Percentage Are Used to Express Either a Whole or Part of a Whole
The term 'per cent' means 'out of a hundred'. In mathematics, percentages are used similar fractions and decimals, as ways to describe parts of a whole. When you are using percentages, the whole is considered to exist made up of a hundred equal parts. The symbol % is used to show that a number is a percentage, and less commonly the abbreviation 'pct' may be used.
You will meet percentages most everywhere: in shops, on the internet, in advertisements and in the media. Being able to understand what percentages mean is a key skill that will potentially save you time and coin and volition also brand y'all more employable.
The Meaning of Percentages
Percentage is a term from Latin, meaning 'out of ane hundred'.
You lot can therefore consider each 'whole' equally broken upward into 100 equal parts, each ane of which is a unmarried per centum.
The box below shows this for a simple filigree, but information technology works the same way for anything: children in a course, prices, pebbles on the beach, and so on.
Visualising Percentages
The grid below has 100 cells.
- Each cell is equal to 1% of the whole (the red cell is ane%).
- Two cells are equal to 2% (the greenish cells).
- Five cells are equal to 5% (the blue cells).
- Twenty 5 cells (purple cells) are equal to 25% of the whole or one quarter (¼).
- Fifty cells (yellow cells) are equal to 50% of the whole or half (½).
How many unshaded (white) cells are there? What is the percentage of unshaded cells?
Respond: There are two means to work this out.
- Count the white cells. There are 17 of them. Out of 100 cells, 17% are therefore white.
- Add up the number of other cells, and have them from 100. There is i ruby cell, two light-green, five bluish, 25 purple, and 50 yellowish. That adds up to 83. 100−83 = 17. Again, out of 100 cells, 17 are white, or 17%.
It is easy to work out the percentage when there are 100 private 'things' making up the whole, as in the grid in a higher place. Just what if there are more than or less?
The answer is that you catechumen the individual elements that brand up the whole into a per centum. For example, if there had been 200 cells in the grid, each percentage (1%) would be two cells, and every prison cell would exist one-half a percent.
We employ percentages to brand calculations easier. It is much simpler to work with parts of 100 than thirds, twelfths and and so on, particularly because quite a lot of fractions do not have an exact (non-recurring) decimal equivalent. Chiefly, this too makes information technology much easier to make comparisons betwixt percentages (which all finer have the common denominator of 100) than information technology is between fractions with unlike denominators. This is partly why so many countries utilise a metric organisation of measurement and decimal currency.
Finding the Percentage
The general rule for finding a given percentage of a given whole is:
Work out the value of 1%, then multiply it by the per centum yous need to find.
This is easiest to understand with an case. Let's suppose that yous desire to buy a new laptop estimator. You accept checked local suppliers and one visitor has offered to give you lot twenty% off the list cost of £500. How much will the laptop cost from that supplier?
In this example, the whole is £500, or the price of the laptop before the disbelieve is applied. The percentage that yous need to find is 20%, or the disbelieve offered by the supplier. You are then going to take that off the full price to find out what the laptop will cost you.
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Start by working out the value of 1%
One percent of £500 is £500 ÷ 100 = £5.
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Multiply it past the percent you are looking for
Once you accept worked out the value of one%, yous but multiply information technology by the percentage you are looking for, in this example xx%.
£v × twenty = £100.
You now know that the disbelieve is worth £100.
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Complete the calculation past adding or subtracting every bit necessary.
The price of the laptop, including the disbelieve, is £500−xx%, or £500−£100 = £400.
The easy fashion to work out 1% of any number
i% is the whole (whatever that may be) divided by 100.
When we split something by 100, we just movement the identify values ii columns to the correct (or move the decimal point two places to the left).
Y'all can detect out more about numbers and place values on our Numbers page, merely here's a quick epitomize:
£500 is fabricated up of 5 hundreds, zero tens and zip units. £500 also has cypher pence (cents if y'all are working in dollars) so could be written equally £500.00, with nil tenths or hundredths.
| Hundreds | Tens | Units | Point | Tenths | Hundredths |
| five | 0 | 0 | . | 0 | 0 |
When we divide past 100, we move our number two columns to the correct. 500 divided by 100 = 005, or 5. Leading zeros (zeros on the 'exterior left' of a number, such every bit those in 005, 02, 00014) have no value, and then we exercise not need to write them.
