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ISEE Quantitative Reasoning study materials
ISEE Quantitative Reasoning study resources

ISEE Quantitative Reasoning practice questions

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Question 1Quantitative Reasoning

A number is multiplied by 4 and then 7 is added. The result is 35. What is the number?

  1. 10
  2. 8
  3. 7 Correct answer.
  4. 12

Explanation

Work backward from the result: subtract 7 to get 28, then divide by 4 to get 7. Checking, 7 × 4 + 7 = 35, so the value is confirmed.

Question 2Quantitative Reasoning

What is the value of 3/4 expressed as a decimal?

  1. 0.75 Correct answer.
  2. 0.43
  3. 0.80
  4. 0.34

Explanation

Dividing the numerator by the denominator gives 3 ÷ 4 = 0.75. This is the standard conversion from a fraction to a decimal.

Question 3Quantitative Reasoning

A rectangle has a length of 12 cm and a width of 5 cm. What is its perimeter?

  1. 60 cm
  2. 30 cm
  3. 17 cm
  4. 34 cm Correct answer.

Explanation

Perimeter of a rectangle is 2(length + width). Substituting gives 2(12 + 5) = 2(17) = 34 cm.

Question 4Quantitative Reasoning

Which list of numbers is ordered from least to greatest?

  1. 0.3, 0.6, 0.098, 0.45
  2. 0.6, 0.45, 0.098, 0.3
  3. 0.45, 0.098, 0.6, 0.3
  4. 0.098, 0.3, 0.45, 0.6 Correct answer.

Explanation

Compare decimals place by place: 0.098 is smallest, then 0.3, then 0.45, then 0.6. Only one list follows this ascending order.

Question 5Quantitative Reasoning

A bag contains 5 red marbles and 15 blue marbles. What is the ratio of red marbles to blue marbles in simplest form?

  1. 1:4
  2. 5:3
  3. 3:1
  4. 1:3 Correct answer.

Explanation

The ratio 5:15 simplifies by dividing both terms by 5, giving 1:3. Ratios are reduced the same way fractions are.

Question 6Quantitative Reasoning

What is 25% of 80?

  1. 32
  2. 25
  3. 20 Correct answer.
  4. 16

Explanation

Convert the percent to a decimal and multiply: 0.25 × 80 = 20. Equivalently, 25% is one quarter, and 80 ÷ 4 = 20.

Question 7Quantitative Reasoning

A sequence begins 3, 7, 11, 15, ... and continues with the same pattern. What is the next term?

  1. 19 Correct answer.
  2. 18
  3. 17
  4. 23

Explanation

Each term increases by a constant difference of 4, so the next term is 15 + 4 = 19. This is an arithmetic sequence with common difference 4.

Question 8Quantitative Reasoning

If x + 9 = 21, what is the value of 2x?

  1. 30
  2. 24 Correct answer.
  3. 42
  4. 12

Explanation

Subtract 9 from both sides to get x = 12, then double it to find 2x = 24.

Question 9Quantitative Reasoning

A train travels at a constant speed of 60 miles per hour. How far does it travel in 2 hours and 30 minutes?

  1. 130 miles
  2. 150 miles Correct answer.
  3. 180 miles
  4. 120 miles

Explanation

Convert the time to hours: 2.5 hours. Multiply by the rate: 60 × 2.5 = 150 miles.

Question 10Quantitative Reasoning

A shirt originally priced at $40 is discounted by 20%. What is the sale price?

  1. $8
  2. $32 Correct answer.
  3. $20
  4. $38

Explanation

The discount is 0.20 × 40 = $8, so the sale price is 40 − 8 = $32. Equivalently, paying 80% of $40 gives $32.

Question 11Quantitative Reasoning

The average of five numbers is 12. Four of the numbers are 8, 10, 14, and 15. What is the fifth number?

  1. 12
  2. 13 Correct answer.
  3. 11
  4. 16

Explanation

The total of all five numbers must be 5 × 12 = 60. The four known numbers sum to 47, so the fifth is 60 − 47 = 13.

Question 12Quantitative Reasoning

A square has an area of 49 square centimeters. What is its perimeter?

  1. 28 cm Correct answer.
  2. 49 cm
  3. 14 cm
  4. 21 cm

Explanation

The side length is the square root of the area, √49 = 7 cm. The perimeter of a square is 4 × 7 = 28 cm.

Question 13Quantitative Reasoning

If 3x − 5 = 2x + 7, what is the value of x?

  1. 7
  2. 24
  3. 2
  4. 12 Correct answer.

Explanation

Subtract 2x from both sides to get x − 5 = 7, then add 5 to get x = 12. Checking, 3(12) − 5 = 31 and 2(12) + 7 = 31.

