Average

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Average & the sum bridge

⏱ 4 min read🧩 5 question types🎯 17 practice Q

The idea in one minute

An average is one number that stands for a whole list: average = sum ÷ count. The exam-useful direction is the reverse: sum = average × count.

Add k to every value and the average rises by k. Multiply every value by k and the average is multiplied by k.

For lists with equal gaps (consecutive numbers, multiples of k) the average is the middle term. Standard lists have fixed averages: first n odd numbers average n, first n even numbers average n + 1.

What an average is

An average is one number that stands for a whole list. It is what each one gets when the total is shared equally.

Five friends have ₹20, ₹30, ₹25, ₹35 and ₹40. Total = ₹150. Shared equally: $150 \div 5 = ₹30$ . So the average is ₹30. Another name for it is the arithmetic mean.

Rule: Average = sum of values ÷ number of values. Turn it around and you get the working tool: sum = average × count.

The sum bridge

Most exam questions live on the turned-around form sum = average × count.

Change every average into a sum.
Add or subtract with the sums.
Divide again only at the last step.

Example: The average of 4 numbers is 25, and three of them are 20, 30 and 18. Sum = $4 \times 25 = 100$ . The known values add to 68. Fourth number = $100 - 68 = 32$ .

Properties that save time

The average always sits between the smallest and the largest value.
Add k to every value → the average rises by k. Subtract k → it falls by k.
Multiply every value by k → the average gets multiplied by k. Divide → it gets divided by k.
Apply changes in the order given: "× 3, then + 5" on an average of 15 gives $15 \times 3 + 5 = 50$ .
Each value's gap from the average (its deviation) adds up to zero across the whole list.

Standard series averages

Learn this table by heart. Here n is the count of terms, not the last term.

Series	Average
First n natural numbers	$\dfrac{n+1}{2}$
First n odd numbers	$n$
First n even numbers	$n+1$
Squares of the first n naturals	$\dfrac{(n+1)(2n+1)}{6}$
Cubes of the first n naturals	$\dfrac{n(n+1)^2}{4}$

Tip: First 30 even numbers average $30 + 1 = 31$ . First 25 odd numbers average 25. No adding needed.

Numbers with equal gaps

Consecutive numbers, consecutive even or odd numbers, and multiples of k all climb by one fixed step. Their average is the middle term, which is also $\dfrac{\text{first} + \text{last}}{2}$ .

Example: 7 consecutive even numbers average 48. The middle (4th) number is 48. Step up by 2, three times: largest = $48 + 6 = 54$ .

Multiples of 6 up to 100 run from 6 to 96. Average = $\dfrac{6 + 96}{2} = 51$ .

Close numbers: the assumed-mean method

To average numbers huddled close together, pick a round value nearby. Write each number's gap (+ or −) from it. Average the gaps. Add the result back.

284, 291, 296, 302, 307 sit around 300. Gaps: $-16, -9, -4, +2, +7$ . Gap total = $-20$ . Average gap = $-20 \div 5 = -4$ . Average = $300 - 4 = 296$ .

Tip: This method replaces one big addition with small ones. Use it for lists of marks, weights or prices.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Sum from average / one missing value

How to spot it:

An average and a count are given. The total is asked, or one value is missing from a list of known values.

\text{Sum} = \text{Average} \times n

Method

Multiply the average by the count to get the sum.
Add up the values you already know.
Subtract: what is left is the missing value (or the asked total).

Why it works:

The average is only the sum shared equally, so the sum carries all the information.

Try this

The average of 5 numbers is 27. Four of them are 20, 31, 25 and 18. Find the fifth number.

Show solution

Sum = $5 \times 27 = 135$ .
Known values add to $20 + 31 + 25 + 18 = 94$ .
Fifth number = $135 - 94 = 41$ .

Answer

Type 2very common3 practice Q

Average of standard series

How to spot it:

'Find the average of the first n natural / odd / even numbers', or of their squares or cubes.

\frac{n+1}{2},\quad n,\quad n+1,\quad \frac{(n+1)(2n+1)}{6},\quad \frac{n(n+1)^2}{4}

Method

Spot the series and n, the count of terms.
Take the average straight from the table.
For even numbers ending at 2n, n is half the last term.

Why it works:

Each table value is the series-sum formula already divided by n.

Try this

Find the average of the first 30 even natural numbers.

Show solution

Series: even numbers. Average = $n + 1$ .
$n = 30$ , so the average is $30 + 1 = 31$ .
Check: sum $= 930$ , and $930 \div 30 = 31$ .

Answer

Type 3very common3 practice Q

Consecutive numbers and equal gaps

How to spot it:

'The average of 7 consecutive even numbers is 48 — find the largest', or the average of the multiples of k in a range.

