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Free SSAT Math Practice Test — Grade 10

Math only, deliberately — the real SSAT (published by the Enrollment Management Association) also has verbal and reading sections, which we don't cover.

20 real practice questions at SSAT Upper Level difficulty, drawn from the same calibrated bank our adaptive practice uses. Work through them right here — every question has the answer and a step-by-step explanation one tap away.

Question 1 · Arithmetic & Number SenseIntro
Pattern sequencePattern sequence: 2, 4, 5, 9, 8, 14, 11, 19, 14, 24, …. What is the 18th term? 2→4→5→9→8→14→11→19→14→24→…What is the 18th term?
The sequence below alternates between two interlocking patterns. What is the 18th term?
A56
B38
C35
D44
E53
Show answer & explanation
Let's work it out
1.Identify the two interlocking subsequences by separating odd-positioned and even-positioned terms.
2.Odd-positioned terms (positions 1, 3, 5, 7, 9, 11, 13, 15, 17, …): 2, 5, 8, 11, 14, 17, 20, 23, 26, … — an arithmetic sequence with first term 2 and common difference 3.
3.Even-positioned terms (positions 2, 4, 6, 8, 10, 12, 14, 16, 18, …): 4, 9, 14, 19, 24, 29, 34, 39, 44, … — an arithmetic sequence with first term 4 and common difference 5.
4.Since 18 is even, the 18th term belongs to the even-positioned subsequence.
5.It is the 9th term of that subsequence.
6.Using the formula for the nth term of an arithmetic sequence: b_n = b_1 + (n − 1)d b_9 = 4 + (9 − 1)(5) = 4 + 40 = 44 The 18th term is 44.
Answer: D) 44
Question 2 · GeometryIntro
Coordinate gridCoordinate grid, x-axis from -1 to 12, y-axis from -1 to 6. Points: P at (0, 0), Q at (6, 0), R at (3, 4). Also shown: a triangle with vertices (0, 0), (6, 0), (3, 4). x y -11357911-1123456PQRPQR
On a coordinate grid, triangle PQR has vertices P(0, 0), Q(6, 0), and R(3, 4). Triangle PQR is reflected across the vertical line x = 5 to form triangle P'Q'R'. What are the coordinates of R'?
A(7, 4)
B(8, 4)
C(13, 4)
D(2, 4)
E(3, 6)
Show answer & explanation
Let's work it out
1.To reflect a point across the vertical line x = 5, use the formula: x' = 2(5) - x = 10 - x, y stays the same.
2.For R(3, 4): x' = 10 - 3 = 7, y' = 4.
3.So R' = (7, 4).
4.Check the others: P(0,0) maps to (10,0) = P', Q(6,0) maps to (4,0) = Q'.
5.The question asks only for R', which is (7,4).
Answer: A) (7, 4)
Question 3 · Arithmetic & Number SenseEasy
Cone Distances in the Six-Round Drill
RoundCone Distance in Meters
212.8
36.4
51.6
A basketball player completes a six-round conditioning drill. In each round, the player sprints from the starting line to a cone and then returns to the starting line. The cone distances form a geometric sequence, as shown in the table. How many meters farther does the player sprint during the first four rounds combined than during the final two rounds combined?
A92.8
B91.2
C45.6
D78.4
E100.8
Show answer & explanation
Let's work it out
1.The common ratio is 6.4 ÷ 12.8 = 0.5, so the six cone distances are 25.6, 12.8, 6.4, 3.2, 1.6, and 0.8 meters.
2.Doubling for each return trip gives 96 meters during the first four rounds and 4.8 meters during the final two rounds.
3.The player therefore sprints 96 - 4.8 = 91.2 meters farther during the first four rounds.