Question 14Quantitative Reasoning

A recipe uses 2 cups of flour for every 3 cups of milk. If 12 cups of milk are used, how many cups of flour are needed?

  1. 18
  2. 9
  3. 8 Correct answer.
  4. 6

Explanation

Set up the proportion 2/3 = f/12. Cross-multiplying gives 3f = 24, so f = 8 cups.

Question 15Quantitative Reasoning

A number is increased by 30% and the result is 91. What was the original number?

  1. 70 Correct answer.
  2. 63
  3. 61
  4. 78

Explanation

An increase of 30% means the new value is 130% of the original, so 1.3n = 91. Dividing gives n = 70.

Question 16Quantitative Reasoning

What is the median of the data set 4, 9, 2, 7, 11, 6?

  1. 7
  2. 6
  3. 6.5 Correct answer.
  4. 7.5

Explanation

Order the values: 2, 4, 6, 7, 9, 11. With an even count, the median is the mean of the two middle values, (6 + 7)/2 = 6.5.

Question 17Quantitative Reasoning

A triangle has a base of 10 inches and a height of 6 inches. What is its area?

  1. 16 square inches
  2. 60 square inches
  3. 30 square inches Correct answer.
  4. 120 square inches

Explanation

Area of a triangle is one half times base times height: (1/2)(10)(6) = 30 square inches.

Question 18Quantitative Reasoning

If a number n is divided by 6 and then 3 is subtracted, the result is 5. What is n?

  1. 18
  2. 42
  3. 12
  4. 48 Correct answer.

Explanation

Translate to n/6 − 3 = 5, so n/6 = 8 and n = 48. Checking, 48 ÷ 6 − 3 = 8 − 3 = 5.

Question 19Quantitative Reasoning

A jar contains 4 green, 6 yellow, and 10 white candies. If one candy is chosen at random, what is the probability that it is yellow?

  1. 1/5
  2. 1/2
  3. 3/10 Correct answer.
  4. 3/5

Explanation

There are 6 yellow candies out of 20 total, giving 6/20. Reducing by a factor of 2 yields 3/10.

Question 20Quantitative Reasoning

A cube has a volume of 216 cubic centimeters. What is the length of one edge?

  1. 12 cm
  2. 18 cm
  3. 36 cm
  4. 6 cm Correct answer.

Explanation

The volume of a cube is the edge length cubed, so the edge is the cube root of 216. Since 6 × 6 × 6 = 216, the edge is 6 cm.

Question 21Quantitative Reasoning

A store sells notebooks for $3 each and pens for $2 each. A student buys a total of 10 items for $26. How many notebooks did the student buy?

  1. 6 Correct answer.
  2. 7
  3. 5
  4. 4

Explanation

Let b be notebooks and p pens, with b + p = 10 and 3b + 2p = 26. Substituting p = 10 − b gives 3b + 20 − 2b = 26, so b = 6.

Question 22Quantitative Reasoning

Two positive integers have a sum of 15 and a product of 56. What is the larger integer?

  1. 7
  2. 8 Correct answer.
  3. 9
  4. 14

Explanation

The factors of 56 that add to 15 are 7 and 8, since 7 × 8 = 56 and 7 + 8 = 15. The larger of the two is 8.

Question 23Quantitative Reasoning

A car travels 150 miles in 2.5 hours and then 90 miles in 1.5 hours. What is its average speed for the whole trip?

  1. 55 mph
  2. 62 mph
  3. 58 mph
  4. 60 mph Correct answer.

Explanation

Total distance is 240 miles and total time is 4 hours. Average speed is total distance divided by total time: 240 ÷ 4 = 60 mph.

Question 24Quantitative Reasoning

A rectangle's length is 3 more than twice its width, and its perimeter is 36 cm. What is the width?

  1. 6 cm
  2. 5 cm Correct answer.
  3. 9 cm
  4. 7 cm

Explanation

Let w be the width, so the length is 2w + 3. Perimeter gives 2(w + 2w + 3) = 36, so 6w + 6 = 36 and w = 5. Checking, length is 13 and 2(5 + 13) = 36.

Question 25Quantitative Reasoning

A bag holds 3 red and 5 blue tokens. Two tokens are drawn without replacement. What is the probability that both are red?

  1. 3/14
  2. 9/64
  3. 3/28 Correct answer.
  4. 5/28

Explanation

The probability the first is red is 3/8, and given that, the second is red with probability 2/7. Multiplying gives 6/56 = 3/28.

Question 26Quantitative Reasoning

A number is multiplied by 4, then 7 is added, giving 43. Which equation represents this relationship?

