignment buys you; that caution belongs to observational studies. The claim covering all plant species in any environment generalizes far beyond the single species and single set of conditions tested. And the claim that every treated plant will be at least 4 cm taller misapplies a difference in group averages to individual plants, which averages never guarantee.

Question 2

A circle has circumference $30\pi$ cm. What is the radius of the circle, in centimeters?

$30$
$15$ (correct answer)
$\frac{15}{\pi}$
$\frac{30}{\pi}$

Explanation: Given a circumference of 30π cm, we need to find the radius. The circumference formula is C = 2πr. Setting up the equation: 30π = 2πr. Dividing both sides by 2π: r = 30π/(2π) = 15 cm. The π cancels out, leaving just 15. A common error is forgetting to divide by 2 or not canceling the π properly. When working backwards from circumference to radius, divide by 2π.

Question 3

A right triangle has legs 5 and 12 (with 12 adjacent to angle $\theta$ ). What is $\tan\theta$ ?

$\dfrac{5}{13}$
$\dfrac{12}{5}$
$\dfrac{5}{12}$ (correct answer)
$\dfrac{12}{13}$

Explanation: We have a right triangle with legs 5 and 12, where 12 is adjacent to angle θ. This means 5 is opposite to angle θ. Using the tangent ratio: tan(θ) = opposite/adjacent = 5/12. A common mistake is confusing which leg is opposite and which is adjacent; the problem explicitly states that 12 is adjacent to θ, making 5 the opposite side. Remember that tangent is the only ratio that doesn't involve the hypotenuse.

Question 4

In the coordinate plane, points $A(-2,1)$ , $B(4,1)$ , and $C(4,6)$ form triangle $ABC$ . What is the measure of $\angle ABC$ ? (Angle $ABC$ is the angle with vertex at $B$ .)

$30^\circ$
$45^\circ$
$60^\circ$
$90^\circ$ (correct answer)

Explanation: This question asks for the measure of angle ABC in a triangle with given coordinates. Since B is at $(4,1)$ and C is at $(4,6)$ , the line segment BC is vertical (same x-coordinate). Since A is at $(-2,1)$ and B is at $(4,1)$ , the line segment AB is horizontal (same y-coordinate). When a horizontal line meets a vertical line, they form a $90^\circ$ angle. A common error is trying to use the distance formula or slope calculations when the answer can be determined by recognizing perpendicular lines. When working with coordinate geometry, always check if lines are horizontal or vertical first.

Question 5

How many solutions does the system have? $$ \begin{cases}4x-2y=10\2x-y=5\end{cases}

$0$
$1$
$2$
infinitely many (correct answer)

Explanation: We need to determine if this system has no solutions, one solution, or infinitely many solutions. Let me examine the relationship between the two equations. The second equation is 2x - y = 5. If I multiply this by 2, I get 4x - 2y = 10, which is exactly the first equation. Since the first equation is just a multiple of the second equation, they represent the same line. When two equations represent the same line, every point on that line is a solution, giving us infinitely many solutions. A common error would be to think these are parallel lines because the coefficients are proportional, but since the constants are also proportional (10/5 = 2), they're the same line. For test-taking, when you see one equation is a multiple of another, immediately check if it's the exact same line (infinitely many solutions) or parallel lines (no solutions).

Question 6

A recipe needs 2.5 gallons of water. How many cups is this, using $1 \text{ gal}=4 \text{ qt}$ and $1 \text{ qt}=4 \text{ cups}$ ?

40 cups (correct answer)
160 cups
10 cups
2.5 cups

Explanation: The question asks how many cups are in 2.5 gallons of water, using conversions $1 \text{ gallon} = 4 \text{ quarts}$ and $1 \text{ quart} = 4 \text{ cups}$ . Set up the conversion factors as $4 \text{ quarts per 1 gallon}$ and $4 \text{ cups per 1 quart}$ . Start with $2.5 \text{ gal} \times (4 \text{ qt} / 1 \text{ gal}) \times (4 \text{ cups} / 1 \text{ qt})$ , where gallons cancel to quarts, and quarts cancel to cups. First, $2.5 \times 4 = 10$ quarts, then $10 \times 4 = 40$ cups; alternatively, $2.5 \times 16 = 40$ cups since $4 \times 4 = 16$ . A key error might be using only one conversion factor instead of both, leading to incorrect units. For multi-step conversions, write out each factor with units to track cancellation properly.

Question 7

A theater has 240 seats. If 68 seats are already sold, and each group reservation buys $g$ seats, the total sold must not exceed 240. Which inequality describes the possible values of $g$ ?

