Average questions with answers and concepts

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Average is a fundamental mathematical concept that represents the central value of a set of numbers. In competitive exams, average problems test your ability to manipulate data sets, understand weighted distributions, and handle scenarios involving additions, removals, and replacements of values in a collection.

Key formulas and rules

10 rules

Average: $Average = \frac{Sum of all values}{Number of values}$

Sum of values: $Sum = Average \times n$

Combined Average: $Average_{(n + m)} = \frac{n A + m B}{n + m}$

Addition property: If each number is increased by $k$ , $New Average = Old Average + k$

Multiplication property: If each number is multiplied by $k$ , $New Average = Old Average \times k$

New member added: $New Average = \frac{Original Sum + New Value}{n + 1}$

Average of first $n$ natural numbers: $Avg = \frac{n + 1}{2}$

Average of first $n$ even numbers: $Avg = n + 1$

Average of first $n$ odd numbers: $Avg = n$

Average of squares of first $n$ natural numbers: $Avg = \frac{( n + 1 ) ( 2 n + 1 )}{6}$

Key concepts

Simple Arithmetic Mean

The basic average is calculated by dividing the sum of all observations by the number of observations. This is the most common form of average and applies to situations where all values have equal importance. For a set {a , a , ..., an}, the average = (a + a + ... + an) / n.

Weighted Average

When different values have different weights or importance, use weighted average. Formula: Weighted Average = (w x + w x + ... + wnxn) / (w + w + ... + wn), where w represents weights and x represents values. Common applications include calculating average marks with different subject weightings, average prices of mixed quantities, and portfolio returns.

Average of Consecutive Numbers

For any set of consecutive numbers (arithmetic progression), the average equals the middle term. If there are an even number of terms, the average is the average of the two middle terms. This property is useful for sequences like consecutive integers, even numbers, odd numbers, or multiples of a constant.

Adding/Removing Members

When a new member joins: New Sum = Old Average x Old Count + New Member's Value; New Average = New Sum / New Count. When a member leaves: New Sum = Old Average x Old Count - Departing Member's Value. The change in average can be found by: Change = (New Member's Value - Old Average) / New Count.

Replacement Scenarios

When one member is replaced by another: Effect on Sum = New Value - Old Value. Effect on Average = (New Value - Old Value) / Total Count. If the average increases by X after replacement, then: New Value = Old Value + (X x Total Count).

Age Problems with Averages

In age-related average problems, remember that everyone ages by the same amount over time. If the average age of a group is A today, it will be (A + n) after n years, assuming no members join or leave. When solving problems about past or future averages, track how each individual's age changes and recalculate accordingly.

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Worked examples

Example 1

The average weight of 8 persons increases by 2.5 kg when a new person replaces one of them who weighs 65 kg. What is the weight of the new person?

Solution

Step 1: Calculate total weight increase across the group ( $n = 8$ ).

Δ W_{total} = 8 \times 2.5 kg = 20 kg

Step 2: Relate total change to incoming and outgoing weights.

W_{new} - W_{replaced} = Δ W_{total}

W_{new} - 65 = 20 ⟹ W_{new} = 65 + 20 = 85 kg

Answer: 85 kg.

Example 2

A batsman in his 17th innings makes a score of 85 and thereby increases his average by 3. What is his average after 17 innings?

Solution

Step 1: Let the average after 16 innings be $A$ .

Total runs after 16 innings = 16 A

Step 2: Equate totals after 17 innings.

16 A + 85 = 17 (A + 3)

16 A + 85 = 17 A + 51

85 - 51 = 17 A - 16 A ⟹ A = 34

Step 3: Calculate new average after 17 innings.

A_{new} = A + 3 = 34 + 3 = 37 runs

Answer: 37 runs.

Example 3

The average of 11 results is 60. If the average of the first six results is 58 and that of the last six is 63, find the sixth result.

Solution

Step 1: Calculate totals.

Total of all 11 results = 11 \times 60 = 660

Sum of first 6 results = 6 \times 58 = 348

Sum of last 6 results = 6 \times 63 = 378

Step 2: Notice that the 6th result ( $x_{6}$ ) is counted in both subgroups.

Sum_{1} + Sum_{2} = 348 + 378 = 726 = Total_{11} + x_{6}

x_{6} = 726 - 660 = 66

Answer: 66.