\text{Average} = \text{middle term} = \frac{\text{first} + \text{last}}{2}

Method

Place the average at the middle term of the list.
Step out from the middle by the gap (1 for consecutive, 2 for even/odd, k for multiples).
For a range of multiples, use (first + last) ÷ 2.

Why it works:

Equal gaps balance on both sides of the middle, so the middle term is the mean.

Try this

The average of 9 consecutive odd numbers is 45. Find the smallest of them.

Show solution

Middle (5th) number = 45.
Step down by 2, four times: $45 - 8 = 37$ .

Answer

Type 4common2 practice Q

Same operation on every value

How to spot it:

'Each number is multiplied by 4 / increased by 6 / divided by 2' — the new average is asked.

\text{new average} = k \times A + c \quad (\text{same order as stated})

Method

Do to the old average exactly what is done to each value.
Keep the stated order of operations.
Addition shifts the average; multiplication scales it.

Why it works:

A uniform change moves the sum and the average exactly the way it moves each single value.

Try this

The average of 7 numbers is 18. Each number is increased by 4 and then divided by 2. Find the new average.

Show solution

Increase: $18 + 4 = 22$ .
Divide: $22 \div 2 = 11$ .

Answer

Type 5common2 practice Q

Assumed mean (deviation method)

How to spot it:

A short list of 4-8 close numbers (weights, marks, runs) is given and their average is asked.

\text{Average} = \text{assumed value} + \frac{\text{sum of deviations}}{n}

Method

Pick a round number near the middle.
Write each value's gap from it, with a + or − sign.
Add the gaps and divide by n.
Add the result to the assumed value.

Why it works:

Shifting every value by the same amount shifts the average by that amount.

Try this

Find the average of 396, 402, 407, 413 and 417.

Show solution

Assume 400. Gaps: $-4, +2, +7, +13, +17$ .
Gap total = $35$ ; average gap = $35 \div 5 = 7$ .
Average = $400 + 7 = 407$ .

Answer

407

Formula sheet

Definition

\bar{x} = \frac{\sum x_i}{n} \iff \sum x_i = n\bar{x}

The sum form is the one you use.

Shift property

\overline{x_i + k} = \bar{x} + k,\quad \overline{k\, x_i} = k\bar{x}

Adding k shifts the average by k; multiplying by k scales it.

First n naturals / odd / even

\frac{n+1}{2},\quad n,\quad n+1

n = how many terms.

Squares / cubes

\frac{(n+1)(2n+1)}{6},\quad \frac{n(n+1)^2}{4}

Equal gaps

\text{average} = \frac{\text{first} + \text{last}}{2} = \text{middle term}

Shortcuts that save time

⚡ Flip to the sum first

Never push averages around. Change them to sums, adjust, divide back at the end.

Example

The average of 8 numbers is 32.5. Find their sum.

Show solution

Sum = average × count.
Sum = $32.5 \times 8 = 260$ .

Answer

260

⚡ Same operation on every value

Do to the average exactly what is done to each value, in the same order.

Example

The average of 6 numbers is 15. Each number is multiplied by 3 and then increased by 5. Find the new average.

Show solution

Multiply first: $15 \times 3 = 45$ .
Then add 5: $45 + 5 = 50$ .

Answer

⚡ Equal gaps: jump from the middle

The average of evenly spaced numbers is the middle term. Step out from it instead of adding everything.

Example

The average of 5 consecutive numbers is 30. Find the largest.

Show solution

Middle (3rd) number = 30.
Largest = $30 + 2 = 32$ .

Answer

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Giving a sum where the question asks for the average.

Read the last line: total or per-item value? Divide by the count before answering.

Mistake 02

Averaging two group averages of different sizes.

Only totals can be combined. Convert each group to size × average first.

Mistake 03

Using the sum formula $\frac{n(n+1)}{2}$ and forgetting to divide by n.

The formula gives the sum. The average is the sum ÷ n.

Mistake 04

Applying just one of two ordered operations (× 3 then + 5).

Apply both, in the stated order: $15 imes 3 + 5 = 50$ .

Mistake 05

Stepping by 1 when the list is of consecutive even or odd numbers.

Consecutive even or odd numbers step by 2. Use the gap of the series.

Quick revision

Read this the night before the exam.

Sum = average × count. Work in sums, divide at the end.
Every value + k → average + k; every value × k → average × k.
Equal gaps → average = middle term = (first + last)/2.
First n naturals $\frac{n+1}{2}$ ; first n odd $n$ ; first n even $n+1$ .
Squares $\frac{(n+1)(2n+1)}{6}$ ; cubes $\frac{n(n+1)^2}{4}$ .
Close numbers → assumed mean + average of the gaps.

Practice: 17 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

Suggested time 4 min · wrong answers go to your mistake notebook automatically.

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