Using units in calculations

Now you have been introduced to units and prefixes, let’s see how they can be used in calculations. When you calculate a quantity from a formula, you can work out the units of that quantity by putting the units into the formula, alongside the numbers. To see how this works, let’s consider the units of volume in more detail.

Suppose you want to calculate the volume of a carton of fruit juice. Let’s say the box is 16.7 cm tall, 10 cm long and 6 cm wide. You can find its volume by multiplying these three numbers together:

Volume = Height × Length × Width so, Volume = 16.7 cm× 10 cm × 6 cm

= 1002 cm× cm × cm

= 1002 cm³

The answer isn’t exactly 1000 cm³, because the original measurements were given to 3 significant figures at most. So, you should round the result to 1000 cm³. Note that the answer is in centimetres cubed, because we multiplied three quantities measured in centimetres.

In any formula, the units on the left-hand side must be the same as the units on the right-hand side. In this case, the left-hand side has the units of volume, and the units on the right-hand side, three lengths multiplied together, combine to give the units of volume. If we had used ‘ten seconds’ as the width of the box, the units would not have balanced in this way. This is an obvious mistake, but the principle applies to all equations involving units, however complicated.

But you don’t have to measure volume in centimetres cubed. There are several other units of volume. The carton holds one litre of fruit juice. So, let’s do a calculation to see how many centimetres cubed there are in one litre (written 1 l).

(Be careful when you read or write these units, because the symbol for litre – l – can look a lot like the number one – 1.)

16.7 cm

10 cm 6 cm

By definition, one litre is the same as a decimetre cubed. Imagine a cube which has all its sides one decimetre (1 dm) long. 1 dm is the same as 10 cm, so we can do this calculation:

1 l = 1 dm³

= 1 dm× 1 dm × 1 dm

= 10 cm× 10 cm × 10 cm

= 1000 cm³

This shows that one litre is the same as 1000 cm³. One litre is also 1000 millilitres, so 1 millilitre (1 ml) is the same as one centimetre cubed (1 cm³). It is also true that there are a thousand litres in one metre cubed (1000 l = 1 m³).

So, there are several different ways to describe a volume of one litre (the amount of fruit juice in a carton):

One litre = 1 l = 1000 ml = 1000 cm³ = 1 dm³= 0.001 m³ However you write it, there is still the same volume of fruit juice. But you’d have a problem if you tried to calculate the volume of the fruit juice box using the same lengths, written with different prefixes:

Height 16.7 cm Length 0.10 m Width 60 mm

There is nothing wrong with these measurements (for

instance, the width of 0.10 m is the same as 10 cm), but see what happens when you use them to calculate the volume:

Volume = Height × Length × Width

so Volume = 16.7 cm × 0.10 m × 60 mm

= 100.2 cm × m × mm

= 100.2 ?

Here, you cannot write down a neat unit for volume, like cm³, because of the mixture of prefixes used in the calculation. Unless you invent a new, jumbled unit called the ‘centimetre-times-metre-times-millimetre’, you have a big problem!

Even worse, if you had been in a hurry, and you hadn’t shown all of the working, it would be easy to write ‘cm³’ after the ‘100.2’, which would constitute an incorrect answer.

What this illustrates is the importance of always using SI units with consistent prefixes. If the calculation just involves lengths, say, then you must make sure

1 dm= 10 cm

1 dm= 10 cm

1 dm= 10 cm

16.7 cm

0.10 m 60 mm

that all of the lengths are calculated in identical units. Convert the original measurements, where necessary. But, don’t forget to convert the corresponding numbers as well! For instance, 20 cm is the same as 0.2 m. This will guarantee that the final answer will be in a consistent SI unit, rather than a jumble.

When you need to do a calculation with letters or symbols instead of words, the same rules apply. You replace the letter with a number and the corresponding unit. (Sometimes, the units are written to the right of the numbers, rather than included in the calculation.) For example:

Question

Find the volume of a box with height 16.7 cm, width 10 cm and breadth 6 cm using the formula V = HLW, where V is the volume, H is the height, L is the length and W is the width.

Answer

In this case, H = 16.7 cm, L = 10 cm and W = 6 cm, so we can find the volume V as follows.

V = HLW

V = 16.7 cm× 10 cm × 6 cm

= 1002 cm³

Here, the units of HLW are cm× cm × cm = cm³, so the volume is 1002 cm³, which can be rounded to 1000 cm³. If the calculation is more complicated – perhaps

involving mass, length and time – then you must be even more careful with the units. Remember that the golden rule is to convert all units to SI. If the original data are in SI, then the answer will be too.

As you have seen, you shouldn’t mix prefixes. If in doubt, convert to units without prefixes – for example, use ‘m’ rather than ‘cm’.

Hints and Tips

Using units in formulas

X Make sure you know what each letter represents. Write it down in words.

X When you substitute a value for a letter, make sure you know what the units are. State what you are substituting for what.

X When you quote the answer, write it out as a sentence that responds to the question. ‘V = 1002’ is not the answer to the question ‘What is the volume?’ The answer is ‘The volume is 1002 cm³.’

X Check the units of the answer by putting the original units into the equation.

H

L W

Key Points

X SI units use standard symbols which have two purposes: to show what is being measured (the unit) and the scale of the

measurement, which is the power of ten (the prefix).

X Always use SI units. Don’t mix different prefixes in the same calculation.

X Check the results of your calculations by confirming that the quantities are in the correct units. This is in addition to estimating the numerical result of the calculation.

X The units must balance in any equation involving physical quantities. This is a useful way to check the units of the result.