Home

Tutoring

Subjects

Live Classes

Study Coach

Essay Review

On-Demand Courses

Colleges

Games

Sign up

Log in

Opening subject page...

Loading your content

Practice

  • All Subjects
  • Algebra Flashcards
  • SAT Math Practice Tests
  • Math Question of the Day
  • Live Classes
  • On-Demand Courses

Varsity Tutors

  • Find a Tutor
  • Test Prep
  • Online Classes
  • K-12 Learning
  • College Search
  • VarsityTutors.com

© 2026 Varsity Tutors. All rights reserved.

← Back to quizzes

ISEE Middle Level Quantitative Reasoning Quiz

ISEE Middle Level Quantitative Reasoning Quiz: Pattern Rules

Practice Pattern Rules in ISEE Middle Level Quantitative Reasoning with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

Question 1 / 19

0 of 19 answered

Each day, a scientist observes that a certain bacterial culture consumes one-third of the nutrients in its petri dish. If the dish started with a full supply of nutrients on Monday morning, what fraction of the original nutrients will remain at the start of business on Thursday morning?

Select an answer to continue

What this quiz covers

This quiz focuses on Pattern Rules, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Middle Level Quantitative Reasoning.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Each day, a scientist observes that a certain bacterial culture consumes one-third of the nutrients in its petri dish. If the dish started with a full supply of nutrients on Monday morning, what fraction of the original nutrients will remain at the start of business on Thursday morning?

  1. 8/27 (correct answer)
  2. 1/3
  3. 2/3
  4. 1/81

Explanation: If the culture consumes one-third of the nutrients, then two-thirds (2/3) of the nutrients remain each day. We need to find the amount remaining on Thursday morning.

  • Monday morning: 1 (full supply)
  • Tuesday morning: 2/3 of the supply remains from Monday.
  • Wednesday morning: 2/3 of Tuesday's amount remains, which is (2/3) * (2/3) = 4/9 of the original.
  • Thursday morning: 2/3 of Wednesday's amount remains, which is (2/3) * (4/9) = 8/27 of the original. So, 8/27 of the original nutrients will remain on Thursday morning.

Question 2

The units digits of the powers of 7 follow a repeating pattern: 7¹=7, 7²=49, 7³=343, 7⁴=2401, 7⁵=16807. The pattern of the units digits is 7, 9, 3, 1, and then it repeats. What is the units digit of 7³⁰?

  1. 7
  2. 9 (correct answer)
  3. 3
  4. 1

Explanation: The pattern of the units digits of powers of 7 is (7, 9, 3, 1). This is a cycle of length 4. To find the units digit of 7³⁰, we need to determine where in the cycle the 30th power falls. We can do this by dividing the exponent, 30, by the length of the cycle, 4. 30 ÷ 4 = 7 with a remainder of 2. The remainder tells us the position in the cycle.

  • A remainder of 1 corresponds to the first digit (7).
  • A remainder of 2 corresponds to the second digit (9).
  • A remainder of 3 corresponds to the third digit (3).
  • A remainder of 0 corresponds to the fourth digit (1). Since the remainder is 2, the units digit of 7³⁰ is the second digit in the pattern, which is 9.

Question 3

A sequence of numbers starts with 1, 2, 4, 7, 11, 16, ... . The pattern continues. What is the rule to find the next number in the sequence?

  1. Add the two previous numbers.
  2. Double the previous number and subtract an increasing amount.
  3. Add a number that is one greater than the number added previously. (correct answer)
  4. Add the term's position number to the previous term.

Explanation: Let's examine the difference between consecutive terms:

  • From 1 to 2, we add 1.
  • From 2 to 4, we add 2.
  • From 4 to 7, we add 3.
  • From 7 to 11, we add 4.
  • From 11 to 16, we add 5. The rule is to add a consecutive integer to each term to get the next term. This is described by choice C: "Add a number that is one greater than the number added previously."

Question 4

A sequence of fractions is 1/3, 2/5, 3/7, 4/9, ... . If the pattern continues, which of the following describes the rule for finding the nth fraction in the sequence?

  1. The numerator is n, and the denominator is the previous denominator plus 2.
  2. The numerator is n, and the denominator is the numerator plus 2n-1.
  3. The numerator is n, and the denominator is n + 2.
  4. The numerator is n, and the denominator is 2n + 1. (correct answer)

Explanation: Let's analyze the numerator and denominator separately in relation to the term number, n.

  • Numerators: 1, 2, 3, 4, ... For the nth term, the numerator is simply n.
  • Denominators: 3, 5, 7, 9, ... This is an arithmetic sequence. Let's see if we can relate it to n. For n=1, denominator is 3. 2(1)+1 = 3. For n=2, denominator is 5. 2(2)+1 = 5. For n=3, denominator is 7. 2(3)+1 = 7. For n=4, denominator is 9. 2(4)+1 = 9. The rule for the denominator is 2n + 1. Therefore, the rule for the nth fraction is n / (2n + 1).

