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Free ISEE Math Practice Test — Grade 10

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

20 real practice questions at ISEE 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 · Equations & InequalitiesIntro
A recipe adjustment formula used in a cooking class relates the adjusted calorie count C to two recipe constants a and b and the serving temperature T (in degrees Celsius, which can be negative for chilled dishes) by C = a - bT. Which of the following expresses T in terms of a, b, and C?
Aa−Cb\frac{a - C}{b}
Bba−C\frac{b}{a - C}
Cab\frac{a}{b} - C
DC−ab\frac{C - a}{b}
Show answer & explanation
Let's work it out
Starting from C = a - bT, subtract a from both sides to get C - a = -bT, then divide both sides by -b to get T = C−a−b\frac{C - a}{-b}, which simplifies to a−Cb\frac{a - C}{b}.
Answer: A) a−Cb\frac{a - C}{b}
Question 2 · MathIntro
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 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}
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 3 · MathIntro
If h(x + 0.5) = x² - 0.25, what is h(1.5)?
A1
B1.25
C0.75
D3.75
Show answer & explanation
Let's work it out
1.For the argument of h to equal 1.5, x + 0.5 must equal 1.5, so x = 1.
2.Therefore, h(1.5) = 1² - 0.25 = 0.75.
Answer: C) 0.75
Question 4 · Coordinate GraphsIntro
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
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 5 · Data & StatisticsIntro
School Supply Shop Sales (in dollars)
MonthOnlineIn-Store
September1,200800
October1,5001,000
November2,100900
December1,8001,500
The table shows monthly sales figures (in dollars) for a school supply shop, broken down by online and in-store purchases over four months. What fraction of the shop's total four-month sales came from online purchases?
A29\frac{2}{9}
B718\frac{7}{18}
C1625\frac{16}{25}
D1118\frac{11}{18}
Show answer & explanation
Let's work it out
1.Add the online sales across all four months: 1,200 + 1,500 + 2,100 + 1,800 = 6,600 dollars.
2.Then add every month's total to get the grand total: (1,200 + 800) + (1,500 + 1,000) + (2,100 + 900) + (1,800 + 1,500) = 2,000 + 2,500 + 3,000 + 3,300 = 10,800 dollars.
3.The fraction of total sales that came from online purchases is 6,60010\frac{600}{10},800, which simplifies by dividing both numerator and denominator by 600 to give 1118\frac{11}{18}.
Answer: D) 1118\frac{11}{18}
Question 6 · Arithmetic & Number SenseEasy
Classroom Reading Challenge
WeekWords Read
214000
526000
During a classroom reading challenge, the number of words read by the class each week forms an arithmetic sequence. According to the table, how many words does the class read altogether during the first 8 weeks?
A352000
B38000
C208000
D192000
Show answer & explanation
Let's work it out
1.From week 2 to week 5, the total increases by 12000 words over 3 equal steps, so the weekly increase is 4000 words.
2.Thus, the first and eighth weekly totals are 10000 and 38000, respectively.
3.Their average is 24000, so the total for 8 weeks is 8 × 24000 = 192000.
Answer: D) 192000
Question 7 · ProbabilityEasy
Two numbers are selected successively at random from the set {-15, -12, -9, -6, -3, 2, 4, 6, 8, 10}. If the first number is negative, it is replaced before the second selection. If the first number is positive, it is not replaced. Given that the second number selected is negative, what is the probability that the first number selected was negative?
A9/19
B9/38
C18/19
D1/2
Show answer & explanation
Let's work it out
1.The probability that both numbers are negative is 5/10 times 5/10, since a negative first number is replaced.
2.The total probability of a negative second number is 5/10 times 5/10 plus 5/10 times 5/9, which equals 19/36.
3.Dividing the probability that both are negative by the probability that the second is negative gives 9/19.
Answer: A) 9/19
Question 8 · Ratios & ProportionsEasy
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}
D12\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: D) 12\frac{1}{2}
Question 9 · Data & StatisticsEasy
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}
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 10 · Equations & InequalitiesStandard
For a cooking contest, each batch makes 2 trays of 4 tasting bites, and 5 bites are set aside for the judges. The team must have fewer than 51 bites left for visitors. The team begins with 25 cups of dough, uses 3 cups per batch, and must have fewer than 10 cups left. Which candidate number of batches meets both requirements?
A6
B4
C7
D8
Show answer & explanation
Let's work it out
1.The number of visitor bites is 8x - 5, so 8x - 5 < 51 gives x < 7.
2.The remaining dough is 25 - 3x, so 25 - 3x < 10 gives x > 5 after the inequality reverses.
3.Thus 5 < x < 7, and the candidate that meets both requirements is 6.
Answer: A) 6
Question 11 · Algebraic ExpressionsStandard
If a is a negative integer and x2 + ax - 72 has x-a as a factor, what is the constant term of the other binomial factor?
A12
B-6
C-12
D-3
Show answer & explanation
Let's work it out
1.Write the other factor as x+k and compare (x-a)(x+k) with x2+ax-72.