Example 4

The average monthly salary of 12 employees is 1,540. When the manager's salary is included, the group average becomes 1,650. What is the manager's salary?

Solution

Step 1: Compute total salary of 12 employees.

Total_{12} = 12 \times 1, 540 = 18, 480

Step 2: Compute total salary of 13 persons including the manager.

Total_{13} = 13 \times 1, 650 = 21, 450

Step 3: Find the manager's salary ( $M$ ).

M = Total_{13} - Total_{12} = 21, 450 - 18, 480 = 2, 970

Answer: 2,970.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

The average of 11 observations is 50. If the average of first 6 observations is 45 and the average of last 6 observations is 55, what is the 6th observation?

Answer: B. 50

Explanation

Sum of 11 observations = 11 x $50 = 550$
Sum of first $6 = 6$ x $45 = 270$
Sum of last $6 = 6$ x $55 = 330$
6th observation = 270 + 330 - 550 = 50

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

A batsman scores 85 runs in his 15th innings and increases his average by 3. What is his new average?

Answer: D. 43

Explanation

Let average after 14 innings = x
Total after 14 innings = 14x
After 15th innings: $(14 x + 85) /15 = x + 3$
$14 x + 85 = 15 x + 45$
$x = 40$
New average = 43

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

The average of 25 results is 18. The average of first 12 results is 15 and the average of last 12 results is 20. What is the 13th result?

Answer: D. 30

Explanation

Sum of $25 = 25$ x $18 = 450$
Sum of first $12 = 12$ x $15 = 180$
Sum of last $12 = 12$ x $20 = 240$
13th result = 450 - 180 - 240 = 30

Sources and review notes

This is an AISEA-authored practice question.

Review status: verified · Reviewed 2026-08-13 · structure and answer-key checks, worked-answer review, human editorial review, duplicate screening

Common mistakes and useful habits

Always verify your answer by working backwards - calculate the total from your answer and check if it matches the given conditions.
In replacement problems, remember: Change in Average = (New Value - Old Value) / Number of items. This shortcut saves time.
For consecutive number averages, use the middle term property to avoid lengthy calculations.
When dealing with overlapping sets (like the 6th result example), use the principle of inclusion-exclusion: Sum of parts = Total + Overlap.
In age problems with averages, note that everyone's age increases equally over time, so the average also increases by the same amount.
For weighted averages, clearly identify which values are the weights and which are the values being averaged - this is a common point of confusion.

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Next topic: Problems on Ages

Average questions with answers and concepts

Learn average with concept notes and worked examples, then practise focused questions with explained answers. Build accuracy for placement and competitive aptitude tests before trying timed mode.

150 practice questionsPage updated 11 Nov 2026Review metadata pendingPractice questions

Practise this concept

Key formulas and rules

10 rules

Average: $Average = \frac{Sum of all values}{Number of values}$

Sum of values: $Sum = Average \times n$

Combined Average: $Average_{(n + m)} = \frac{n A + m B}{n + m}$

Addition property: If each number is increased by $k$ , $New Average = Old Average + k$

Multiplication property: If each number is multiplied by $k$ , $New Average = Old Average \times k$

New member added: $New Average = \frac{Original Sum + New Value}{n + 1}$

Average of first $n$ natural numbers: $Avg = \frac{n + 1}{2}$

Average of first $n$ even numbers: $Avg = n + 1$

Average of first $n$ odd numbers: $Avg = n$

Average of squares of first $n$ natural numbers: $Avg = \frac{( n + 1 ) ( 2 n + 1 )}{6}$

Key concepts

Simple Arithmetic Mean

Weighted Average

Average of Consecutive Numbers

Adding/Removing Members

Replacement Scenarios

Age Problems with Averages

Check the idea while it is fresh.Open a focused exercise without losing this article.Practise this concept

Worked examples

Example 1

The average weight of 8 persons increases by 2.5 kg when a new person replaces one of them who weighs 65 kg. What is the weight of the new person?

Solution

Step 1: Calculate total weight increase across the group ( $n = 8$ ).

Δ W_{total} = 8 \times 2.5 kg = 20 kg

Step 2: Relate total change to incoming and outgoing weights.

