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Free SSAT Math Practice Test — Grade 8

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 SenseEasy
A biology student is observing bacterial growth in a petri dish. She models the fraction of the dish covered as n60\frac{n}{60}, where n is a whole number between 1 and 60. She records two facts about n: n is divisible by both 3 and 4, and when n is divided by 4, the result is a prime number. What fraction of the petri dish is covered?
A14\frac{1}{4}
B120\frac{1}{20}
C15\frac{1}{5}
D13\frac{1}{3}
E215\frac{2}{15}
Show answer & explanation
Let's work it out
1.Since n must be divisible by both 3 and 4, it must be divisible by LCM(3, 4) = 12, so n is a multiple of 12 in the range 1 to 60: that gives candidates 12, 24, 36, 48, and 60.
2.The second condition requires n divided by 4 to be prime.
3.Checking each: 124\frac{12}{4} = 3 (prime), 244\frac{24}{4} = 6 (not prime), 364\frac{36}{4} = 9 (not prime), 484\frac{48}{4} = 12 (not prime), 604\frac{60}{4} = 15 (not prime).
4.Only n = 12 satisfies both conditions, so the fraction covered is 1260\frac{12}{60}, which simplifies to 15\frac{1}{5}.
Answer: C) 15\frac{1}{5}
Question 2 · Fractions & DecimalsEasy
In a science experiment on bacterial decline, a colony is losing exactly 23\frac{2}{3} of its total starting population every hour due to an introduced antibacterial agent. A researcher wants to work backwards: she knows the colony has so far lost only 16\frac{1}{6} of its starting population in total, and she needs to find what fraction of an hour has elapsed. She sets up the complex fraction whose numerator is the fraction already lost and whose denominator is the hourly loss rate, then simplifies it to find the elapsed time. What fraction of an hour has elapsed?
A19\frac{1}{9}
B49\frac{4}{9}
C14\frac{1}{4}
D12\frac{1}{2}
E56\frac{5}{6}
Show answer & explanation
Let's work it out
1.The researcher needs elapsed time, which equals total loss divided by hourly loss rate.
2.The total loss so far is 16\frac{1}{6} of the starting population, and the hourly loss rate is 23\frac{2}{3} of the starting population per hour.
3.Setting up the complex fraction gives (16\frac{1}{6}) ÷ (23\frac{2}{3}).
4.To divide by a fraction, multiply by its reciprocal: (16\frac{1}{6}) × (32\frac{3}{2}) = 312\frac{3}{12} = 14\frac{1}{4}.
5.Therefore, 14\frac{1}{4} of an hour has elapsed.
Answer: C) 14\frac{1}{4}
Question 3 · MathEasy
A scientist tracks a bacterial colony whose population relative to a baseline changes at a constant rate. At 4 hours before the start of an experiment (t = -4), the population is 20 units below baseline. At the exact start of the experiment (t = 0), the population is 20 units above baseline. Assuming the population changes as a linear function of time, what is the constant rate of change in units per hour?
A3
B10
C5
D8
E2
Show answer & explanation
Let's work it out
1.The rate of change of a linear function equals the change in output divided by the change in input.
2.The population goes from −20 units (below baseline) at t = −4 to +20 units (above baseline) at t = 0.
3.The change in population is 20 − (−20) = 40 units.
4.The change in time is 0 − (−4) = 4 hours.
5.Dividing gives 40 ÷ 4 = 10 units per hour.
Answer: B) 10
Question 4 · Coordinate GraphsEasy
The graphs of y = 0.5x - 2 and y = 1.5x - 4 intersect at a single point. What is that point?
A(3, -0.5)
B(2, -1)
C(0, -2)
D(6, 1)
E(1.5, -1.25)
Show answer & explanation
Let's work it out
1.To find where the two lines meet, set their right-hand sides equal: 0.5x - 2 = 1.5x - 4.
2.Subtracting 0.5x from both sides gives -2 = x - 4, and adding 4 gives x = 2.
3.Substituting back into either equation yields y = 0.5(2) - 2 = -1, so the intersection point is (2, -1).
Answer: B) (2, -1)
Question 5 · PercentagesEasy
Regional Soccer Tournament Attendance
YearAttendance
0?
1?
212,100
The table shows the attendance at a regional soccer tournament for three years. The tournament director says attendance grew by exactly 10% each year from Year 0 to Year 2 through compound annual growth. If Year 2 attendance was 12,100 fans, what was the attendance at Year 0?