Question 5

The terms in a sequence are generated by the rule V = n³ - 2n², where V is the value of the term and n is its position number (n=1, 2, 3...). What is the first term to have a value greater than 50?

  1. The 4th term
  2. The 5th term (correct answer)
  3. The 6th term
  4. The 7th term

Explanation: We need to test values of n starting from 1 until the value of the term, V, is greater than 50.

  • For n=1: V = 1³ - 2(1²) = 1 - 2 = -1
  • For n=2: V = 2³ - 2(2²) = 8 - 8 = 0
  • For n=3: V = 3³ - 2(3²) = 27 - 18 = 9
  • For n=4: V = 4³ - 2(4²) = 64 - 32 = 32
  • For n=5: V = 5³ - 2(5²) = 125 - 50 = 75 Since 75 is the first value in the sequence greater than 50, the 5th term is the correct answer.

Question 6

The first five terms of a sequence are 2, 5, 10, 17, 26. Which rule describes the relationship between a term's value (V) and its position number (n) in the sequence?

  1. V = 3n - 1
  2. V = n² + 1 (correct answer)
  3. Add the next consecutive odd number, starting with 3, to the previous term's value.
  4. V = 2n + (n - 2)

Explanation: The question asks for an explicit rule that connects the value of a term (V) directly to its position number (n). Let's test the terms:

  • For n=1, V=2
  • For n=2, V=5
  • For n=3, V=10
  • For n=4, V=17
  • For n=5, V=26 The pattern is that each term is one more than the square of its position number. For example, for n=3, 3² + 1 = 9 + 1 = 10. For n=5, 5² + 1 = 25 + 1 = 26. So, the rule is V = n² + 1.

Question 7

A pattern starts with 100 and decreases by 7 at each step (Pattern X). Another pattern starts with 5 and doubles at each step (Pattern Y).

Column A: The value of the 6th term in Pattern X. Column B: The value of the 6th term in Pattern Y.

  1. The quantity in Column A is greater.
  2. The quantity in Column B is greater. (correct answer)
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.

Explanation: To find the value in each column, we list the terms of each pattern. Column A (Pattern X): The sequence is 100, 93, 86, 79, 72, 65, ... The 6th term is 65. Column B (Pattern Y): The sequence is 5, 10, 20, 40, 80, 160, ... The 6th term is 160. Comparing the two values, 160 is greater than 65. Therefore, the quantity in Column B is greater.

Question 8

A sequence of numbers follows the rule: "multiply the previous term by 2, then add 3." The 4th term of the sequence is 37. What is the first term?

  1. 2 (correct answer)
  2. 3
  3. 5
  4. 7

Explanation: To find the first term, we must work backward from the 4th term. The rule is T_n = 2 * T_(n-1) + 3. The inverse operation is to subtract 3, then divide by 2.

  • 4th term = 37
  • 3rd term = (37 - 3) / 2 = 34 / 2 = 17
  • 2nd term = (17 - 3) / 2 = 14 / 2 = 7
  • 1st term = (7 - 3) / 2 = 4 / 2 = 2 The first term is 2. We can check by working forward: 2 -> (22+3)=7 -> (72+3)=17 -> (17*2+3)=37. This confirms the answer.

Question 9

A pattern is defined by the rule "multiply the previous term by x, then add 2." The first term is 4. The variable x is a positive integer.

Column A: The value of the third term if x = 3. Column B: 50.

  1. The quantity in Column A is greater.
  2. The quantity in Column B is greater. (correct answer)
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.

Explanation: Let's calculate the value for Column A. The first term is 4 and the rule is T_n = x * T_(n-1) + 2. We are given x = 3.

  • First term: 4
  • Second term: (4 * 3) + 2 = 12 + 2 = 14
  • Third term: (14 * 3) + 2 = 42 + 2 = 44 So, the quantity in Column A is 44. The quantity in Column B is 50. Since 50 is greater than 44, the quantity in Column B is greater.

Question 10

Leo is building pyramids with blocks. A Level 1 pyramid is a single block. For each subsequent level, he builds a square base and places the previous level's pyramid on top. A Level 2 pyramid has a 2x2 base, and a Level 3 pyramid has a 3x3 base. The total number of blocks in a pyramid is the sum of the blocks in its base and all the blocks in the levels above it. Following this pattern, how many total blocks are in a Level 4 pyramid?

