Approximate measurement

No measurement is completely accurate as there will always be some degree of error when measuring. You may say you are 163 cm tall but you could really be 163.42 cm or even 163.4234323432 cm. The first figure, 163 cm, has been rounded to the nearest cm. In mathematics we round all the time but this can cause problems when measuring.

Rounding

When we have a rounded answer, we cannot say for sure what the value of the number was before it was rounded. We can give a range of answers by thinking about the rules of rounding.

Number line from 162 cm to 164 cm

When we are rounding to the nearest cm we decide which cm the value is closest to.

Example

Round 162.8 cm to the nearest cm.

Number line from 162 cm to 164 cm with a marker at 162.8 cm

162.8 cm is closest to 163 cm. In fact any numbers above 162.5 would be closer to 163 cm.

Example

Round 163.2 cm to the nearest cm.

Number line from 162 cm to 164 cm with a marker at 163.2 cm

163.2 cm is closest to 163 cm. In fact any number below 163.5 cm would be closer to 163 cm.

When we are exactly halfway, eg 162.5 cm, we could round either way. However, the rule around the world is that we always round up in this situation. Therefore, 162.5 cm would round up to 163 cm.

A height of 163 cm could have started out as any measurement between 162.5 cm and 163.5 cm.

Upper and lower bounds

We have already discovered that if a value has been rounded, there is a range of answers it could have been before rounding. The biggest number in this range is called our upper bound and the smallest number is called our lower bound.

Example

If we take our previous example of 163 cm correct to the nearest cm, we know our range of values before rounding was 162.5 cm to 163.5 cm. So our lower bound is 162.5 cm and our upper bound is 163.5 cm.

Question

What is the upper and lower bound of 5 m when it is measured to the nearest m?

Lower bound = 4.5 m

Upper bound = 5.5 m

You may have noticed that it is always half the unit we are correcting to above and below.

Example

Find the upper and lower bound of 50 if corrected to the nearest 10.

Solution

We are correcting to the nearest 10.

½ of 10 = 5

So we need to go 5 above and 5 below to find our upper and lower bounds:

Lower bound = 50 – 5 = 45

Upper bound = 50 + 5 = 55

Question

Find the upper and lower bound of 200 if corrected to the nearest 100.

Lower bound = 200 – 50 = 150

Upper bound = 200 + 50 = 250

Upper and lower bounds of decimals

Find the upper and lower bounds of 3.4 cm to the nearest mm.

We are rounding to the nearest tenth as 1 mm is one tenth of a cm. A tenth as a decimal is 0.1.

½ of 0.1 = 0.05.

So we need to go 0.05 above and 0.05 below.

Lower bound = 3.4 – 0.05 = 3.35 cm

Upper bound = 3.4 + 0.05 = 3.45 cm

Question

Find the upper and lower bound of 10.2 if corrected to the nearest tenth.

Lower bound = 10.2 – 0.05 = 10.15

Upper bound = 10.2 + 0.05 = 10.25