A11,000
B10,000
C14,641
D10,890
E9,680
Show answer & explanation
Let's work it out
1.Because attendance grew by 10% each year through compound growth, each year’s attendance equals the previous year’s attendance multiplied by 1.10.
2.To reverse two years of 10% compound growth, divide by 1.10 twice.
3.Dividing 12,100 by 1.10 gives 11,000 (Year 1 attendance), then dividing 11,000 by 1.10 gives 10,000.
4.The original Year 0 attendance was 10,000 fans.
Answer: B) 10,000
Question 6 · MathStandard
Model Cell Dimensions
Volume (cu cm)Side Length (cm)
11
82
273
18\frac{1}{8}?
A science class is modeling cells using cubes. The table below shows the volume of each model cell and the side length students need to compute. For the cell with a volume of 18\frac{1}{8} cubic centimeter, what is the side length in centimeters?
A2
B12\frac{1}{2}
C124\frac{1}{24}
D1512\frac{1}{512}
E38\frac{3}{8}
Show answer & explanation
Let's work it out
1.The side length of a cube equals the cube root of its volume.
2.To find the cube root of 18\frac{1}{8}, take the cube root of the numerator and the cube root of the denominator separately: the cube root of 1 is 1, and the cube root of 8 is 2, giving a side length of 12\frac{1}{2}.
3.This is confirmed by the pattern in the table, where the side length cubed always equals the volume.
Answer: B) 12\frac{1}{2}
Question 7 · MathStandard
Coordinate gridCoordinate grid, x-axis from -4 to 4, y-axis from -20 to 12. Points: survey stake (0, -5), x = -3 at (-3, -17). Also shown: a line through (-4, -21) and (4, 11), labeled E(x) = 4x - 5. x y -4-3-2-11234-20-16-12-8-44812survey stake (0, -5)x = -3
A surveyor models the ground elevation (in feet relative to the base of a retaining wall) along a sloped hillside using the function E(x) = 4x - 5, where x is the horizontal distance in feet from a survey stake. Negative elevation values indicate ground that is below the wall base. What is the ground elevation at a point 3 feet to the left of the survey stake, where x = -3?
A-7
B-12
C7
D-17
E-22
Show answer & explanation
Let's work it out
1.Substitute x = -3 into the function E(x) = 4x - 5.
2.First compute the slope term: 4 times -3 equals -12.
3.Then apply the constant: -12 - 5 = -17.
4.The ground elevation at that point is -17 feet, meaning 17 feet below the base of the wall.
Answer: D) -17
Question 8 · GeometryStandard
Kite Frame Strut Lengths
StrutLength (cm)
Short strut 12.5
Short strut 26.0
Long outer strut6.5
A crafter is building a diamond-shaped kite frame. The table shows the planned lengths of three wooden struts. The two shorter struts meet at a right angle at the center of the kite. The crafter wants to add one straight diagonal brace connecting the outer ends of those two shorter struts. What length, in centimeters, should the diagonal brace be so that the triangle formed by the two shorter struts and the brace is a perfect right triangle?
A7.0
B8.5
C42.25
D6.5
E6.0
Show answer & explanation
Let's work it out
1.The two shorter struts (2.5 cm and 6.0 cm) meet at a right angle, making them the two legs of the right triangle.
2.The diagonal brace connecting their outer ends is the hypotenuse.
3.Applying the Pythagorean theorem: 2.5² + 6.0² = 6.25 + 36.00 = 42.25, and 42.25\sqrt{42.25} = 6.5 cm.
4.The correct answer is D.
Answer: D) 6.5
Question 9 · Coordinate GraphsStandard
A biology student records the height (in centimeters) of five plant seedlings at two different times and models each seedling's growth as a linear function of days elapsed. The student wants the seedling whose growth rate is exactly 1.5 centimeters per day -- that is, whose line has a slope of 1.5. The two recorded (day, height) measurements for each seedling are listed as the five options. Which seedling has a growth rate of exactly 1.5 cm per day?
ADay 0: 2.0 cm; Day 4: 7.0 cm
BDay 0: 0.6 cm; Day 3: 2.6 cm
CDay 0: 0.4 cm; Day 4: 8.4 cm
DDay 0: 1.8 cm; Day 2: 4.8 cm
EDay 0: 3.0 cm; Day 4: 7.0 cm
Show answer & explanation
Let's work it out
1.Slope equals the change in height divided by the change in days elapsed.