Answer: B) 91.2
Question 4 · PercentagesEasy
Changes in Snow Depth
StageSnow depth information
Before the storm16.00 centimeters
New snowfallIncrease of 37.5%
Snow settlesDecrease of 10%
Afternoon meltingDecrease of 20%
The table shows how the snow depth beside a school weather station changes during a storm and the following afternoon. After all the stages shown, another snowfall begins. How many centimeters of new snow must accumulate to bring the depth to exactly 18.00 centimeters?
A17.56
B6.21
C1.80
D2.16
E0.80
Show answer & explanation
Let's work it out
1.The new snowfall changes the depth to 16.00 × 1.375 = 22.00 centimeters.
2.Successive decreases then give 22.00 × 0.90 × 0.80 = 15.84 centimeters.
3.Therefore, 18.00 − 15.84 = 2.16 centimeters of additional snow must accumulate.
Answer: D) 2.16
Question 5 · Data & StatisticsEasy
Mini-Pizza Recipe Votes
RecipeFirst TastingFinal Tasting
Pesto Pocket68
Taco Twist75
Apple Crisp9?
A school cooking club held two tastings to choose a mini-pizza recipe for family night. Each student voted for one recipe at each tasting. Across both tastings, Apple Crisp received 4 fewer votes than Pesto Pocket and Taco Twist received together. How many votes did Apple Crisp receive at the final tasting?
A21
B1
C17
D13
E9
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Let's work it out
1.Pesto Pocket received 14 total votes, and Taco Twist received 12, so together they received 26.
2.Apple Crisp received 26 - 4 = 22 votes across both tastings.
3.Subtracting its 9 first-tasting votes gives 22 - 9 = 13 final-tasting votes.
Answer: D) 13
Question 6 · Data & StatisticsEasy
Data Sets
PositionSet PSet Q
17.83.9
24.68.4
39.36.2
45.75.1
How much greater is the sum of the two largest entries in Set P than the sum of the two smallest entries in Set Q?
A1.3
B26.1
C0.3
D13.2
E8.1
Show answer & explanation
Let's work it out
1.The two largest entries in Set P sum to 9.3 + 7.8 = 17.1.
2.The two smallest entries in Set Q sum to 3.9 + 5.1 = 9.0.
3.The difference is 17.1 − 9.0 = 8.1.
Answer: E) 8.1
Question 7 · Equations & InequalitiesStandard
If (34\frac{3}{4})[x - 2y−z3\frac{2y - z}{3}] - x+z5\frac{x + z}{5} = (23\frac{2}{3})(y - x2\frac{x}{2}), which relation must also hold?
Ax = 70y−27z53\frac{70y - 27z}{53}
Bx = 70y−3z63\frac{70y - 3z}{63}
Cx = 70y−3z53\frac{70y - 3z}{53}
Dx = 70y−3z13\frac{70y - 3z}{13}
Ex = 70y+27z53\frac{70y + 27z}{53}
Show answer & explanation
Let's work it out
1.Multiply the relation by 60 and distribute to obtain 45x - 30y + 15z - 12x - 12z = 40y - 20x.
2.Combining like terms and collecting the x-terms gives 53x = 70y - 3z.
3.Therefore, x = 70y−3z53\frac{70y - 3z}{53}.
Answer: C) x = 70y−3z53\frac{70y - 3z}{53}
Question 8 · Algebraic ExpressionsStandard
For nonzero x and y, simplify [−2x−3y24xy−1\frac{-2x^-3y^2}{4xy^-1}]-2 times [x−18y−2\frac{x^-1}{8y^-2}].
Ay82x9\frac{y^8}{2x^9}
Bx732y4\frac{x^7}{32y^4}
Cy832x9\frac{y^8}{32x^9}
Dy52x5\frac{y^5}{2x^5}
Ex72y4\frac{x^7}{2y^4}
Show answer & explanation
Let's work it out
1.The expression inside the first brackets simplifies to (-12\frac{1}{2})x-4y3, so raising it to -2 gives 4x8y-6.
2.The second factor is (18\frac{1}{8})x-1y2.
