**Upper **and **lower bounds** show us where an actual value of a rounded number can lie between.

Suppose we have a number that has been rounded to the nearest 10 and got 20, the numbers that could round up to give 20 include:

15, 16, 17 , 18, 19

The smallest value the number could be is 15. This is the **lower bound.**

The numbers larger than 20 that could round down to 20 include:

21, 22, 23, 24

The largest value that could round down to 20 would be 24.999999...

This means that the **upper bound** would be 25. It includes all numbers up to, but not including 25.

The general rule for finding the **upper bound **is to **add half** the** rounding unit;** and the **lower bound **would be found by **subtracting half** the** rounding unit**.

**Example 1:**

£420 has been rounded to the nearest £10.

Give the upper and lower bounds for the price.

Let's first find half of the rounding unit.

£10 ÷ 2 = £5

To find the smallest value we need to **subtract **half of the rounding unit.

**Lower bound **= £420 - £5 = £415

To find the largest value we need to **add** half of the rounding unit.

£10 ÷ 2 = £5

**Upper bound **= £420 + £5 = £425

**Example 2:**

175 cm has been rounded to the nearest cm.

Give the upper and lower bounds for this measurement.

Let's first find half of the rounding unit.

1 cm ÷ 2 = 0.5 cm

The smallest number that would round to 175 cm would be 0.5 cm less.

175 - 0.5 = 174.5

The biggest number that would round to 175 would be 0.5 cm more.

175 + 0.5 = 175.5

The **lower bound** = 174.5

The **upper bound** = 175.5

Does this make sense?

Let's make a start on the questions. If you get stuck, you can look back at this introduction by clicking on the red help button on the screen.