2.For seedling D, measured at Day 0 with height 1.8 cm and Day 2 with height 4.8 cm: slope = 4.8−1.82−0\frac{4.8 − 1.8}{2 − 0} = 3.0 / 2.0 = 1.5 cm per day, which matches the required growth rate exactly.
Answer: D) Day 0: 1.8 cm; Day 2: 4.8 cm
Question 10 · IntegersStandard
Positions Along the Trail
LocationSigned Position in Miles
Tunnel-78\frac{7}{8}
Overlook?
Visitor Center0
The table gives signed positions along an east-west trail, measured in miles from the visitor center. A guide defines one ridge unit as the distance from Tunnel to Overlook, which is east of Tunnel. A shuttle route measuring 158\frac{15}{8} miles is exactly 3 ridge units long. What signed position should replace the question mark?
A58\frac{5}{8}
B194\frac{19}{4}
C-32\frac{3}{2}
D2940\frac{29}{40}
E-14\frac{1}{4}
Show answer & explanation
Let's work it out
1.If the Overlook position is x, one ridge unit is x - (-78\frac{7}{8}) = x + 78\frac{7}{8} miles.
2.Since 158x+78\displaystyle \frac{\frac{15}{8}}{x + \frac{7}{8}} = 3, the unit length is 58\frac{5}{8} mile, so x = 58\frac{5}{8} - 78\frac{7}{8} = -14\frac{1}{4}.
Answer: E) -14\frac{1}{4}
Question 11 · Ratios & ProportionsStandard
Live Streaming Event -- Listener Flow
StreamRate (listeners per hour)
Listeners joining+90
Listeners leaving-30
The table shows the rate at which listeners join and leave a live music streaming event each hour. If the event starts with no audience and must reach a net gain of 360 listeners, how many hours will that take?
A3
B6
C4
D300
E12
Show answer & explanation
Let's work it out
1.The net gain per hour is found by adding the two rates in the table: 90 + (-30) = 60 listeners per hour.
2.Dividing the target audience gain by this combined net rate gives 360 / 60 = 6 hours.
Answer: B) 6
Question 12 · MathStandard
Material Thickness (meters)
ExpressionValue
5225
515
501
5-115\frac{1}{5}
5-2?
A carpenter records the thickness of various materials as powers of 5 meters. The table below shows several entries. What is the missing value in the table?
A-125\frac{1}{25}
B-25
C25
D132\frac{1}{32}
E125\frac{1}{25}
Show answer & explanation
Let's work it out
A negative exponent indicates a reciprocal: 5-2 = 1 divided by 52 = 1 divided by 25, which equals 125\frac{1}{25}.
Answer: E) 125\frac{1}{25}
Question 13 · Coordinate GraphsStandard
Cookie Baking Cost
Batches (x)Total Cost in dollars (y)
1350
2500
3650
4800
The table shows the total ingredient cost (in dollars) for baking different numbers of batches of cookies for a school fundraiser. Which equation models the total cost y as a linear function of the number of batches x?
Ay = 150x + 250
By = 150x + 350
Cy = 200x + 150
Dy = 150x + 200
Ey = 800x + 200
Show answer & explanation
Let's work it out
1.The slope is found by computing the change in cost per additional batch: 500−3502−1\frac{500 - 350}{2 - 1} = 150, confirmed consistently across all rows.
2.Using the point (1, 350) in y = mx + b gives 350 = 150(1) + b, so b = 200.
3.The equation is y = 150x + 200, which correctly predicts all four values in the table.
Answer: D) y = 150x + 200
Question 14 · Coordinate GraphsStandard
Coordinate gridCoordinate grid, x-axis from -1 to 7, y-axis from -1 to 6. Points: (0, 2.0) at (0, 2), (4, 4.0) at (4, 4). Also shown: a line with slope 0.5 and y-intercept 2. x y -11234567-1123456(0, 2.0)(4, 4.0)
The graph shows the total monthly cost (in dollars) of a music streaming plan based on the number of premium song downloads purchased. Two points on the line are marked: (0, 2.0) and (4, 4.0). Which equation represents this linear relationship?
Ay = 0.5x + 2.0
By = 2.0x + 2.0
Cy = 1.5x + 1.5
Dy = 0.5x + 1.0
Ey = 2.0x + 0.5
Show answer & explanation
Let's work it out
1.Using the two marked points (0, 2.0) and (4, 4.0), the slope is (4.0 - 2.0) divided by (4 - 0), which equals 2.04\frac{0}{4} = 0.5.