3.Multiplying the factors and combining exponents gives x72y4\frac{x^7}{2y^4}.
Answer: E) x72y4\frac{x^7}{2y^4}
Question 9 · MathStandard
In a music app, the sound-processing rule q is applied before the rule p, where p(x) = √20−xx−12\frac{20 - x}{x - 12} and q(n) = 129−n−2\displaystyle \frac{12}{\sqrt{9 - n} - 2}. Which whole-number setting n can be entered into q but produces a value that cannot then be entered into p?
A1
B9
C8
D0
E5
Show answer & explanation
Let's work it out
1.The rule q accepts 0 because its radicand is nonnegative and its denominator is not zero.
2.Substitution gives q(0) = 12.
Answer: D) 0
Why the other choices are tempting: The value 12 cannot be entered into p because it makes the denominator of p equal zero, so the required setting is 0.
Question 10 · IntegersStandard
What is the value of -3.7 + (-8.4) - (-5.2) + 6.1?
A2.4
B-11.6
C-0.8
D0.8
E-10.8
Show answer & explanation
Let's work it out
1.Rewrite each operation: -3.7 + (-8.4) - (-5.2) + 6.1 becomes -3.7 - 8.4 + 5.2 + 6.1.
2.Grouping the positive terms gives 5.2 + 6.1 = 11.3, and the negative terms give -3.7 - 8.4 = -12.1.
3.Adding these together, 11.3 + (-12.1) = -0.8.
Answer: C) -0.8
Question 11 · ProportionsStandard
For each of two school copiers, the number of worksheets produced varies directly with the number of minutes the copier runs. Copier A produces 28,000 worksheets in 40 minutes, while copier B produces 18,000 worksheets in 30 minutes. To produce 100,000 worksheets, copier A starts first and copier B starts 20 minutes later. Both copiers then run until the job is finished. About how many minutes after copier A starts will the job be finished?
A80
B85
C65
D95
E75
Show answer & explanation
Let's work it out
1.The direct-variation constants are 700 worksheets per minute for copier A and 600 worksheets per minute for copier B.
2.Copier A produces 14,000 worksheets before copier B starts, leaving 86,000 worksheets to be produced at a combined rate of 1,300 per minute.
3.The remaining work takes about 66.15 minutes, so the total of about 86.15 minutes is closest to 85 minutes.
Answer: B) 85
Question 12 · Equations & InequalitiesStandard
If -32a−b4\frac{2a - b}{4} + 5a+2b6\frac{a + 2b}{6} = c - d, which expression is equal to b?
A14a+12c−12d32\frac{14a + 12c - 12d}{32}
B−8a+12c−12d29\frac{-8a + 12c - 12d}{29}
C8a+12c−12d29\frac{8a + 12c - 12d}{29}
D8a+12c−12d11\frac{8a + 12c - 12d}{11}
E8a+12c−d29\frac{8a + 12c - d}{29}
Show answer & explanation
Let's work it out
1.Multiply the equation by 12 to obtain -9(2a - b) + 10(a + 2b) = 12c - 12d.
2.Distributing and combining like terms gives -8a + 29b = 12c - 12d.
3.Isolating b yields b = 8a+12c−12d29\frac{8a + 12c - 12d}{29}.
Answer: C) 8a+12c−12d29\frac{8a + 12c - 12d}{29}
Question 13 · MathStandard
Values of P, Q, and R
xP(x)Q(x)R(x)
018\frac{1}{8}18\frac{1}{8}18\frac{1}{8}
138\frac{3}{8}28\frac{2}{8}48\frac{4}{8}
258\frac{5}{8}48\frac{4}{8}98\frac{9}{8}
378\frac{7}{8}88\frac{8}{8}168\frac{16}{8}
498\frac{9}{8}168\frac{16}{8}258\frac{25}{8}
The table gives values of three functions. If each displayed pattern continues, which lists the functions in order from slowest eventual growth to fastest eventual growth?
