# Averages — Formula, Shortcuts and Placement Exam Questions

Source: https://skills101.in/learn/aptitude/averages — Skills101 free tutorial (Quantitative Aptitude). Updated 2026-11-09.

The **average** of some numbers is the value each one would have if you **shared the total equally**.

> **Average = Sum of values ÷ Number of values**   so   **Sum = Average × Number**

![Five bars 40, 70, 50, 80, 60 with a dashed line at the average 60: the parts above the line exactly fill the gaps below](https://skills101.in/learn-img/aptitude/average-levelling.svg)

The picture shows the key idea: the amounts **above** the average exactly balance the amounts **below** it. That idea gives us a powerful shortcut.

## 1. The basic formula

### Example 1
*The marks of 5 students are 40, 70, 50, 80 and 60. Find the average.*

Sum = 300, count = 5 → average = **60**.

Most average questions are really about the **sum**. Whenever you are stuck, convert averages to sums.

## 2. Shortcut: deviation from an assumed average

Guess an average, then fix it with the deviations.

### Example 2
*Find the average of 47, 52, 49, 55 and 57.*

Assume 50. Deviations: −3, +2, −1, +5, +7 → total = +10 → average = 50 + 10/5 = **52**.

## 3. Someone joins or leaves

### Example 3
*The average age of 30 students is 15. When the teacher's age is included, the average becomes 16. Find the teacher's age.*

- Old sum = 30 × 15 = 450
- New sum = 31 × 16 = 496
- Teacher = 496 − 450 = **46 years**

**Shortcut thinking:** the teacher brings everyone up by 1 year (31 people × 1 = 31) on top of the old average 15 → 15 + 31 = 46 ✓.

### Example 4
*The average weight of 8 people rises by 2.5 kg when one person weighing 65 kg is replaced by a new person. Find the new person's weight.*

Total rise = 8 × 2.5 = 20 kg → new person = 65 + 20 = **85 kg**.

## 4. Consecutive numbers

For numbers in a regular sequence (equal steps), **average = (first + last) ÷ 2 = the middle number**.

- Average of 1 to 50 = (1 + 50)/2 = **25.5**
- Average of the first n natural numbers = (n + 1)/2
- Average of the first n odd numbers = n

### Example 5
*The average of 5 consecutive even numbers is 36. What is the largest?*

The middle (3rd) number is 36 → the numbers are 32, 34, 36, 38, 40 → largest = **40**.

## 5. Combined average

### Example 6
*Section A (40 students) averages 70 marks; section B (60 students) averages 80. Find the overall average.*

(40 × 70 + 60 × 80) ÷ 100 = (2800 + 4800) ÷ 100 = **76**.

(Notice it is closer to 80 because section B is bigger. It is **not** (70 + 80)/2 = 75.)

## 6. Average speed — a classic trap

If you travel the **same distance** at speeds a and b:

> **Average speed = 2ab ÷ (a + b)**   (not (a + b)/2)

### Example 7
*A car goes from Jaipur to Ajmer at 60 km/h and returns at 40 km/h. Find the average speed.*

2 × 60 × 40 ÷ 100 = **48 km/h**.

Why not 50? You spend **more time** at the slower speed, so the average is pulled towards 40.

## 7. Cricket averages

### Example 8
*A batsman's average after 16 innings is 40. How many runs must he score in the 17th innings to raise his average to 42?*

New total = 17 × 42 = 714. Old total = 16 × 40 = 640. Needed = **74**.

## Common mistakes

- Averaging two averages directly when the groups have different sizes.
- Using (a + b)/2 for average speed.
- Forgetting that the number of people changes when someone joins (n becomes n + 1).

## Quick recap

- Sum = average × count. Work with sums.
- Assumed average + average of deviations = true average.
- Consecutive numbers: average = middle = (first + last)/2.
- Same distance at a and b: average speed = 2ab/(a + b).


Practice quiz and free certificate: https://skills101.in/learn/aptitude/averages