GMAT® Problem Solving : Averages

Concept: Computing arithmetic mean of consecutive integers.

This GMAT practice question is a problem solving question in statistics. Concept tested: Basics of computing arithmetic mean (average).

Question: The arithmetic mean of the 5 consecutive integers starting with 's' is 'a'. What is the arithmetic mean of 9 consecutive integers that start with s + 2?

  1. 2 + s + a
  2. 22 + a
  3. 2s
  4. 2a + 2
  5. 4 + a

Video Explanation

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Explanatory Answer

Use these hints and try

  1. Consecutive integers are equally and symmetrically distributed with respect to the middle number.
  2. i.e., the middle number is the average of a set of consecutive integers.
  3. Determine which term is 'a' and then work out the average of the second set relative to that term.

Step 1: Data given

The first sequence of 5 consecutive numbers starts with 's' and its mean is 'a'

The mean of 5 consecutive numbers is the 3rd term - the middle term.

Hence, 'a' the mean is the middle (3rd) term.

Step 2: Relating 'a' to 's'

The first sequence starts with 's'

Hence, the terms are s, s + 1, s + 2, s + 3, and s + 4

The middle term is s + 2

Therefore, a = s + 2

Step 3: The second series and its mean

The second series of 9 consecutive numbers starts from s + 2

The terms will therefore, be s + 2, s + 3, s + 4, s + 5, s + 6, s + 7, s + 8, s + 9, and s + 10

The middle term of the second series is the 5th term = s + 6

If a = s + 2, then s + 6 will be a + 4

The average of the second sequence is a + 4

Alternative Approach

The fastest way to solve such questions is to assume a value for 's'.

Let s be 1

Therefore, the 5 consecutive integers that start with 1 are 1, 2, 3, 4 and 5

The average of these 5 numbers is the middle term, which is 3. Hence, a = 3

9 consecutive integers that start with s + 2 will start from 1 + 2 = 3

The second sequence is therefore, 3, 4, 5, 6, 7, 8, 9, 10 and 11

The average of these 9 number is the middle term, which is 7

If the average of the first sequence 3 = a, the average of the second sequence 7 = 4 + a

Complete Explanation

The sequence starts with 's' and its mean is 'a'

The mean of 5 consecutive numbers is the 3rd term - the middle term.

Hence, 'a' the mean is the middle (3rd) term.

The terms of the sequence are s, s + 1, s + 2, s + 3, and s + 4

The middle term is s + 2

Therefore, a = s + 2

The second series starts from s + 2

The terms will therefore, be s + 2, s + 3, s + 4, s + 5, s + 6, s + 7, s + 8, s + 9, and s + 10

The middle term of the second series is the 5th term = s + 6

If a = s + 2, then s + 6 will be a + 4

The average of the second sequence is a + 4

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