AQ, P, R
BQ, R, P
CR, P, Q
DP, Q, R
EP, R, Q
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Let's work it out
1.To determine eventual growth rates, identify what type of function each one is.
2.For P: compute consecutive differences of the outputs.
3.If the table shows outputs increasing by a constant amount each step, P has a constant first difference — making it a **linear** function.
4.For R: check whether the outputs follow a pattern where the numerators are perfect squares (1, 4, 9, 16, …).
5.If so, R grows as a **quadratic** function.
6.For Q: check whether consecutive outputs have a constant ratio (each output is a fixed multiple of the previous one).
7.A constant multiplicative factor means Q is an **exponential** function.
8.Once the function types are identified, apply the fundamental growth-rate hierarchy: linear functions are eventually outpaced by quadratic functions, which are eventually outpaced by exponential functions — regardless of how the values compare in the early rows of the table.
9.Therefore, from slowest to fastest eventual growth: **P (linear), R (quadratic), Q (exponential)**, which corresponds to answer choice **E**.
Answer: E) P, R, Q
Question 14 · IntegersStandard
The product of a number and -34\frac{3}{4} is 98\frac{9}{8}. What is the number?
A43\frac{4}{3}
B32\frac{3}{2}
C-43\frac{4}{3}
D-32\frac{3}{2}
E-2732\frac{27}{32}
Show answer & explanation
Let's work it out
1.If the unknown number is n, then n multiplied by -34\frac{3}{4} equals 98\frac{9}{8}.
2.Dividing both sides by -34\frac{3}{4} gives n = 98\frac{9}{8} divided by -34\frac{3}{4} = 98\frac{9}{8} multiplied by -43\frac{4}{3} = -3624\frac{36}{24} = -32\frac{3}{2}.
Answer: D) -32\frac{3}{2}
Question 15 · MathHard
During a sprint drill, a coach checks five proposed stride factors x. Which fraction has the property that x1/2 / x2 equals the reciprocal of (16/81)3/4?
A4/9
B8/27
C9/4
D64/729
E729/64
Show answer & explanation
Let's work it out
1.The fourth root of 16/81 is 2/3, so (16/81)3/4 = (2/3)3 = 8/27, whose reciprocal is 27/8.
2.For x = 4/9, the numerator is 2/3 and the denominator is 16/81.
3.Their quotient is (2/3)(81/16) = 27/8, so the required fraction is 4/9.
Answer: A) 4/9
Question 16 · Algebraic ExpressionsHard
If x is negative and x2+5x+6x+2\frac{x² + 5x + 6}{x + 2} = -4, what is the value of x2−49x+7\frac{x² - 49}{x + 7}?
A-7
B14
Cundefined
D-14
E0
Show answer & explanation
Let's work it out
1.Factoring the first numerator gives (x + 2)(x + 3), so the relation simplifies to x + 3 = -4 and therefore x = -7.
2.Substituting -7 into the denominator x + 7 gives 0, so the requested expression is undefined.
Answer: C) undefined
Question 17 · GeometryHard
A bakery prices each wedge of a uniformly thick circular tart according to the fraction of the tart contained in the wedge. A whole tart has a marked price of 26 14\frac{1}{4} dollars. After a discount of 15\frac{1}{5} of its marked price, one wedge costs 7 dollars. A point P is chosen on the major arc between the endpoints of the wedge, and straight lines are drawn from P to those endpoints. What is the measure of the angle formed at P?
A48°
B30°
C60°
D36°
E120°
Show answer & explanation
Let's work it out
1.Because the customer paid 45\frac{4}{5} of the marked price, the wedge was marked at 7 ÷ 45\frac{4}{5} = 354\frac{35}{4} dollars.
2.Compared with the whole-tart price of 1054\frac{105}{4} dollars, the wedge is 13\frac{1}{3} of the tart, so its arc measures 120 degrees.