  1. 4(n + 7) = 43
  2. 4n − 7 = 43
  3. 4n + 7 = 43 Correct answer.
  4. 7n + 4 = 43

Explanation

Translating words into symbols, 'a number multiplied by 4' becomes 4n, and adding 7 afterward gives 4n + 7. Setting this equal to the stated result 43 yields 4n + 7 = 43, which is the only equation matching the order of operations described.

Question 27Quantitative Reasoning

Which value of x makes the inequality 5x − 3 < 12 true?

  1. 5
  2. 4
  3. 3
  4. 2 Correct answer.

Explanation

Solving the inequality gives 5x < 15, so x < 3. Among the choices, only 2 is less than 3, so it is the sole value satisfying the inequality.

Question 28Quantitative Reasoning

A right triangle has legs measuring 9 cm and 12 cm. What is the length of its hypotenuse?

  1. 18 cm
  2. 13 cm
  3. 21 cm
  4. 15 cm Correct answer.

Explanation

The Pythagorean theorem states that the hypotenuse squared equals the sum of the squares of the legs: 9² + 12² = 81 + 144 = 225. Taking the square root of 225 gives 15 cm.

Question 29Quantitative Reasoning

What is the greatest common factor of 36 and 60?

  1. 18
  2. 4
  3. 12 Correct answer.
  4. 6

Explanation

Factoring both numbers gives 36 = 2² · 3² and 60 = 2² · 3 · 5. The shared prime factors are 2² and 3, whose product is 12, the greatest common factor.

Question 30Quantitative Reasoning

A circle has a radius of 5 inches. What is its area, in square inches, using π ≈ 3.14?

  1. 157
  2. 78.5 Correct answer.
  3. 15.7
  4. 31.4

Explanation

The area of a circle is πr². Substituting r = 5 gives 3.14 × 25 = 78.5 square inches.

Question 31Quantitative Reasoning

A bag contains 3 red marbles and 7 blue marbles. One marble is drawn and kept, then a second is drawn. What is the probability that both marbles are blue?

  1. 1/5
  2. 7/10
  3. 49/100
  4. 7/15 Correct answer.

Explanation

The first draw is blue with probability 7/10; because that marble is not replaced, the second draw is blue with probability 6/9. Multiplying gives (7/10)(6/9) = 42/90 = 7/15.

Question 32Quantitative Reasoning

A sequence is defined by a₁ = 2 and aₙ = 3aₙ₋₁ − 1 for n ≥ 2. What is a₄?

  1. 14
  2. 32
  3. 5
  4. 41 Correct answer.

Explanation

Applying the recursive rule: a₂ = 3(2) − 1 = 5, a₃ = 3(5) − 1 = 14, and a₄ = 3(14) − 1 = 41. Each term depends on the one immediately before it.

Question 33Quantitative Reasoning

A data set consists of the values 3, 5, 5, 8, 10, 12, 15. What is the mean?

  1. 8
  2. 8.29 Correct answer.
  3. 5
  4. 58

Explanation

The sum of the seven values is 58, and dividing by 7 gives approximately 8.29. The mean is the total divided by the count, not the middle value.

Question 34Quantitative Reasoning

If 2/3 of a number is 18, what is 5/6 of the same number?

  1. 27
  2. 22.5 Correct answer.
  3. 30
  4. 15

Explanation

Let n be the number. From (2/3)n = 18, multiplying both sides by 3/2 gives n = 27. Then (5/6)(27) = 22.5.

Question 35Quantitative Reasoning

The sum of three consecutive even integers is 84. What is the largest of the three integers?

  1. 30 Correct answer.
  2. 28
  3. 26
  4. 32

Explanation

Let the integers be n, n + 2, and n + 4. Their sum is 3n + 6 = 84, so n = 26, and the largest is 26 + 4 = 30.

Question 36Quantitative Reasoning

A rectangle has a perimeter of 40 cm. If its length is 4 cm greater than its width, what is its area?

  1. 64 cm²
  2. 96 cm² Correct answer.
  3. 84 cm²
  4. 100 cm²

Explanation

With width w and length w + 4, the perimeter is 2w + 2(w + 4) = 40, giving 4w + 8 = 40 and w = 8. The length is 12, so the area is 8 × 12 = 96 cm².

Question 37Quantitative Reasoning

A machine produces 240 parts in 8 hours. At the same rate, how many parts does it produce in 5 hours?

  1. 150 Correct answer.
  2. 200
  3. 180
  4. 120

Explanation

The unit rate is 240 ÷ 8 = 30 parts per hour. Multiplying by 5 hours gives 30 × 5 = 150 parts.

Question 38Quantitative Reasoning

Simplify the expression 3(2x − 4) − 2(x − 5).

  1. 4x − 22
  2. 8x − 22
  3. 4x − 2 Correct answer.
  4. 4x + 2

Explanation

Distributing gives 6x − 12 − 2x + 10. Combining like terms yields 4x − 2.