$68g\le 240$
$68+g\le 240$ (correct answer)
$68+g\ge 240$
$g-68\le 240$

Explanation: This problem involves setting up an inequality for theater seating capacity. The theater has 240 total seats, 68 are already sold, and g more seats will be sold. The total seats sold is 68 + g, which cannot exceed 240, giving us 68 + g ≤ 240. The key insight is that we add the already-sold seats to the group reservation seats. A common mistake is multiplying 68 by g or using the wrong inequality direction. When dealing with capacity constraints, the total used must be less than or equal to the maximum capacity.

Question 8

A phone battery has 100% charge at $t=0$ and then decreases by 12 percentage points every hour. Which equation models the battery percentage, $p$ , after $t$ hours (assuming the model continues)?

$p=100(0.88)^t$
$p=100-12t$ (correct answer)
$p=12-100t$
$p=100+12t$

Explanation: A phone battery starts at 100% and decreases by 12 percentage points each hour. Since the decrease is constant (12 percentage points per hour), this is linear decay, not exponential. The equation is p = 100 - 12t, where we subtract 12t from the initial 100%. We can verify: at t=0, p=100%; at t=1, p=88%; at t=2, p=76%. The key distinction is "percentage points" (linear) versus "percent" (exponential) - losing 12 percentage points means going from 100% to 88%, not to 88% of 100%. Watch for this language carefully in word problems.

Question 9

Two linear equations are shown as lines on a coordinate plane: $y=x+1$ and $y=-2x+7$ . What is the $x$ -coordinate of their intersection point?

$1$
$2$ (correct answer)
$3$
$4$

Explanation: We need to find where the lines y = x + 1 and y = -2x + 7 intersect. Since both expressions equal y, I can set them equal: x + 1 = -2x + 7. Adding 2x to both sides: 3x + 1 = 7. Subtracting 1: 3x = 6, so x = 2. To verify, when x = 2: y = 2 + 1 = 3 from the first equation, and y = -2(2) + 7 = -4 + 7 = 3 from the second equation. The key insight is that at the intersection point, both equations give the same y-value for the same x-value. For test-taking, setting the equations equal is faster than graphing when finding intersection points algebraically.

Question 10

Consider $h(x)=-3x^4+2x^2-7$ . Which statement best describes the end behavior of $h(x)$ as $x\to\infty$ and $x\to-\infty$ ?

Both ends rise to $\infty$
Left falls, right rises
Left rises, right falls
Both ends fall to $-\infty$ (correct answer)

Explanation: The question asks for the end behavior of h(x) = -3x⁴ + 2x² - 7 as x approaches ±∞. The leading term -3x⁴ dominates, with even degree 4 and negative coefficient, so both ends fall to -∞. Lower terms like 2x² become negligible for large |x|. This illustrates how degree and leading sign dictate long-term trends, connecting polynomial form to asymptotic behavior. A common mistake is treating the degree as odd or ignoring the negative sign, predicting rising ends. Some focus on positive terms only. In tests, isolate the leading term to quickly determine end behavior without full graphing.

Question 11

A set of 6 numbers has median 10 and range 8. The minimum value is 6. What is the maximum value? (Use $\text{range}=\text{max}-\text{min}$ .)

12
14 (correct answer)
16
18

Explanation: Given: 6 numbers with median 10, range 8, and minimum 6. Using range = max - min, we have 8 = max - 6, so max = 14. This uses the fundamental relationship between range, maximum, and minimum values. The median information confirms this is reasonable - with 6 values, the median of 10 falls between the 3rd and 4th values when ordered, which is consistent with a minimum of 6 and maximum of 14. When given range and one extreme value, you can always find the other extreme value by adding or subtracting appropriately.

Question 12

A square tile has side length $30\text{ cm}$ . What is the tile's area in square meters? Use $100\text{ cm}=1\text{ m}$ . (Area requires squaring the length conversion.)

0.9 m $^2$
0.09 m $^2$ (correct answer)
9 m $^2$
900 m $^2$

Explanation: We need to find the area of a 30 cm × 30 cm square in square meters. First find area in cm²: 30 × 30 = 900 cm². Then convert using (100 cm)² = 10,000 cm² = 1 m²: 900 cm² × (1 m² / 10,000 cm²) = 0.09 m². The key insight is that for area conversions, we must square the linear conversion factor: 100² = 10,000. A common error is using just 100 instead of 10,000 as the conversion factor.

Question 13

A librarian recorded, for 10 students, the number of books checked out (books) and the number of minutes spent reading that week (minutes). The scatterplot is shown. Which statement best describes the strength and direction of the association?

Strong negative association
Weak negative association
Moderate positive association (correct answer)
No association

Explanation: This question asks about the strength and direction of association between books checked out and reading time. Looking at the scatterplot, as the number of books increases (moving right), reading time generally increases (points trend upward), indicating a positive association. The points show some scatter but follow a general upward trend - not tightly clustered (which would be strong) nor widely scattered (which would be weak), making this a moderate association. The upward trend is clear enough to rule out "no association" but not tight enough for "strong." When assessing association strength, consider how closely points follow the overall pattern: tight clustering = strong, loose clustering = moderate, no pattern = weak or none.