  1. 16
  2. 21
  3. 30 (correct answer)
  4. 64

Explanation: Let's determine the total number of blocks for each level:

  • Level 1: 1² = 1 block total.
  • Level 2: Has a 2x2 base (4 blocks) plus the Level 1 pyramid on top (1 block). Total = 4 + 1 = 5 blocks. (This is 1² + 2²).
  • Level 3: Has a 3x3 base (9 blocks) plus the Level 2 pyramid on top (5 blocks). Total = 9 + 5 = 14 blocks. (This is 1² + 2² + 3²). The pattern is that the total number of blocks in a Level n pyramid is the sum of the squares from 1 to n. For a Level 4 pyramid, the total is 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30 blocks.

Question 11

A numerical sequence follows an alternating two-step rule. The sequence begins 2, 6, 3, 9, 6, 18, 15, ... If the pattern continues, what is the next number in the sequence?

  1. 12
  2. 30
  3. 42
  4. 45 (correct answer)

Explanation: Let's analyze the steps between consecutive terms:

  • From 2 to 6: Multiply by 3.
  • From 6 to 3: Subtract 3.
  • From 3 to 9: Multiply by 3.
  • From 9 to 6: Subtract 3.
  • From 6 to 18: Multiply by 3.
  • From 18 to 15: Subtract 3. The pattern is to alternately multiply by 3 and subtract 3. The last operation was subtracting 3 to get 15. The next operation must be to multiply by 3. So, the next term is 15 * 3 = 45.

Question 12

A sequence is defined by the rule that each term after the second is the sum of the two preceding terms.

Column A: The 8th term if the first two terms are 1 and 4. Column B: The 7th term if the first two terms are 2 and 5.

  1. The quantity in Column A is greater. (correct answer)
  2. The quantity in Column B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.

Explanation: We need to generate the terms for each sequence. For Column A, the sequence starts with 1, 4. The subsequent terms are the sum of the previous two: 1, 4, 5, 9, 14, 23, 37, 60. The 8th term is 60. For Column B, the sequence starts with 2, 5: 2, 5, 7, 12, 19, 31, 50. The 7th term is 50. Comparing the quantities, 60 (Column A) is greater than 50 (Column B).

Question 13

Sequence S starts with 1 and is generated by the rule "add 8 to the previous term." Sequence T starts with 1 and is generated by the rule "multiply the previous term by 2."

Column A: The amount that is added to the 99th term to get the 100th term in Sequence S. Column B: The amount that is added to the 7th term to get the 8th term in Sequence T.

  1. The quantity in Column A is greater.
  2. The quantity in Column B is greater. (correct answer)
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.

Explanation: Column A asks for the difference between the 100th and 99th terms of Sequence S. Sequence S is an arithmetic sequence where 8 is always added. So, the amount added is always 8. Column B asks for the difference between the 8th and 7th terms of Sequence T. Sequence T is a geometric sequence.

  • 7th term: T₇ = 1 * 2⁷⁻¹ = 2⁶ = 64.
  • 8th term: T₈ = 1 * 2⁸⁻¹ = 2⁷ = 128. The amount added is T₈ - T₇ = 128 - 64 = 64. Comparing the two quantities, 8 (Column A) is less than 64 (Column B).

Question 14

Two alarms beep at regular intervals. Alarm A beeps every 6 minutes. Alarm B beeps every 10 minutes. They both beep together at 1:00 PM. What is the first time after 2:00 PM that they will beep together again?

  1. 2:00 PM
  2. 2:10 PM
  3. 2:30 PM (correct answer)
  4. 3:00 PM

Explanation: To find when the alarms beep together, we need to find the least common multiple (LCM) of their intervals, 6 minutes and 10 minutes.

  • Multiples of 6: 6, 12, 18, 24, 30, 36, ...
  • Multiples of 10: 10, 20, 30, 40, ... The LCM of 6 and 10 is 30. So, the alarms beep together every 30 minutes. They start at 1:00 PM. The times they beep together are:
  • 1:00 PM
  • 1:30 PM
  • 2:00 PM
  • 2:30 PM
  • 3:00 PM The question asks for the first time after 2:00 PM. The next time on the list after 2:00 PM is 2:30 PM.

Question 15

A car rental company charges a flat fee plus a cost per mile. A 100-mile trip costs 80.A150−miletripcosts80. A 150-mile trip costs 80.A150−miletripcosts95. What is the rule used to calculate the total cost (C) for a trip of m miles?

