Fractions, decimals and percentages are all forms for expressing parts of a whole and each can be represented in any of the three forms. For example, the fraction 1/2 can be represented as a decimal (0.5) or as a percentage (50%).
Fractions express parts of a whole. Fractions are given in the form of a numerator and a denominator. The denominator indicates what size part is being expressed and the numerator indicates how many of those parts are involved. For example, in 6/7 the denominator (7) indicates that each part is one seventh of the whole. The numerator (6) indicates that there are six of these parts.
Example
To find 6/7
Converting fractions to decimals and vice versa
Converting a fraction to a decimal
A fraction can always be converted to a decimal by dividing the top (numerator) by the bottom (denominator).
So the fraction 60/150 can be converted to a decimal by division: 60 ÷ 150, or 6 ÷ 15, or 2 ÷ 5.
Each of these will give the same answer 0.4, because all three fractions have the same value (or are equivalent). The on-screen calculator can be used to carry out the division.
Another way of doing this more quickly is to use the simplified fraction 2/5 = 0.4
It is worthwhile remembering some common equivalents.
3/4 = 0.75
1/2 = 0.5
1/4 = 0.25
Converting a decimal to a fraction
A decimal such as 0.7 can be written as a fraction by remembering that the first decimal represents tenths, the second hundredths and so on.
So 0.7 = 7/10, 0.46 = 46/100 and so on.
Sometimes the result of division gives more digits after the decimal point than are needed or are appropriate. In this case the number would be rounded up or down, remembering always to state the degree of accuracy that has been given. So an answer 63.472418 might be rounded up to 63.5 to one decimal place or to 63.47 to two decimal places.
Converting fractions and decimals to percentages and vice versa
Converting a percentage to a fraction
Because a percentage, as its name implies, represents parts out of a hundred, it is easy to convert a percentage to a fraction.
| So 60% |
= 60/100 |
|
= 30/50 (by dividing top and bottom numbers by 2) |
|
= 3/5 (by dividing top and bottom numbers by 10) |
| 75% |
= 75/100 |
|
= 3/4 (by dividing top and bottom numbers by 25) |
It is worthwhile remembering the more common percentages:
| 10% |
= 1/10 |
(divide by 10) |
| 12 1/2% |
= 1/8 |
(divide by 8) |
| 20% |
= 1/5 |
(divide by 5) |
| 25% |
= 1/4 |
(divide by 4) |
| 40% |
= 2/5 |
(divide by 5, multiply by 2) |
| 50% |
= 1/2 |
(divide by 2) |
| 75% |
= 3/4 |
(divide by 4, multiply by 3) |
| 100% |
= 1 |
(divide by 1) |
Converting a decimal to a percentage
It is easy to convert a decimal to a percentage, because, as already seen above, decimals can always be easily written with a denominator of a hundred.
| 0.5 |
= 5/10 |
|
= 50/100 |
|
= 50% |
| 0.413 |
= 413/1000 |
|
= 41.3/100 |
|
= 41.3% |
Converting a fraction to a percentage
Fractions with denominators of 10 or closely related to 10 can be easily converted to a percentage.
| 3/5 |
= 6/10 |
|
= 60/100 |
|
= 60% |
Example
A pupil scores 71/80 in a test; what percentage is this?
71÷ 80 using a calculator gives 88.75%.
For most fractions it is easier to convert them first into decimals (using a calculator if appropriate) and then convert that decimal into a percentage.
| 4/9 |
= 4 ÷ 9 |
|
= 0.444444 |
|
= 44.4444% |
|
= 44% to the nearest whole number |
| 13/17 |
= 13 ÷ 17 |
|
= 0.7647058 |
|
= 76.47058% |
|
= 76% to the nearest whole number, or 76.5% to 1 decimal place |
Calculating with percentages
Some of the most common fractions, decimals and their percentage equivalents:
| 1% |
= 1/100 |
= 0.01 |
(divide by 100) |
| 5% |
= 1/20 |
= 0.05 |
(divide by 20) |
| 10% |
= 1/10 |
= 0.1 |
(divide by 10) |
| 12.5% |
= 1/8 |
= 0.125 |
(divide by 8) |
| 20% |
= 1/5 |
= 0.2 |
(divide by 5) |
| 25% |
= 1/4 |
= 0.25 |
(divide by 4) |
| 50% |
= 1/2 |
= 0.5 |
(divide by 2) |
| 75% |
= 3/4 |
= 0.75 |
(divide by 4, multiply by 3) |
You should learn all of these and be able to switch between equivalent forms. This knowledge could be particularly helpful when answering mental arithmetic questions.
Examples
- A mathematics department ordered books to the value of £96. The supplier gave a discount of 12.5%. 1/8 x 96 =12
The saving is £12, so the books actually cost £84. - Three eighths of a class of 30 pupils have school dinners. What percentage do not have school dinners?
3/8 = 37.5%, so 62.5% do not have school dinners.
Simple percentage calculations
Some percentages are very easy to calculate both in your head and on a calculator. For example, it is always straightforward to find 10% of a quantity in your head by remembering that 10% is 1/10. So, to find 10% of anything, all you need to do is divide by 10.
What is 10% of 632?
10% of 632 = 632 ÷ 10 = 63.2
Finding 10% can often help with finding other percentages in your head.
Look at the following two examples.
- What is 30% of 40?
10% of 40 = 40 ÷ 10 = 4
30% of 40 = 4 x 3 = 12 - What is 15% of 30?
