# Upper and Lower Bounds

This section explores upper and lower bounds. We shall explore numbers rounded to the nearest 10, 100 or 1 and work out the lowest and highest values of the original rounded values.
World’s strongest man claims he can lift 70kg with his little finger rounded to the nearest 10kg. What is the smallest amount he can lift? Here we have to think of all the numbers that can be rounded up to give 70. A number line would be useful here to observe these numbers. The numbers shown in red can be rounded up to give 70. This means the smallest amount the world’s strongest man claims to lift is 65kg which is the smallest number shown on the number line. This number is called the lower bound How about the greatest amount that he can lift? Here we need to think about the numbers that can round down to give 70. These numbers have been indicated on the number line below. These numbers are between 70 and 74.99999999... and so on. Notice that 75 is not included as this would round to 80. The world’s strongest man claims to lift a greatest amount of 74. This is called the upper bound. If you looked carefully about you will have realised that to find the upper and lower bounds of a number rounded to the nearest 10 you could divide 10 by 2 and then add or subtract from 70 to find the upper and lower bound as shown below. Find the upper and lower bounds of 70 rounded off to the nearest 10.
Upper bound   75    +5
70    10
Lower bound   65    -5

## To the nearest 100

Example The same principle described above applies to this section as well.

The distance from London to Chester is 700km rounded to the nearest 100km. What are the lower and upper bounds?

Explanation
Upper bound   750    +50
700    100
Lower bound   650    -50
And this can also be represented on the number line as shown below;

## To the nearest whole number

Above we’ve seen that when a value is given to the nearest 10 we find the upper and lower bounds by adding and subtracting 5. We’ve also seen that when a number is rounded to the nearest 100 we find the upper and lower bounds by adding or subtracting 50.

To round to the nearest whole number we go through the same steps above.

ExampleA pen is 16 cm to the nearest cm, find the upper and lower bounds.
Explanation As we did above; we’re going to divide 1 by 2 to get 0.5.
1 ÷ 2 = 0.5
This is because we’re asked to round 16 to the nearest 1. Now we subtract or add 0.5 to 16 to find the lower and upper bounds as shown below.
Upper bound  16.5    +0.5
16    1
Lower bound   15.5    -0.5
This is also shown on the number line below. The lower bound is 15.5 and the upper bound is 16.5.
;
This section explores upper and lower bounds. We shall explore numbers rounded to the nearest 10, 100 or 1 and work out the lowest and highest values of the original rounded values.
World’s strongest man claims he can lift 70kg with his little finger rounded to the nearest 10kg. What is the smallest amount he can lift? Here we have to think of all the numbers that can be rounded up to give 70. A number line would be useful here to observe these numbers. The numbers shown in red can be rounded up to give 70. This means the smallest amount the world’s strongest man claims to lift is 65kg which is the smallest number shown on the number line. This number is called the lower bound How about the greatest amount that he can lift? Here we need to think about the numbers that can round down to give 70. These numbers have been indicated on the number line below. These numbers are between 70 and 74.99999999... and so on. Notice that 75 is not included as this would round to 80. The world’s strongest man claims to lift a greatest amount of 74. This is called the upper bound. If you looked carefully about you will have realised that to find the upper and lower bounds of a number rounded to the nearest 10 you could divide 10 by 2 and then add or subtract from 70 to find the upper and lower bound as shown below. Find the upper and lower bounds of 70 rounded off to the nearest 10.
Upper bound   75    +5
70    10
Lower bound   65    -5

## To the nearest 100

Example The same principle described above applies to this section as well.

The distance from London to Chester is 700km rounded to the nearest 100km. What are the lower and upper bounds?

Explanation
Upper bound   750    +50
700    100
Lower bound   650    -50
And this can also be represented on the number line as shown below;

## To the nearest whole number

Above we’ve seen that when a value is given to the nearest 10 we find the upper and lower bounds by adding and subtracting 5. We’ve also seen that when a number is rounded to the nearest 100 we find the upper and lower bounds by adding or subtracting 50.

To round to the nearest whole number we go through the same steps above.

ExampleA pen is 16 cm to the nearest cm, find the upper and lower bounds.