W_{new} - W_{replaced} = Δ W_{total}

W_{new} - 65 = 20 ⟹ W_{new} = 65 + 20 = 85 kg

Answer: 85 kg.

Example 2

A batsman in his 17th innings makes a score of 85 and thereby increases his average by 3. What is his average after 17 innings?

Solution

Step 1: Let the average after 16 innings be $A$ .

Total runs after 16 innings = 16 A

Step 2: Equate totals after 17 innings.

16 A + 85 = 17 (A + 3)

16 A + 85 = 17 A + 51

85 - 51 = 17 A - 16 A ⟹ A = 34

Step 3: Calculate new average after 17 innings.

A_{new} = A + 3 = 34 + 3 = 37 runs

Answer: 37 runs.

Example 3

The average of 11 results is 60. If the average of the first six results is 58 and that of the last six is 63, find the sixth result.

Solution

Step 1: Calculate totals.

Total of all 11 results = 11 \times 60 = 660

Sum of first 6 results = 6 \times 58 = 348

Sum of last 6 results = 6 \times 63 = 378

Step 2: Notice that the 6th result ( $x_{6}$ ) is counted in both subgroups.

Sum_{1} + Sum_{2} = 348 + 378 = 726 = Total_{11} + x_{6}

x_{6} = 726 - 660 = 66

Answer: 66.

Example 4

The average monthly salary of 12 employees is 1,540. When the manager's salary is included, the group average becomes 1,650. What is the manager's salary?

Solution

Step 1: Compute total salary of 12 employees.

Total_{12} = 12 \times 1, 540 = 18, 480

Step 2: Compute total salary of 13 persons including the manager.

Total_{13} = 13 \times 1, 650 = 21, 450

Step 3: Find the manager's salary ( $M$ ).

M = Total_{13} - Total_{12} = 21, 450 - 18, 480 = 2, 970

Answer: 2,970.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

The average of 11 observations is 50. If the average of first 6 observations is 45 and the average of last 6 observations is 55, what is the 6th observation?

Answer: B. 50

Explanation

Sum of 11 observations = 11 x $50 = 550$
Sum of first $6 = 6$ x $45 = 270$
Sum of last $6 = 6$ x $55 = 330$
6th observation = 270 + 330 - 550 = 50

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

A batsman scores 85 runs in his 15th innings and increases his average by 3. What is his new average?

Answer: D. 43

Explanation

Let average after 14 innings = x
Total after 14 innings = 14x
After 15th innings: $(14 x + 85) /15 = x + 3$
$14 x + 85 = 15 x + 45$
$x = 40$
New average = 43

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

The average of 25 results is 18. The average of first 12 results is 15 and the average of last 12 results is 20. What is the 13th result?

Answer: D. 30

Explanation

Sum of $25 = 25$ x $18 = 450$
Sum of first $12 = 12$ x $15 = 180$
Sum of last $12 = 12$ x $20 = 240$
13th result = 450 - 180 - 240 = 30

Sources and review notes

This is an AISEA-authored practice question.

Review status: verified · Reviewed 2026-08-13 · structure and answer-key checks, worked-answer review, human editorial review, duplicate screening

Common mistakes and useful habits

Always verify your answer by working backwards - calculate the total from your answer and check if it matches the given conditions.
In replacement problems, remember: Change in Average = (New Value - Old Value) / Number of items. This shortcut saves time.
For consecutive number averages, use the middle term property to avoid lengthy calculations.
When dealing with overlapping sets (like the 6th result example), use the principle of inclusion-exclusion: Sum of parts = Total + Overlap.
In age problems with averages, note that everyone's age increases equally over time, so the average also increases by the same amount.
For weighted averages, clearly identify which values are the weights and which are the values being averaged - this is a common point of confusion.

Ready to test your understanding?

Work through 150 questions with explanations after each answer.

Start practice

Next topic: Problems on Ages

Key formulas and rules

Key concepts

Simple Arithmetic Mean

Weighted Average

Average of Consecutive Numbers

Adding/Removing Members

Replacement Scenarios

Age Problems with Averages

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Keep building the same skill family.

Key formulas and rules

Key concepts

Simple Arithmetic Mean

Weighted Average

Average of Consecutive Numbers

Adding/Removing Members

Replacement Scenarios

Age Problems with Averages

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Keep building the same skill family.