  1. C = 0.30m + 50 (correct answer)
  2. C = 0.50m + 30
  3. C = 0.25m + 55
  4. C = 0.40m + 40

Explanation: This problem describes a linear relationship. First, find the cost per mile. The difference in cost for the two trips is 95−95 - 95−80 = 15.Thedifferenceinmilesis150−100=50miles.So,thecostpermileis15. The difference in miles is 150 - 100 = 50 miles. So, the cost per mile is 15.Thedifferenceinmilesis150−100=50miles.So,thecostpermileis15 / 50 miles = 0.30permile.Now,findtheflatfee.LettheflatfeebeF.ThetotalcostisC=0.30m+F.Usingthe100−miletrip:0.30 per mile. Now, find the flat fee. Let the flat fee be F. The total cost is C = 0.30m + F. Using the 100-mile trip: 0.30permile.Now,findtheflatfee.LettheflatfeebeF.ThetotalcostisC=0.30m+F.Usingthe100−miletrip:80 = 0.30(100) + F 80=80 = 80=30 + F F = 50.TheruleisC=0.30m+50.Wecancheckthiswiththe150−miletrip:C=0.30(150)+50=45+50=50. The rule is C = 0.30m + 50. We can check this with the 150-mile trip: C = 0.30(150) + 50 = 45 + 50 = 50.TheruleisC=0.30m+50.Wecancheckthiswiththe150−miletrip:C=0.30(150)+50=45+50=95. This is correct.

Question 16

Pattern A follows the rule Tₙ = 2n + k, where k is a constant. The 5th term of Pattern A is 13. Pattern B follows the rule Tₙ = 4n + j, where j is a constant. The 3rd term of Pattern B is 11.

Column A: The value of the 10th term of Pattern A. Column B: The value of the 6th term of Pattern B.

  1. The quantity in Column A is greater.
  2. The quantity in Column B is greater.
  3. The two quantities are equal. (correct answer)
  4. The relationship cannot be determined from the information given.

Explanation: First, determine the complete rule for each pattern. For Pattern A: Tₙ = 2n + k. Given T₅ = 13, we have 13 = 2(5) + k, which gives 13 = 10 + k, so k = 3. The rule for Pattern A is Tₙ = 2n + 3. For Pattern B: Tₙ = 4n + j. Given T₃ = 11, we have 11 = 4(3) + j, which gives 11 = 12 + j, so j = -1. The rule for Pattern B is Tₙ = 4n - 1. Now calculate the required terms: Column A: The 10th term of Pattern A is T₁₀ = 2(10) + 3 = 23. Column B: The 6th term of Pattern B is T₆ = 4(6) - 1 = 23. The two quantities are equal.

Question 17

A manufacturing machine has a rule for converting an input value, x, into an output value, y. The rule is: "multiply the input by -4, then subtract the result from 10." Which equation correctly represents this rule?

  1. y = 10 - 4x
  2. y = -4x - 10
  3. y = 10 - (-4x) (correct answer)
  4. y = 10 - (x / -4)

Explanation: Let's break down the rule into algebraic steps.

  1. "multiply the input by -4": This gives -4x.
  2. "subtract the result from 10": This means we start with 10 and subtract the result from step 1. So, it's 10 - (-4x). Combining these, we get the equation y = 10 - (-4x), which simplifies to y = 10 + 4x. Choice C matches the unsimplified, direct translation of the rule.

Question 18

A pattern is described by the rule V = -5n + 32, where V is the value of the term and n is its position number (1, 2, 3, ...). What is the position number of the first term in the pattern to have a negative value?

  1. The 5th term
  2. The 6th term
  3. The 7th term (correct answer)
  4. The 8th term

Explanation: We need to find the smallest integer n for which the value V is less than 0. -5n + 32 < 0 Add 5n to both sides: 32 < 5n Divide by 5: 32/5 < n 6.4 < n Since n must be an integer representing the position number, the smallest integer n that is greater than 6.4 is 7. So, the 7th term is the first to have a negative value. Let's check: For n=6, V = -5(6) + 32 = -30 + 32 = 2. For n=7, V = -5(7) + 32 = -35 + 32 = -3.

Question 19

The number of diagonals in a polygon follows a pattern. A quadrilateral (4 sides) has 2 diagonals. A pentagon (5 sides) has 5 diagonals. A hexagon (6 sides) has 9 diagonals. A heptagon (7 sides) has 14 diagonals. Based on this pattern, how many diagonals does a nonagon (9 sides) have?

  1. 20
  2. 27 (correct answer)
  3. 35
  4. 44

Explanation: Let's analyze the pattern of increase in the number of diagonals.

  • From 4 sides to 5 sides, the diagonals increase from 2 to 5 (an increase of 3).
  • From 5 sides to 6 sides, the diagonals increase from 5 to 9 (an increase of 4).
  • From 6 sides to 7 sides, the diagonals increase from 9 to 14 (an increase of 5). The pattern is that for each additional side, the number of diagonals increases by an amount that is one greater than the previous increase.
  • For an octagon (8 sides), the increase would be 6. Diagonals = 14 + 6 = 20.
  • For a nonagon (9 sides), the increase would be 7. Diagonals = 20 + 7 = 27.