2.The y-intercept is the value when x = 0, which is 2.0 directly from the first point.
3.Substituting into slope-intercept form gives y = 0.5x + 2.0, confirmed by plugging in x = 4: 0.5(4) + 2.0 = 4.0.
Answer: A) y = 0.5x + 2.0
Question 15 · Equations & InequalitiesHard
If 2(x - y) = y - 4 and 3(x + y) = 2x - y - 13, what is the value of x - y?
A-5
B-3
C3
D-2
E-7
Show answer & explanation
Let's work it out
1.Expand and collect terms in each equation: 2(x - y) = y - 4 becomes 2x - 2y = y - 4, or 2x - 3y = -4, and 3(x + y) = 2x - y - 13 becomes 3x + 3y = 2x - y - 13, or x + 4y = -13.
2.From the second equation, x = -13 - 4y; substituting into the first gives -26 - 8y - 3y = -4, so -11y = 22 and y = -2, which makes x = -13 + 8 = -5.
3.Therefore x - y = -5 - (-2) = -3.
Answer: B) -3
Question 16 · GeometryHard
A baker has a rectangular cooling rack with one side measuring 0.9 meters. When the baker stretches a measuring tape from one corner to the opposite corner, the diagonal measures 1.5 meters. The baker needs to know the length of the other side of the rack to check whether it will fit in the storage cabinet. What is the length, in meters, of the other side of the cooling rack?
A0.6
B1.44
C1.2
D1.0
E2.4
Show answer & explanation
Let's work it out
1.The two sides of a rectangle and its diagonal form a right triangle, with the diagonal as the hypotenuse.
2.Here the known leg is 0.9 m and the hypotenuse is 1.5 m, so the missing leg b satisfies 0.9² + b² = 1.5².
3.Computing: 0.81 + b² = 2.25, so b² = 1.44, and b = 1.44\sqrt{1.44} = 1.2 m.
4.This is the 3-4-5 Pythagorean triple scaled by 0.3 (0.9 = 3×0.3, 1.2 = 4×0.3, 1.5 = 5×0.3).
Answer: C) 1.2
Question 17 · PercentagesHard
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 18 · Ratios & ProportionsHard
For a set of handmade bracelets, Nia has one bead mix that is 13\frac{1}{3} metallic beads and another that is 34\frac{3}{4} metallic beads. She begins with 1 18\frac{1}{8} cups of the first mix but uses 38\frac{3}{8} cup on a different project. She combines all of the remaining first mix with some of the second mix so that exactly 12\frac{1}{2} of the finished blend is metallic beads. How many cups of the second mix does she use?
A98\frac{9}{8}
B16\frac{1}{6}
C34\frac{3}{4}
D23\frac{2}{3}
E12\frac{1}{2}
Show answer & explanation
Let's work it out
1.After the other project, 34\frac{3}{4} cup of the first mix remains, containing 14\frac{1}{4} cup of metallic beads.
2.If x is the amount of the second mix, equating the metallic portion to half of the finished blend gives 14\frac{1}{4} + 3x/4 = 12\frac{1}{2} times the quantity 34\frac{3}{4} + x.
3.Solving this equation gives x = 12\frac{1}{2} cup.
Answer: E) 12\frac{1}{2}
Question 19 · Data & StatisticsHard
Set P contains half as many numbers as Set Q, and the mean of Set P is 34\frac{3}{4}. One-fourth of the numbers in Set Q, with a mean of 12\frac{1}{2}, are removed. The mean of the numbers in Set P together with the numbers remaining in Set Q is 710\frac{7}{10}. What was the original mean of Set Q?
A710\frac{7}{10}
B58\frac{5}{8}
C12\frac{1}{2}
D23\frac{2}{3}
E712\frac{7}{12}
Show answer & explanation
Let's work it out
1.Let Set Q contain 8 numbers, so Set P contains 4 and 2 are removed from Set Q.
2.The combined sum after removal is 10 times 710\frac{7}{10}, or 7; subtracting the sum of Set P leaves 4 for the remaining part of Set Q.
3.Restoring the removed sum gives an original Set Q total of 5, so its mean is 58\frac{5}{8}.
Answer: B) 58\frac{5}{8}
Question 20 · Data & StatisticsHard
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

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