2.The coefficient relationships give k=2a and -ak=-72, so a2=36.
3.Since a is negative, a=-6 and the other constant is k=-12.
Answer: C) -12
Question 12 · MathStandard
During a classroom activity, students compare the exponential function E(x) = (12\frac{1}{2})rx with the linear function L(x) = 12\frac{1}{2} + 5x/16. Which candidate value of r makes both E(2) = L(2) and E(3) > L(3) true?
A516\frac{5}{16}
B94\frac{9}{4}
C32\frac{3}{2}
D98\frac{9}{8}
Show answer & explanation
Let's work it out
1.Evaluating the linear function gives L(2) = 98\frac{9}{8}, so (12\frac{1}{2})r2 = 98\frac{9}{8} and r2 = 94\frac{9}{4}.
2.Testing the resulting candidates at x = 3 shows that r = 32\frac{3}{2} gives E(3) = 2716\frac{27}{16}, which is greater than L(3) = 2316\frac{23}{16}.
Answer: C) 32\frac{3}{2}
Question 13 · Equations & InequalitiesStandard
What is the sum of the squares of all real numbers x that satisfy |2|x - 3| - 5| = 1?
A62
B61
C26
D36
Show answer & explanation
Let's work it out
1.The outer absolute value produces the cases 2|x - 3| - 5 = 1 and 2|x - 3| - 5 = -1.
2.These simplify to |x - 3| = 3 and |x - 3| = 2, giving x = 6, 0, 5, and 1.
3.The sum of their squares is 36 + 0 + 25 + 1 = 62.
Answer: A) 62
Question 14 · MathStandard
For h(x)=x+7\sqrt{x+7}+−x−2\sqrt{-x-2}+1x+4\frac{1}{x+4}, what is the sum of all integer inputs in the domain of h?
A-23
B-27
C-2
D-14
Show answer & explanation
Let's work it out
1.The square roots require x ≥ -7 and x ≤ -2, so their combined restriction is -7 ≤ x ≤ -2.
2.Since the denominator cannot be zero, -4 must be removed.
3.The remaining integer inputs are -7, -6, -5, -3, and -2, whose sum is -23.
Answer: A) -23
Question 15 · MathHard
If x - {[(0.008)2/3 + (0.0001)1/2] / (0.25)1/2} = 0.7, what is the value of x?
A0.6
B0.725
C0.8
D1.12
Show answer & explanation
Let's work it out
1.The cube root of 0.008 is 0.2, so (0.008)2/3 = 0.22 = 0.04.
2.The other fractional powers are (0.0001)1/2 = 0.01 and (0.25)1/2 = 0.5, making the fraction (0.04 + 0.01)/0.5 = 0.1.
3.Therefore, x - 0.1 = 0.7, so x = 0.8.
Answer: C) 0.8
Question 16 · Algebraic ExpressionsHard
If x is a real number and x2−0.36x2+1.6x+0.6\frac{x^2 - 0.36}{x^2 + 1.6x + 0.6} = 0.2, which value of x satisfies the relation?
A0.8
B-0.6
C2.0
D1.0
Show answer & explanation
Let's work it out
1.Factor the rational expression and cancel the common factor x + 0.6, giving x−0.6x+1\frac{x - 0.6}{x + 1} = 0.2.
2.Cross-multiplication gives x - 0.6 = 0.2x + 0.2, so 0.8x = 0.8 and x = 1.0.
3.This value does not make the original denominator zero.
Answer: D) 1.0
Question 17 · GeometryHard
A circle has radius 12. A sector has area 48π. An inscribed angle intercepting the sector's arc measures (2x + 100)°. What is x?
A-20
B10
C-852\frac{85}{2}
D80
Show answer & explanation
Let's work it out
1.The entire circle has area 144π, so the sector with area 48π occupies 13\frac{1}{3} of the circle and intercepts a 120° arc.
2.An inscribed angle intercepting that arc measures 60°.
3.Thus, 2x + 100 = 60, so x = -20.
Answer: A) -20
Question 18 · Coordinate GraphsHard
If M=(3,5) is the midpoint of AB, N=(7,4) is the midpoint of BC, and A=(1,2), what is the distance AC?
A2*17\sqrt{17}
B17\sqrt{17}
C101\sqrt{101}
D68
Show answer & explanation
Let's work it out
1.Reflect A across midpoint M to obtain B=(5,8), and then reflect B across midpoint N to obtain C=(9,0).
2.Thus, AC=(9−1)2+(0−2)2\sqrt{(9-1)^2+(0-2)^2}=68\sqrt{68}=2*17\sqrt{17}.
Answer: A) 2*17\sqrt{17}
Question 19 · GeometryHard
A sector of a circle has area 100000π square units and arc length 400π units. What is the measure of the central angle of the sector?
A144 degrees
B72 degrees
C288 degrees
D216 degrees
Show answer & explanation
Let's work it out
1.Use the sector relationship A = ½ · r · L, where A is the area, r is the radius, and L is the arc length.
2.Solving for the radius: r = 2A/L = 2×100000π400π\frac{2 × 100000π}{400π} = 200000π / 400π = 500 units.
3.The full circumference of the circle is 2πr = 2π(500) = 1000π units.
4.The arc length of 400π is therefore 400π/1000π = 25\frac{2}{5} of the full circumference, meaning the sector spans 25\frac{2}{5} of the full circle.
5.The central angle is (25\frac{2}{5}) × 360° = 144°.
Answer: A) 144 degrees
Question 20 · Coordinate GraphsHard
On a music editing grid, the horizontal coordinate gives the beat number and the vertical coordinate gives an automation level. A melody automation line passes through (1, 2) and (5, 4). A bass automation line passes through (2, 3) and is parallel to the melody line. A cymbal fade begins at (4, 9), follows a line perpendicular to the bass automation, and ends when it meets the bass automation. At which beat does the cymbal fade end?
A315\frac{31}{5}
B9
C23\frac{2}{3}
D6
Show answer & explanation
Let's work it out
1.The melody line has slope 12\frac{1}{2}, so the parallel bass line through (2, 3) is y = 12\frac{1}{2}x + 2.
2.The perpendicular cymbal fade has slope -2, giving y = -2x + 17 through (4, 9).
3.Setting the two equations equal gives 12\frac{1}{2}x + 2 = -2x + 17, so the cymbal fade ends at beat 6.
Answer: D) 6

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