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Average

Competitive Mathematics · Arithmetic

8 Part3 worked examples10 practice questions

Five children have 12, 15, 18, 20 and 25 marbles. Pool them and share equally, and each gets . That is their average — one number that stands for the whole group. Every average question in the exam rests on one idea: total = average × count.

The formula to remember
1

Properties of average

Definition · properties · effect of adding and multiplying

Where the average lies, and what happens when something is added to or multiplied into every number.

Finding an average

Sum ÷ count

Four direct properties and finding the missing number.

  • The average always lies between the smallest and the biggest number. The average of 12, 15, 25 is 17.33 — above 12 and below 25.
  • If all numbers are equal the average is that number: the average of 7, 7, 7, 7 is 7.
  • Zero is counted. The average of 4, 8, 0 is , not .
  • Given the average and the count you know the total: total = average × count — every other formula comes from it.
Example 1 · Basic
A shopkeeper sold ₹4,200, ₹3,850, ₹5,100 and ₹4,650 on four days. What is the average daily sale?
Solution, step by step
  1. 1Sum .
  2. 2Average
Answer ₹4,450
Example 2 · Standard
The average of seven numbers is 24. The first six are 18, 22, 25, 31, 20 and 27. What is the seventh?
Solution, step by step
  1. 1Total .
  2. 2The six add to ⟹ the seventh
Answer 25
BasicTry it yourself first
Find the mean of 14, 22, 18, 26, 30.
BasicTry it yourself first
Find the mean of 7.5, 9.5, 12, 11.
StandardTry it yourself first
Find the mean of 248, 305, 192, 221, 284, 260.
BasicTry it yourself first
The average of 5 numbers is 15. Of them 4 are 12, 16, 14, 18. What is the remaining number?
StandardTry it yourself first
The average of 6 numbers is 38. Of them 5 are 36, 41, 29, 44, 38. What is the remaining number?
AdvancedTry it yourself first
The average of 7 numbers is 80. Of them 6 are 82, 76, 91, 68, 79, 85. What is the remaining number?

When every number changes

The same operation on the average

Whatever you add, subtract, multiply or divide on every number, the average follows.

On every numberOn the average
add is added
subtract is subtracted
multiply by multiplied by
divide by divided by
Example 3 · Standard
The average of eight numbers is 15. If every number is multiplied by 4 and then 3 is added, what is the new average?
Solution, step by step
  1. 1Multiply by 4: average .
  2. 2Then add 3:
Answer 63
BasicTry it yourself first
The average of 10 numbers is 24. If every number is increased by 6, what is the new average?
BasicTry it yourself first
The average of 12 numbers is 35. If every number is decreased by 8, what is the new average?
StandardTry it yourself first
The average of 15 numbers is 18. If each number is multiplied by 5, what is the new average?
StandardTry it yourself first
The average of 9 numbers is 96. If every number is divided by 4, what is the new average?
2

Averages of series

An average without adding

The direct average of natural, even and odd numbers, squares and cubes, and equally spaced series.

Average of the first numbers

Six formulas

Averages of natural, even, odd numbers, squares, cubes and multiples.

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Equally spaced series

Average = (first + last) ÷ 2

When the terms are equally spaced the average is the average of the first and last term.

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3

Weighted averages

The combined average of two or more groups

When the groups differ in size do not average the averages — take total ÷ total count.

Combined average of groups

First the total of each group

Add the group totals and divide by the total count.

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Removing a part, pass / fail

Subtract the part from the whole

The whole’s average and one part’s average are given — the rest’s average; and pass-fail counts.

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4

Changes in an average

Adding · removing · replacing · ages

How the average moves when a number is added, removed or replaced — and recovering that number.

Adding, removing and replacing

Change in total = change in average × count

In all three cases the change in the total does the work.

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Average age

Moving with time

Everyone ages together so the average rises by the same amount; if someone joins or leaves, correct the total.

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5

Average speed

Total distance ÷ total time

Not the plain mean of speeds — a matter of distance and time.

Three forms of average speed

Equal distances · equal times

Average speed over two or three equal distances and over equal times.

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6

Traps and fast tricks

Slips · saving time

How marks leak in average questions and how to solve them quickly.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

Assume · deviation · pairs

Tricks that avoid long addition.

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7

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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8

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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