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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 · 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?
A38
B35
C44
D53
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: C) 44
Question 2 · Equations & InequalitiesIntro
Bacterial Growth Measurements
DishP (thousands)A (thousands)r
1200250?
2400500?
3800960?
412001500?
A scientist studying bacterial growth models the total colony size after one growth cycle using the formula A = P(1 + r), where P is the initial population (in thousands), r is the fractional growth rate, and A is the population after one cycle. The table shows measurements from four culture dishes. For any dish, which expression correctly gives r in terms of A and P?
Ar = 1 - AP\frac{A}{P}
Br = AP\frac{A}{P}
Cr = A−1P\frac{A - 1}{P}
Dr = AP\frac{A}{P} - 1
Show answer & explanation
Let's work it out
Starting from A = P(1 + r), divide both sides by P to get AP\frac{A}{P} = 1 + r, then subtract 1 from both sides to isolate r, giving r = AP\frac{A}{P} - 1.
Answer: D) r = AP\frac{A}{P} - 1
Question 3 · IntegersIntro
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}
D-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: D) -14\frac{1}{4}
Question 4 · Equations & InequalitiesEasy
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
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 5 · Fractions & DecimalsEasy
Which value is equal to 134+12+13\displaystyle \frac{1}{\frac{3}{4} + \frac{1}{2 + \frac{1}{3}}}?
A3328\frac{33}{28}
B113\frac{11}{3}
C2833\frac{28}{33}
D1219\frac{12}{19}
Show answer & explanation
Let's work it out
1.Evaluate from the innermost fraction outward: 2 + 13\frac{1}{3} = 73\frac{7}{3}, so 12+13\displaystyle \frac{1}{2 + \frac{1}{3}} = 37\frac{3}{7}.
2.Then 34\frac{3}{4} + 37\frac{3}{7} = 3328\frac{33}{28}, and taking the outer reciprocal gives 2833\frac{28}{33}.
Answer: C) 2833\frac{28}{33}
Question 6 · ProbabilityEasy
A swim meet replay station can create a clip by choosing one of 5 race recordings, one camera angle, one of 4 playback speeds, and one starting point. The playback speeds are 0.5, 0.75, 1.0, and 1.25 times normal speed. Starting points occur every 0.8 second from 0.0 through 2.4 seconds, inclusive. If the station offers about 610 distinct clips, which is closest to the number of camera angles available?
A6
B38
C8
D10
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Let's work it out
1.There are 4 starting points: 0.0, 0.8, 1.6, and 2.4 seconds.
2.Each camera angle therefore produces 5 × 4 × 4 = 80 possible clips.
3.Since 610 ÷ 80 = 7.625, the number of camera angles is closest to 8.
Answer: C) 8
Question 7 · 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}
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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 8 · ProbabilityEasy
At a school music festival, a listening challenge uses 1,000 prompt cards, 500 about melody and 500 about rhythm. After a melody card is used in round 1, the judge returns it to the card box because melody prompts may be repeated. After a rhythm card is used, the player keeps it as a point token. The cards are mixed before a card is chosen for round 2. What is the probability that the round 2 card is about melody?
A9992000\frac{999}{2000}
B19973996\frac{1997}{3996}
C500999\frac{500}{999}
D19993996\frac{1999}{3996}
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Let's work it out
1.Separate the outcome into two cases based on the first card.
2.The melody-then-melody case has probability 5001000\frac{500}{1000} times 5001000\frac{500}{1000}, while the rhythm-then-melody case has probability 5001000\frac{500}{1000} times 500999\frac{500}{999}.
3.Adding these probabilities gives 14\frac{1}{4} + 250999\frac{250}{999} = 19993996\frac{1999}{3996}.
Answer: D) 19993996\frac{1999}{3996}
Question 9 · 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 10 · 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
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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 11 · 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
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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 12 · 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
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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 13 · Algebraic ExpressionsStandard
If p and q are nonzero real numbers and (p-3q2)/(pq-2) = 1681\frac{16}{81}, what is the value of (p-6q6)/(pq)0?
A827\frac{8}{27}
B72964\frac{729}{64}
C2566561\frac{256}{6561}
D64729\frac{64}{729}
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Let's work it out
1.The given expression is (qp\frac{q}{p})4, so the magnitude of qp\frac{q}{p} is 23\frac{2}{3}.
2.The target expression is (qp\frac{q}{p})6 because (pq)0 equals 1.
3.Since the sixth power is even, either possible sign of qp\frac{q}{p} gives 64729\frac{64}{729}.
Answer: D) 64729\frac{64}{729}
Question 14 · MathStandard
For every nonnegative whole number x, f(x + 1) - f(x) = 2x + 3. If f(0) = 1, what is f(5)?
A35
B46
C36
D94
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Let's work it out
1.The successive increases are 3, 5, 7, 9, and 11.
2.Starting from 1 and adding these increases gives 1 + 3 + 5 + 7 + 9 + 11 = 36.
Answer: C) 36
Question 15 · MathHard
A school weather club compares fractional storm-model indices. Which candidate has a value strictly between 14\frac{1}{4} and 13\frac{1}{3}?
A(49\frac{4}{9})32\frac{3}{2}
B(1243\frac{1}{243})15\frac{1}{5}
C(132\frac{1}{32})25\frac{2}{5}
D(271000\frac{27}{1000})23\frac{2}{3}
Show answer & explanation
Let's work it out
1.For (49\frac{4}{9})32\frac{3}{2}, first take the square root of 49\frac{4}{9} to obtain 23\frac{2}{3}, and then cube the result to obtain 827\frac{8}{27}.
2.Since 14\frac{1}{4} < 827\frac{8}{27} < 13\frac{1}{3}, the qualifying candidate is (49\frac{4}{9})32\frac{3}{2}.
Answer: A) (49\frac{4}{9})32\frac{3}{2}
Question 16 · 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
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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 17 · 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 18 · MathHard
For a class warm-up, which expression has a value that is a perfect cube greater than 10 and less than 100?
A2532\frac{3}{2}
B8134\frac{3}{4}
C1634\frac{3}{4}
D3245\frac{4}{5}
Show answer & explanation
Let's work it out
1.The fourth root of 81 is 3, so 8134\frac{3}{4} = 33 = 27.
2.Since 27 is a perfect cube greater than 10 and less than 100, the required expression is 8134\frac{3}{4}.
Answer: B) 8134\frac{3}{4}
Question 19 · GeometryHard
A school art club is making a circular display for a classroom. A blue sector occupies 38\frac{3}{8} of the display and has area 54π square inches. An adjacent gold sector has a central angle that is 23\frac{2}{3} as large as the central angle of the blue sector. What is the length, in inches, of the curved outer edge of the gold sector?
A36π
B16π
C3π
D6π
Show answer & explanation
Let's work it out
1.Since the blue sector is 38\frac{3}{8} of the circle, its area gives (38\frac{3}{8})πr² = 54π, so the radius is 12 inches.
2.The gold sector occupies (23\frac{2}{3})(38\frac{3}{8}) = 14\frac{1}{4} of the circle.
3.Therefore, its curved edge is (14\frac{1}{4})(2π)(12) = 6π inches long.
Answer: D) 6π
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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