Averages Notes

Last reviewed 22 Sept 2026 · 6 min read

Meaning

The average (arithmetic mean) of $n$ values is

Average = \frac{Sum of the values}{Number of values}, Sum = Average \times n

Almost every average question is solved by moving between these two forms: turn each average into a sum, do the arithmetic on sums, then turn the answer back into an average.

How an average behaves

Change to every value	Effect on the average
Add $k$	Average increases by $k$
Subtract $k$	Average decreases by $k$
Multiply by $k$	Average is multiplied by $k$
Divide by $k$	Average is divided by $k$

The average always lies between the smallest and the largest value.

Averages of standard series

FormulaStandard results

First $n$ natural numbers: $\frac{n + 1}{2}$
First $n$ odd numbers ( $1, 3, 5, \dots$ ): $n$
First $n$ even numbers ( $2, 4, 6, \dots$ ): $n + 1$
Squares of the first $n$ naturals: $\frac{( n + 1 ) ( 2 n + 1 )}{6}$
Cubes of the first $n$ naturals: $\frac{n ( n + 1 ) ^{2}}{4}$
Any set of consecutive numbers (or numbers in AP): $\frac{first + last}{2}$ , which is also the middle term when the count is odd

Combined and weighted average

FormulaGroups put together

For two groups of sizes $n_{1}, n_{2}$ with averages $A_{1}, A_{2}$ :

A = \frac{n _{1} A _{1} + n _{2} A _{2}}{n _{1} + n _{2}}

The same formula with $n_{i}$ read as "weights" is the weighted average. Note that it is not $\frac{A _{1} + A _{2}}{2}$ unless the groups are equal in size.

Replacement, inclusion and correction

FormulaThe three standard moves

Someone joins: new average $= \frac{n A + x}{n + 1}$
Someone is replaced: if the average of $n$ values changes by $d$ when a value $x$ is replaced by $y$ , then $y = x + n d$
A value was read wrongly: correct average $=$ wrong average $+ \frac{correct value - wrong value}{n}$

Average speed

FormulaTwo legs of a journey

Equal distances at speeds $x$ and $y$ : $Average speed = \frac{2 x y}{x + y}$ (the harmonic mean)
Equal times at speeds $x$ and $y$ : $Average speed = \frac{x + y}{2}$
In general, average speed $= \frac{total distance}{total time}$ — never the plain average of the speeds when the distances differ

Worked examples

Worked ExampleExample 1 — sum from average

The average of 11 results is 50. The average of the first six is 49 and of the last six is 52. Find the sixth result.

Solution. The sixth is counted in both groups: $6 \times 49 + 6 \times 52 - 11 \times 50 = 294 + 312 - 550 =$ 56.

Worked ExampleExample 2 — one more person

The average age of 30 students is 14 years. Including the teacher, the average becomes 15. Find the teacher's age.

Solution. $30 \times 14 + T = 31 \times 15 \Rightarrow 420 + T = 465 \Rightarrow T =$ 45 years.

Worked ExampleExample 3 — replacement

The average weight of 8 people increases by 2.5 kg when a new person replaces one weighing 65 kg. Find the new person's weight.

Solution. $y = x + n d = 65 + 8 \times 2.5 =$ 85 kg.

Worked ExampleExample 4 — a wrong reading

The average of 10 numbers is 23. One number, 32, was read as 23. Find the correct average.

Solution. Correct average $= 23 + \frac{32 - 23}{10} = 23 + 0.9 =$ 23.9.

Worked ExampleExample 5 — two classes

Class A has 30 students averaging 60 marks and class B has 20 averaging 70. Find the combined average.

Solution. $\frac{30 \times 60 + 20 \times 70}{50} = \frac{1800 + 1400}{50} =$ 64 marks.

Worked ExampleExample 6 — a standard series

Find the average of the first 50 natural numbers.

Solution. $\frac{n + 1}{2} = \frac{51}{2} =$ 25.5.

Worked ExampleExample 7 — consecutive numbers

The average of five consecutive even numbers is 26. Find the largest.

Solution. The middle number is 26, so the numbers are 22, 24, 26, 28, 30 and the largest is 30.

Worked ExampleExample 8 — a batsman's average

A batsman averages 40 runs in 20 innings. He scores 82 in the 21st. Find the new average.

Solution. $\frac{20 \times 40 + 82}{21} = \frac{882}{21} =$ 42.

Worked ExampleExample 9 — average speed, equal distances

A man travels a distance at 40 km/h and returns at 60 km/h. Find the average speed.

Solution. $\frac{2 \times 40 \times 60}{100} =$ 48 km/h — not 50 km/h, because he spends longer on the slower leg.

Worked ExampleExample 10 — average speed in general

Someone covers 120 km at 60 km/h and the next 120 km at 40 km/h, then rests for 1 hour. Find the average speed for the whole outing.

Solution. Time $= 2 + 3 + 1 = 6$ h for 240 km, so $\frac{240}{6} =$ 40 km/h. Resting time counts.

Worked ExampleExample 11 — average of a group after a change

The average salary of 20 workers is ₹ 15,000. Two workers earning ₹ 12,000 and ₹ 18,000 leave. Find the new average.

Solution. Total $= 300, 000$ ; the two removed total 30,000, so 18 workers share ₹ 270,000 → ₹ 15,000. Removing values whose own average equals the group average leaves the average unchanged.

Frequently tested points

Sum $=$ average $\times$ count; convert every average into a sum first.
Adding $k$ to every value adds $k$ to the average; multiplying by $k$ multiplies it.
First $n$ naturals: $(n + 1) /2$ ; first $n$ odd: $n$ ; first $n$ even: $n + 1$ .
Consecutive numbers or an AP: average $=$ (first + last)/2 $=$ middle term.
Combined average is weighted by group size, never the plain mean of two averages.
Replacement rule: $y = x + n d$ .
Wrong reading: correct the average by (correct − wrong)/ $n$ .
Equal distances: average speed $= 2 x y / (x + y)$ ; equal times: $(x + y) /2$ .

Common MistakeCommon mistakes

Averaging two averages when the groups are of different sizes.
Using $(x + y) /2$ for a there-and-back journey; equal distance needs $2 x y / (x + y)$ .
Forgetting that the overlapping value is counted twice in "first six / last six" problems.
Ignoring halts when asked for average speed over a whole journey.

Revision SummaryChapter summary

An average is a sum divided by a count, and most problems are solved on the sums.
Shifting or scaling every value shifts or scales the average by the same amount.
Standard series have standard averages, and any set of consecutive numbers averages to its middle term.
Groups combine by a weighted average, and joining, replacing or correcting a value each has a one-line rule.
Average speed is total distance over total time — the harmonic mean for equal distances, the arithmetic mean for equal times.

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Averages