# Averages

> Find the mean, use the total to find a missing number, and see what happens when a number joins or leaves.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/averages/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The cricket captain
Arjun plays cricket. In his first three games he scored 20, 35 and 14 runs. "What's my average?" he asks.

His coach adds them up: $20 + 35 + 14 = 69$ runs. Then he shares them out equally over the 3 games: $69 \div 3 = 23$. "Your average is 23 runs a game."

"What do I need next game to make my average 30?" Arjun asks. Coach thinks: for an average of 30 over 4 games, the total must be $4 \times 30 = 120$. Arjun has 69, so he needs 51 runs. The trick is to work with **totals**.

**In short:** The average (mean) is the total divided by how many there are. Turning it around, total = average × how many, and most hard average problems are solved by finding totals first.

## Before you start

- Adding a list of numbers and dividing.
- Simple equations, such as $39 + n = 48$ (see Solving Equations).

## The mean

### Key idea: Average (mean)
$$\text{average} = \frac{\text{total of the numbers}}{\text{how many numbers}}$$

The average is what each would be if everything were shared out **equally**.

### Worked example: Simple average
The average of 4, 7, 9 and 12 is $\frac{32}{4} = 8$.

## The total is the key

Turn the formula around:

$$\text{total} = \text{average} \times \text{how many}$$

### Worked example: Finding a missing number
The average of 3 numbers is 13. A fourth number is added and the new average is 12. What is the fourth number?

Old total: $3 \times 13 = 39$. New total: $4 \times 12 = 48$. Fourth number: $48 - 39 = 9$.

### Worked example: Two groups
The average mark of 10 students is 76. The average of the first 7 students is 81. What is the average of the other 3?

Total for 10: $760$. Total for 7: $567$. The other 3 have $760 - 567 = 193$ marks, so their average is $193 \div 3 = 64\frac{1}{3}$.

### Watch out: Do not average the averages
If one group of 7 averages 81 and a group of 3 averages $64\frac{1}{3}$, the overall average is **not** halfway between them. The bigger group pulls harder. Always go back to totals.

## Adding or removing a number

### Exam tip: Quick check
- Adding a number **above** the average raises the average.
- Adding a number **below** the average lowers it.
- Adding a number **equal** to the average leaves it the same.

### Worked example: Raising the average
Five tests average 70. What must the sixth test be to raise the average to 72?

Old total: 350. New total: $6 \times 72 = 432$. Sixth test: $432 - 350 = 82$.

## Averages with letters

The average of Lisa's 38 kg and Gina's $K$ kg is $\frac{38 + K}{2}$ kg.

The total of 5 numbers is half of 10 dozen, which is 60. Their average is $60 \div 5 = 12$.

### Challenge: Selective level: consecutive numbers and changes
The average of 5 consecutive whole numbers is 17. The middle one equals the average, so they are 15, 16, 17, 18 and 19.

A class of 24 has an average height of 140 cm. A new student who is 165 cm tall joins. The total grows by 165 and there are 25 students: new average $= \frac{24 \times 140 + 165}{25} = \frac{3525}{25} = 141$ cm.

The average age of a mother and her 3 children is 16. The children are 5, 8 and 11. The total is 64, so the mother is $64 - 24 = 40$.

**Mind map**

```text
Averages
  Mean
    Total ÷ how many
    Shared out equally
  Total
    Average x how many
    Use totals to find missing numbers
  Changes
    Above the average raises it
    Below lowers it
  Traps
    Never average two averages
```

## Quick summary

- Average (mean) $=$ total $\div$ how many.
- Total $=$ average $\times$ how many. Most hard average questions are solved with totals.
- To find a missing number, find the new total and the old total, then subtract.
- Do not average two averages unless the groups are the same size.
- The middle of an odd list of consecutive numbers is the average.

## Quiz

Try these questions from the chapter quiz.

1. What is the average of 4, 7, 9 and 12?
   - A. 7
   - B. 8
   - C. 9
   - D. 32
   - Answer: B. 8. Total 32, divided by 4 is 8.

2. The average of 3 numbers is 10. What is their total?
   - A. 13
   - B. 3.3
   - C. 30
   - D. 10
   - Answer: C. 30. Total $=$ average $\times$ how many $= 30$.

3. Arjun scored 20, 35 and 14 runs. What is his average?
   - A. 23
   - B. 69
   - C. 21
   - D. 35
   - Answer: A. 23. $69 \div 3 = 23$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/averages/#quiz