Question 39Quantitative Reasoning

If y varies directly with x, and y = 24 when x = 6, what is y when x = 10?

  1. 4
  2. 240
  3. 60
  4. 40 Correct answer.

Explanation

Direct variation means y = kx. Using the given pair, k = 24/6 = 4, so y = 4x. When x = 10, y = 40.

Question 40Quantitative Reasoning

A square is inscribed in a circle of radius 5. What is the area of the square?

  1. 25
  2. 100
  3. 50 Correct answer.
  4. 25π

Explanation

The diagonal of the inscribed square equals the diameter of the circle, which is 10. For a square, diagonal = s√2, so s = 10/√2, and the area is s² = 100/2 = 50.

Question 41Quantitative Reasoning

What is the slope of the line perpendicular to the line passing through (2, 3) and (6, 11)?

  1. 1/2
  2. −2
  3. −1/2 Correct answer.
  4. 2

Explanation

The slope of the given line is (11 − 3)/(6 − 2) = 8/4 = 2. Perpendicular lines have slopes whose product is −1, so the perpendicular slope is −1/2.

Question 42Quantitative Reasoning

A jacket is marked down 25% and then an additional 20% is taken off the reduced price. What single percent discount is equivalent to these two reductions?

  1. 45%
  2. 55%
  3. 35%
  4. 40% Correct answer.

Explanation

After the first markdown the price is 0.75 of the original, and the second markdown leaves 0.80 of that, giving 0.75 × 0.80 = 0.60. The customer pays 60%, so the equivalent discount is 40%.

Question 43Quantitative Reasoning

If |2x − 5| = 11, what is the sum of all possible values of x?

  1. 8
  2. 5 Correct answer.
  3. 11
  4. −3

Explanation

The absolute value equation splits into 2x − 5 = 11 and 2x − 5 = −11. These give x = 8 and x = −3, whose sum is 5.

Question 44Quantitative Reasoning

A chemistry mixture is 30% acid. How many liters of pure acid must be added to 20 liters of this mixture so that the result is 50% acid?

  1. 6
  2. 4
  3. 8 Correct answer.
  4. 10

Explanation

The original mixture contains 6 liters of acid. Let a be the liters of pure acid added; then (6 + a)/(20 + a) = 0.5. Solving gives 6 + a = 10 + 0.5a, so 0.5a = 4 and a = 8.

Question 45Quantitative Reasoning

A fair six-sided die is rolled twice. What is the probability that the sum of the two rolls is 8?

  1. 1/9
  2. 5/36 Correct answer.
  3. 7/36
  4. 1/6

Explanation

There are 36 equally likely ordered outcomes. The pairs summing to 8 are (2,6), (3,5), (4,4), (5,3), and (6,2), giving 5 favorable outcomes and probability 5/36.

Question 46Quantitative Reasoning

The expression x² − 7x + 12 can be factored as (x − a)(x − b). What is the value of a + b?

  1. −12
  2. 12
  3. −7
  4. 7 Correct answer.

Explanation

Factoring gives (x − 3)(x − 4), so a = 3 and b = 4. Their sum is 7, which equals the coefficient of x with the sign reversed, consistent with the sum of roots.

Question 47Quantitative Reasoning

A geometric sequence has a first term of 3 and a common ratio of 2. What is the sum of the first 6 terms?

  1. 189 Correct answer.
  2. 96
  3. 126
  4. 192

Explanation

The sum of a geometric series is a(rⁿ − 1)/(r − 1). Substituting a = 3, r = 2, and n = 6 gives 3(64 − 1)/(2 − 1) = 3 × 63 = 189.

Question 48Quantitative Reasoning

If x + y = 10 and xy = 21, what is the value of x² + y²?

  1. 100
  2. 58 Correct answer.
  3. 79
  4. 42

Explanation

Using the identity x² + y² = (x + y)² − 2xy, substitute to get 100 − 42 = 58. This avoids solving for x and y individually.

Question 49Quantitative Reasoning

A cone has a radius of 3 cm and a height of 8 cm. What is its volume, in cubic centimeters, in terms of π?

  1. 72π
  2. 12π
  3. 36π
  4. 24π Correct answer.

Explanation

The volume of a cone is (1/3)πr²h. Substituting r = 3 and h = 8 gives (1/3)π(9)(8) = 24π cubic centimeters.

Question 50Quantitative Reasoning

For what value of k does the system kx + 3y = 6 and 4x + 6y = 12 have infinitely many solutions?

  1. 4
  2. 6
  3. 2 Correct answer.
  4. 3

Explanation

Infinitely many solutions occur when the two equations describe the same line, so the coefficients must be proportional. Comparing 4x and kx with 6y and 3y gives k/4 = 3/6 = 1/2, so k = 2.