10% of 30 = 3
5% of 30 = 3 ÷ 2 = 1.5
15% of 30 = 3 + 1.5 = 4.5
The other common percentages can be used in the same way to find answers in your head.
What is 75% of 80p?
75% of 80 = 3/4 of 80
1/4 = 20
3/4 of 80 = 20 x 3 = 60
Other more complex percentages can also be found easily by converting the percentages to a decimal, using a calculator if appropriate.
What is 73% of 84?
Convert 73% into a decimal in your head.
73% = 0.73
0.73 x 84 on the calculator, or by other means
= 61.32
You can check whether your answer is about right by estimation, eg 73% is about 75% which you know is 3/4. So you know the answer will certainly be more than 1/2 of 84 but less than 84.
Using percentages to compare data
Percentages are often useful when making comparisons between data.
Example
| Total number of half days of unauthorised absences each term |
|---|
|
Term 1 |
Term 2 |
Term 3 |
Total absences |
| Year 9 (1998/99) |
46 |
60 |
44 |
150 |
| Year 9 (1997/98) |
34 |
78 |
18 |
130 |
In which year, 1997/98 or 1998/99, was the percentage of term two unauthorised absences out of total absences for the year the greater?
From the table, the term two absences for the year 1998/99 are 60 out of 150.
For the year 1997/8 the absences for term two were 78 out of a total absences of 130.
Convert both of these fractions to percentages. The first can be done mentally:
| 60/150 |
= 6/15 |
|
= 2/5 |
|
= 4/10 |
|
= 40% |
For the second, convert the fraction first to a decimal, using a calculator if you need to, and then to a percentage.
| 78/100 |
= 78 ÷ 130 |
|
= 0.60 |
|
= 60% |
So:
The percentage of unauthorised absences in term two for 1997/98 = 60%
The percentage of unauthorised absences in term two for 1998/99 = 40%
This means that the percentage of term two unauthorised absences out of total absences was greater for 1997/98.
Worked examples
Example one
A pupil obtained the following marks in three tests.
| Pupil | Test 1 | Test 2 | Test 3 |
|---|
| Actual mark |
95 |
39 |
61 |
| Possible mark |
150 |
60 |
100 |
In which test did the pupil a achieve the best result?
The results can be written as:
61/100
To answer this question using percentages, first convert each fraction to a decimal, using a calculator where appropriate:
| 95/100 |
= 95 ÷ 150 |
>= 0.63 to 2 decimal places |
| 39/60 |
= 39 ÷ 60 |
>= 0.65 |
| 61/100 |
= 61 ÷ 100 |
>= 0.61 |
Now convert each decimal result above to a percentage in your head.
0.63 = 63%
0.65 = 65%
0.61 = 61%
65% is the highest percentage. So the pupil achieved the best result in test 2.
Selecting the best method of comparing data is a matter of judgment. However, with data presented as in the question above percentages would normally be used.
Example two
Four schools had the following proportion of pupils receiving free school meals.
| School | Proportion of pupils receiving free school meals |
|---|
| A |
1/9 |
| B |
0.12 |
| C |
11% |
| D |
11 pupils out of 110 |
Which school had the highest proportion of pupils receiving free school meals?
Converting all the proportions to percentages enables a comparison to be made easily.
The proportion of pupils having free school meals in school A is given as the fraction 1/9.
1÷ 9 = 0.11 to two decimal places, or 11%
The proportion for school B is given as a decimal: 0.12 = 12%
The proportion for school C is given at 11%
The proportion for school D is given as 11 out of 110. This is the same as 11/110. First convert this to a decimal:
11 ÷ 110 = 0.1 = 10%
| School A | School B | School C | School D |
|---|
| 11% |
12% |
11% |
10% |
So school B had the highest proportion of pupils receiving free school meals because 12% is the highest percentage.
Example three
The numbers of year 13 pupils who left school to enter employment at the end of 1998 and at the end of 1999 are given in the table below.
| Destination |
1998 |
1999 |
|---|
| Employment |
50 |
60 |
|---|
| Total number of year 13 pupils |
525 |
540 |
|---|
| Percentage entering employment (to 1 decimal place) |
9.5% |
11.1% |
|---|
By how many percentage points had the number of pupils entering employment increased between 1998 and 1999?
Finding the increase in percentage points is just a matter of comparing the percentages.
The percentage entering employment in 1998 was 9.5% and in 1999 was 11.1%.
The increase was 11.1% - 9.5% = 1.6%
The increase was 1.6 percentage points.
By what percentage did the numbers entering employment increase between 1998 and 1999?
To find the percentage increase, we need to think what any increase is to be compared to. There was an actual increase of pupils entering employment of 10. Compare this increase with the original number of pupils to get a measure of the significance of the increase.
So, to calculate the percentage increase that 10 more pupils represents, it is necessary to find 10 as a percentage of the number of pupils in employment last year, ie 10 as a percentage of 50.
10/50 = 10 ÷ 50 = 0.2 = 20%
So, the number entering employment increased by 20% between 1998 and 1999.
Note: when entering answers which are fractions, for example 5 eighths, the answer will appear as 5/8.
Avoiding common errors
Most errors can be avoided by:
- memorising equivalent values for commonly used percentages, fractions and decimals
- recognising the equivalence between decimals, fractions and percentages, eg. 35/100, 0.35 and 35%
- knowing how to convert from fraction into decimals and percentages into another form and back again
- simplifying fractions to ease understanding and use, and
- clearly understanding the difference between a percentage increase or decrease and a percentage point increase or decrease.