3.The angle at P is an inscribed angle intercepting that arc, so it measures 120 ÷ 2 = 60 degrees.
Answer: C) 60°
Question 18 · Coordinate GraphsHard
A microscope image uses a coordinate grid in which each coordinate unit represents 0.4 millimeter. One endpoint of a needle-shaped organism is A=(1.5,2.0), and its center is M=(3.5,3.5), the midpoint of its two endpoints. A food particle is located at F=(7.5,6.5). How many millimeters separate the food particle from the other endpoint of the organism?
A2.5
B1
C1.4
D6.25
E2
Show answer & explanation
Let's work it out
1.Reflecting A across center M gives the other endpoint B=(5.5,5.0).
2.Its coordinate-grid distance from F is (7.5−5.5)2+(6.5−5.0)2\sqrt{(7.5-5.5)^2+(6.5-5.0)^2}=2.5 units, which represents 2.5*0.4=1 millimeter.
Answer: B) 1
Question 19 · Algebraic ExpressionsHard
If x = 30,000, evaluate x2−100,000,000x2−20,000x+100,000,000\frac{x^2 - 100,000,000}{x^2 - 20,000x + 100,000,000} - x2−400,000,000x2−40,000x+400,000,000\frac{x^2 - 400,000,000}{x^2 - 40,000x + 400,000,000}.
A1
B7
C-3
D0
E3
Show answer & explanation
Let's work it out
1.The first fraction factors and simplifies to x+10,000x−10,000\frac{x + 10,000}{x - 10,000}, which has value 2 at x = 30,000.
2.The second simplifies to x+20,000x−20,000\frac{x + 20,000}{x - 20,000}, which has value 5.
3.Their difference is 2 - 5 = -3.
Answer: C) -3
Question 20 · Coordinate GraphsHard
Coordinate gridCoordinate grid, x-axis from 0 to 3, y-axis from 0 to 3. Points: A (1/2, 1/2) at (0.5, 0.5), B (5/2, 3/2) at (2.5, 1.5), P (1/2, 5/2) at (0.5, 2.5), Q (9/4, 3/2) at (2.25, 1.5). Also shown: a line labeled AB. x y 123123A (1/2, 1/2)B (5/2, 3/2)P (1/2, 5/2)Q (9/4, 3/2)
Mira is laying out a quilt design on a coordinate grid. The figure shows the fabric grain guide AB and two marked points, P and Q. She draws one chalk line through P parallel to AB and a second chalk line through Q perpendicular to AB. A decorative square will be centered where the two chalk lines meet. What is the x-coordinate of that center?
A1310\frac{13}{10}
B920\frac{9}{20}
C38\frac{3}{8}
D72\frac{7}{2}
E32\frac{3}{2}
Show answer & explanation
Let's work it out
1.Identify the slope of grain guide AB from the figure.
2.Using the two marked points on AB, the slope is 12\frac{1}{2}.
3.A line through P parallel to AB shares this slope; using P's coordinates from the figure, that chalk line has equation y = (12\frac{1}{2})x + 94\frac{9}{4}.
4.A line through Q perpendicular to AB has slope equal to the negative reciprocal of 12\frac{1}{2}, which is −2; using Q's coordinates, that chalk line has equation y = −2x + 6.
5.The center of the decorative square is where these two lines meet, so set the expressions equal: (12\frac{1}{2})x + 94\frac{9}{4} = −2x + 6.
6.Add 2x to both sides: (52\frac{5}{2})x + 94\frac{9}{4} = 6.
7.Subtract 94\frac{9}{4}: (52\frac{5}{2})x = 6 − 94\frac{9}{4} = 154\frac{15}{4}.
8.Multiply both sides by 25\frac{2}{5}: x = (154\frac{15}{4})(25\frac{2}{5}) = 3020\frac{30}{20} = 32\frac{3}{2}.
9.The x-coordinate of the center is 32\frac{3}{2}.
Answer: E) 32\frac{3}{2}

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