Question 14

In triangle DEF, $\angle D = 90^\circ$ , $\angle E = 45^\circ$ , and side $DE = 5$ units. What is the length of side $DF$ ?

5 units (correct answer)
7.07 units
3.54 units
6 units

Explanation: We need to find the length of side DF in a right triangle where ∠D = 90°, ∠E = 45°, and DE = 5 units. Since ∠D is the right angle, sides DE and DF are the legs, and EF is the hypotenuse. In a 45-45-90 triangle, the two legs are equal in length. Since DE = 5 units and this is a 45° triangle, DF must also equal 5 units. A common error is assuming one of the legs is the hypotenuse or forgetting the special properties of 45-45-90 triangles. In isosceles right triangles, always remember that the two legs are congruent.

Question 15

A nutritionist wants to compare two diets on cholesterol levels. She randomly assigns 100 volunteers to Diet A and 100 volunteers to Diet B for 3 months and measures the mean change in LDL cholesterol. Diet A's mean change is $-12$ mg/dL and Diet B's is $-5$ mg/dL. Which statement is most appropriate?

Diet A likely caused a larger average reduction in LDL than Diet B for these volunteers, because random assignment supports a causal interpretation. (correct answer)
Diet A is associated with lower LDL, but random assignment prevents concluding Diet A caused the reduction.
Diet A will reduce every person's LDL by exactly 12 mg/dL, because the mean change for Diet A is $-12$ mg/dL.
Diet B is better than Diet A for all populations, because Diet B was tested on the same number of volunteers as Diet A.

Explanation: This question presents results from a randomized experiment comparing two diets' effects on LDL cholesterol. With 100 volunteers randomly assigned to each diet, Diet A showed a mean reduction of 12 mg/dL while Diet B showed 5 mg/dL. Choice A correctly concludes that Diet A likely caused a larger reduction for these volunteers, which is appropriate given random assignment in an experiment. Choice B incorrectly suggests random assignment prevents causal conclusions when it actually enables them. Choice C unrealistically claims identical effects for every person. Choice D makes inappropriate population generalizations and ignores that A performed better. Randomized experiments support causal conclusions about average treatment effects.

Question 16

Lines $l$ and $m$ are parallel and cut by a transversal. A corresponding angle on line $l$ measures $119^\circ$ . Angle $x$ is the corresponding angle on line $m$ . What is $x$ ?

$121^\circ$
$119^\circ$ (correct answer)
$239^\circ$
$61^\circ$

Explanation: This problem asks for the measure of angle x, which is a corresponding angle to a 119° angle on parallel lines. When parallel lines are cut by a transversal, corresponding angles are congruent (equal in measure). Corresponding angles are in the same position relative to their intersection point - both above or both below the transversal, and both on the same side. Since the angle on line l measures 119°, the corresponding angle x on line m also measures 119°. A common mistake is confusing corresponding angles with other angle relationships like alternate interior angles or same-side interior angles.

Question 17

Integers $x$ and $y$ are inserted into the set 4, 6, 8, 12, 15 to form a 7-number set whose median is 10, with $x < y$ . What is the least possible value of $y$ ?

9
10
12
11 (correct answer)

Explanation: To make the 4th value 10, one inserted value must be 10 and the other must be just above 10; taking $x = 10$ gives the least $y = 11$ . Smaller choices place the median below 10 or violate $x < y$ .

Question 18

A ball is thrown upward and its height (in meters) after $t$ seconds is modeled by $H(t)=-5(t-2)^2+20$ . What is the maximum height of the ball?

$15$
$20$ (correct answer)
$25$
$-20$

Explanation: This question asks for the maximum height of a quadratic function modeling projectile motion. The function $H(t) = -5(t-2)^2 + 20$ is in vertex form, where the vertex is at $(2, 20)$ . Since the coefficient of the squared term is negative $(-5)$ , the parabola opens downward, making the vertex the maximum point. The maximum height is the y-coordinate of the vertex, which is 20 meters. A common error is confusing the time when maximum height occurs $(t = 2)$ with the maximum height itself (20). For quadratics in vertex form $a(x-h)^2 + k$ , the vertex is always at $(h, k)$ .

Question 19

A club requires members to be at least 16 years old and younger than 21. Let $x$ be a member's age in years. Which inequality represents the requirement?

