Working with percentages

This study guide explains the various ways percentages can be used to compare values, quantify change and how to increase or decrease values.

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This guide is designed to help you use percentages in your work and daily life. It explains how to use percentages to compare values, how to quantify changes using percentages, and how to calculate the amount represented by a percentage increase or decrease.

Other Useful Guides:  Using Averages, Measures of Variability

Introduction

The term percentage or symbol % is used frequently in everyday language and life. For example, it is common to see sales with 20% discount, or restaurant bills with 10% service charges, as well as reports in newspapers discussing tax, unemployment and other values in percentage terms. Percentages are used in these ways as a simplified means of conveying size or scale or value

Percentage means parts out of 100 and is the same as a fraction with a denominator (bottom) of 100. Therefore:

15% means 15 parts out of 100 and is the same as the fraction 15/100

87% means 87 parts out of 100 and is the same as the fraction 87/100

A further way of expressing parts out of 100 is using a decimal and so percentages can also be expressed as decimals:

15% is the same as 0.15 or 15/100

87% is the same as 0.87 or 87/100

There are three main ways in which percentages are frequently used:

  • to enable data with different sample sizes or totals to be compared;
  • to quantify the amount of change over time;
  • to express an increase or reduction relative to initial size.

Using percentages to compare information

Whilst researching for an essay or dissertation you may come across many sources of data in tables, graphs or reports which you would like to incorporate into your work. However, this can be difficult if they do not share a common base line. Percentages are useful for comparing information where the sample sizes or totals are different. By converting different data to percentages you can readily compare them.

For example, the following table shows sales of CDs, LPs, cassettes and singles in 1996 and 1997.

 

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Table 1

 

Because  the total music sales in 1996 and 1997 were different it is difficult to compare the data for the two years and to determine whether or not there was any notable change in the format in which music was bought. However, if the number of sales in each format (LP, CD etc.) is expressed as a percentage of the total sales, then it is easier to compare the data for the two years.

For example, the conversion from the actual number of CD sales in 1996 to a percentage can be done in the following way:

Determine the fraction of the total music sales represented by CDs in 1996:

159.7 million out of a total of 286.6 million sales were of CDs

Re-write this as a fraction: 159.7/286.6

Convert the fraction to a decimal by dividing 159.7 by 286.6

159.7 ÷ 286.6 = 0.5572 (this is the decimal format explained above)

Convert the decimal to a percentage by multiplying by 100

0.5572 x 100 = 55.72

The result indicates that sales of CDs accounted for 55.72% of all music sales in 1996.

These four stages can be summarised by the following formula: (159.7/289.6)x(100/1)

This process can be repeated for all the music sales in 1996 and 1997 to produce table 2:

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Table 2

 

Converting the music sales to percentages makes it easier to compare the sales according to the music format. For example, table 2 shows that in 1996 16.1% (or 16.1 out of every 100 music sales) was in the form of cassettes, but in 1997 this had fallen to 12.8%.

Calculating percentage change

Percentages are also very useful if you wish to quantify change. This is because they provide a result in the form of parts per hundred that is usually more readily understandable and comparable than when the information is presented as raw values.  

Using the information presented in table 1 the decrease in cassette sales between 1996 and 1997 can be calculated as a percentage.

Determine the difference between the 1996 and 1997 cassette sales:

46.2 million (1996 sales) − 36.6 million (1997 sales) = 9.6 million

Express this difference as a fraction of the starting value (the 1996 sales of 46.2 million) and multiply the result by 100: (9,600,000/46,200,000)x(100/1) = 20.78%

This means that there was a fall of 20.78% in the number of cassette sales in 1997 as compared to 1996.

When calculating the percentage change between two values it is important to determine the correct base from which to calculate the percentage change (i.e. the appropriate starting value). This is because the percentage change from a low number to a higher number is not the same as the percentage change from the same higher number to the same lower number. For example:

The percentage increase from 50 to 75 = (25/50)x(100/1) = 50%

However, the percentage decrease from 75 to 50 = (25/75)x(100/1) = 33.3%

Calculating percentage increase or reductions

In the previous example, the percentage change between two values was calculated. The reverse process is where the actual amount represented by a particular percentage is to be calculated. Typical examples of the use of percentages in this way are in shop sales, where the prices have been reduced by a certain percentage, or bank charges where interest rates are expressed as a percentage. Again the use of the term percentage expresses the amount in terms of parts per hundred. For example, a bank interest rate of 6% is the same as saying the interest on each £1.00 invested is 6 parts out of 100 or 6p and a sale reduction of 12% means that an item will be reduced by 12p for every £1.00 it originally cost.

There are two ways of calculating the new value of the item following a percentage increase or reduction. For example you may wish to calculate the sale price of a TV normally priced at £180.00 that has been reduced by 15%.

The first method is to calculate the amount that the item has been reduced by:

15% of £180 = 15 ÷ 100 x £180 = £27

This amount is then subtracted from the original price:

£180 – £27 = £153

The second method is to calculate the percentage of the original price you are now paying for the item:

100% (full price) - 15% (discount) = 85% (discounted price)

This value is multiplied by the original price:

0.85 x £180 = £153 (this uses the decimal format explained above)

The result (£153) is the same whichever method you use.

You can use the same methods to calculate the value of percentage increases. For example, if you had savings of £112.72 a year ago that you put in an account earning 4.3% interest, you can calculate the current value of your savings. Again there are two methods.

Either calculate the actual amount by which your savings have increased:

4.3% of £112.72 = 4.3 ÷ 100 x £112.72 = £4.85

and add this to the original value of the savings:

£112.72 + £4.85 = £117.57

or calculate the percentage of your original savings you now have:

100% + 4.3% = 104.3%

and multiply this by the original value of your savings:

1.043 x £112.72 = £117.57 (this uses the decimal format explained on page 1)

Again the two methods produce the same result.

Summary

This guide has shown three main uses of percentages and how to perform calculations for each of these. Percentages are related to fractions and decimals and express values in the common currency of parts per hundred. Expressing values in the form of percentages in your work will enable you to readily compare information from different sources, quantify change over time and find the amount by which something has increased or decreased following a percentage change. Whilst you will come across data that is already presented as percentages remember that you can also convert data from tables and graphs that you find in books, journals and other publications into percentages in order to make useful comparisons.

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