## 5. Ratios, proportions and percentages

In section 4.2, you will have seen the concept of precision expressed by analogy with a clock that gains or loses a certain amount within a period of time. Taking an extreme example of a clock that gains a minute a day, we can say that the clock gains a minute every 1440 minutes. This can be written as a 'ratio', usually in the form '1:1440', '1/1440' or 'one part in 1440'. Note that the change (the gain) and the base measurement (elapsed time) are both expressed in the same unit of measurement, so that the ratio has no units and is said to be 'dimensionless'. This is not always the case, and if units for the two quantities differ, the unit(s) should be indicated.

You may also meet the term ratio used erroneously to describe the recipe for a mixture. For instance, cement mortar for bricklaying is made by mixing one part of Portland cement with five parts of sand. This is written as 1:5 (e.g. http://www.diydata.com/materials/cement/cement.php), but this is not the same as the ratio described in the previous paragraph.

If you take the ratio 1:1440 described earlier, and divide the gain by the elapsed time, the result (0.0007) expresses the gain as a 'proportion' of the elapsed time. This is another way of writing the same relationship, and again it is dimensionless in this case. The proportion can be used to calculate how much the clock gains over other periods, for instance over a week (10 080 minutes) it will gain 0.0007 × 10 080 = 7 minutes.

A ratio or proportion is normally a small thing expressed as part of a larger thing, but it doesn't have to be so. A ratio of 2:1 or a proportion of 2 both express a change that is twice the value of the base measurement. However, in most cases, proportions tend to be less than one, and are often quite small numbers. For instance, if a bank account pays £5 interest for each £100 invested over a period of one year, the interest expressed as a proportion is 0.05 per year (note that this proportion has a unit). For a variety of reasons, proportions like this are often multiplied by 100 to yield a quantity called a 'percentage' - in this example the interest would be advertised by the bank as '5% per year'. Spreadsheets and most calculators will allow you to work directly with percentages, but be careful that you understand fully how you are using them. If you are in any doubt, it is better to use proportions.

percentage=proportion × 100

and:
 proportion = percentage 100

Another way that you will need to use percentages or proportions is to work out the final value or cost of something subject to increase. What does a 3.25% pay award mean to someone earning £19 565 per year? What will be the price of a power drill marked at £34-99 plus VAT? For the VAT example, value-added tax is charged at 17.5% of the purchase price, so that:

price including VAT = price excluding VAT + 17.5% of the ex-VAT price

Many people will work out the VAT payable and then add this to the price. However, if you understand the relationship between percentages and proportions, you can do the calculation in one go. Remember that a percentage of 17.5% is equal to a proportion of 0.175. So the equation above can be re-written as:

price including VAT = ex-VAT price + 0.175 × ex-VAT price = 1.175 × ex-VAT price

so that:

£34-99 × 1.175 = £41-11

Returning to the pay increase, the new salary will be £19 565 × 1.0325 = £20 201.

The way that things like interest rates are reported in the media can cause confusion. An increase in the mortgage interest rate of 'half a percentage point' sounds fairly benign. However, if the current rate is 3.75%, the rate has now increased to 4.25% - this represents an increase of 13% over the previous rate, which would be reflected in monthly payments.

All ratios, proportions and percentages are interconvertable:

Ratio Proportion Percentage
1 : R
 1 R
 ( 1 ) × 100 R
1:20 0.2 20%
1:200 0.02 2%
1:2000 0.002 0.2%
2:1 2 200%