$16 < x < 21$
$16 \le x \le 21$
$16 \le x < 21$ (correct answer)
$16 < x \le 21$

Explanation: Members must be at least 16 years old (x ≥ 16) AND younger than 21 (x < 21). Combining these requirements gives us 16 ≤ x < 21. The phrase "at least 16" includes 16, so we use ≤ on the left. The phrase "younger than 21" excludes 21, so we use < on the right. This creates a half-open interval. The common error is misinterpreting "younger than" as "21 or younger" - remember that "younger than 21" means strictly less than 21.

Question 20

A right triangle has legs of lengths $9$ cm and $12$ cm. What is the area of the triangle in square centimeters?

21 cm $^2$
54 cm $^2$ (correct answer)
108 cm $^2$
216 cm $^2$

Explanation: This question asks for the area of a right triangle given the lengths of its two legs. The area of a right triangle is A = ½(base)(height), where the legs serve as base and height. Substituting: A = ½(9)(12) = ½(108) = 54 square centimeters. A common error is forgetting the factor of ½ in the triangle area formula, which would give 108 cm². For right triangles, the legs are perpendicular, making them perfect choices for base and height.

Question 21

In a right triangle, one acute angle is $\theta$ and the other is $90^\circ-\theta$ . If $\tan(\theta)=\frac{2}{3}$ , what is $\tan(90^\circ-\theta)$ ?

$\frac{5}{3}$
$\frac{3}{5}$
$\frac{2}{3}$
$\frac{3}{2}$ (correct answer)

Explanation: We need $\tan(90^\circ - \theta)$ when $\tan(\theta) = \frac{2}{3}$ . Using the cofunction identity, $\tan(90^\circ - \theta) = \cot(\theta) = \frac{1}{\tan(\theta)}$ . Since $\tan(\theta) = \frac{2}{3}$ , we have $\tan(90^\circ - \theta) = \frac{1}{\frac{2}{3}} = \frac{3}{2}$ . This makes sense geometrically: if $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{2}{3}$ in a right triangle, then for the complementary angle, these sides switch roles, giving $\frac{3}{2}$ . The key insight is that complementary angles in a right triangle swap the roles of opposite and adjacent sides. Remember: $\tan$ and $\cot$ are reciprocals for complementary angles.

Question 22

In right triangle $\triangle RST$ , $\angle T=90^\circ$ and $\angle R=41^\circ$ . If $ST=18$ (opposite $\angle R$ ), what is the length of $RT$ (adjacent to $\angle R$ ), to the nearest tenth?

$13.7$
$15.6$
$20.7$ (correct answer)
$23.8$

Explanation: Given angle R = 41° and ST = 18 (opposite to angle R), we need to find RT (adjacent to angle R). Since we have the opposite side and need the adjacent side, we use tangent: tan(θ) = opposite/adjacent. Setting up: tan(41°) = 18/RT, so RT = 18/tan(41°) ≈ 18/0.869 ≈ 20.7. A common error is using sine or cosine when you have one leg and need the other leg - remember that tangent directly relates the two legs. When solving for the adjacent side, divide the opposite by the tangent of the angle.

Question 23

A circle is centered at the origin and passes through $(0,-9)$ . What is the equation of the circle?

$(x-9)^2+y^2=81$
$x^2+y^2=81$ (correct answer)
$x^2+y^2=9$
$x^2+(y+9)^2=81$

Explanation: This question asks for the equation of a circle centered at the origin passing through (0, -9). Since the center is (0, 0) and the circle passes through (0, -9), the radius is the distance from origin to (0, -9), which is 9. The equation is x² + y² = r² = 81. A common error is using the y-coordinate (-9) as the radius without taking its absolute value. Remember that radius is always positive, representing distance.

Question 24

A right triangle has hypotenuse $13\text{ in}$ and one leg $5\text{ in}$ . What is the area of the triangle?

$60\text{ in}^2$
$30\text{ in}^2$ (correct answer)
$24\text{ in}^2$
$65\text{ in}^2$

Explanation: This problem asks for the area of a right triangle with hypotenuse 13 in and one leg 5 in. Using the Pythagorean theorem to find the other leg: a² + 5² = 13², so a² = 169 - 25 = 144, giving a = 12 in. Area = (1/2) × 5 × 12 = 30 in². A common error is trying to use hypotenuse as a side in the area formula, or miscalculating with the Pythagorean theorem. Always find both legs before calculating area of a right triangle.

Question 25

The weights (in kg) of a group of 10 students are recorded as follows: 45, 48, 52, 50, 49, 48, 51, 47, 46, 48. What is the mode of the data?

48 (correct answer)
47
50
49

Explanation: This question asks for the mode, which is the value that appears most frequently in a data set. Looking at the weights 45, 48, 52, 50, 49, 48, 51, 47, 46, 48, we need to count how often each value appears. The value 48 appears three times, while all other values appear only once. Therefore, 48 is the mode since it has the highest frequency.