Yous can also think of this as moving the decimal point ii places to the left.
| Hundreds | Tens | Units | Point | Tenths | Hundredths |
| 0 | 0 | 5 | . | 0 | 0 |
This rule applies to all numbers, and then £327 divided by 100 is £iii.27. This is the same every bit proverb that £three.27 is i% of £327. £one divided by 100 = £0.01, or one pence. In that location are i hundred pence in a pound (and one hundred cents in a dollar). 1p is therefore 1% of £1.
In one case you have calculated one% of the whole, yous can and then multiply your reply to the pct you are looking for (see our page on multiplication for help).
Mental Maths Hacks
As your maths skills develop, you can brainstorm to see other means of arriving at the same answer. The laptop example higher up is quite straightforward and with practise, yous tin can use your mental maths skills to call up most this problem in a dissimilar way to arrive easier. In this example, you are trying to discover 20%, then instead of finding 1% and then multiplying it by 20, you tin notice 10% and and so simply double information technology. Nosotros know that x% is the aforementioned every bit ane/10th and we tin can divide a number by ten by moving the decimal place ane place to left (removing a zero from 500). Therefore 10% of £500 is £fifty and 20% is £100.
A useful mental maths hack is that percentages are reversible, so 16% of 25 is the same as 25% of 16. Invariably, one of those will be much easier to work out in our head…try information technology!
Utilize our Percentage Calculators to quickly solve your percent bug.
Working with Percentages
We calculated a xx% discount in the case above then subtracted this from the whole to piece of work out how much a new laptop would cost.
Equally well as taking a percentage abroad, we can also add a percentage to a number. Information technology works exactly the aforementioned way, but in the last stride, you simply add together instead of subtracting.
For example: George is promoted and gets a 5% pay rising. George currently earns £24,000 a yr, so how much will he earn after his pay rise?
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Work out 1% of the whole
The whole in this case is George'south electric current salary, £24,000. 1% of £24,000 is 24,000 ÷ 100 = £240.
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Multiply that by the percentage you are looking for
George is getting a five% pay ascension, so nosotros need to know the value of 5%, or 5 times 1%.
£240 × 5 = £1,200.
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Complete the calculation by adding to the original amount
George's pay rise is £1,200 per year. His new bacon will therefore be £24,000 + £1,200 = £25,200.
Percentages over 100%
It is possible to have percentages over 100%. This example is i: George's new salary is actually 105% of his old ane.
However, his former bacon is not 100% of his new one. Instead, it is only over 95%.
When you are calculating percentages, the key is to check that you are working with the right whole. In this case, the 'whole' is George's one-time salary.
Percentages as Decimals and Fractions
One percent is one hundredth of a whole. It can therefore exist written equally both a decimal and a fraction.
To write a percentage as a decimal, simply divide it by 100.
For case, 50% becomes 0.5, 20% becomes 0.two, ane% becomes 0.01 so on.
We can calculate percentages using this knowledge. fifty% is the same as a half, so fifty% of x is 5, because 5 is half of 10 (10 ÷ 2). The decimal of fifty% is 0.5. And then another way of finding fifty% of 10 is to say 10 × 0.5, or ten halves.
20% of l is the aforementioned as saying fifty × 0.two, which equals ten.
17.5% of 380 = 380 × 0.175, which equals 66.5.
George's bacon increase in a higher place was five% of £24,000. £24,000 × 0.05 = £one,200.
The conversion from decimal to pct is just the reverse calculation: multiply your decimal by 100.
0.v = l%
0.875 = 87.five%
To write a pct as a fraction, put the percentage value over a denominator of 100, and dissever information technology down into its everyman possible form.
50% = l/100 = 5/10 = ½
20% = 20/100 = 2/10 = 1/5
30% = thirty/100 = 3/ten
Warning!
It is possible to convert fractions to percentages by converting the denominator (the bottom number of the fraction) into 100.
Nevertheless, it is harder to convert fractions to percentages than percentages to fractions because not every fraction has an exact (non-recurring) decimal or percentage.
If the denominator of your fraction does non divide a whole number of times into 100, then in that location will non be a simple conversion. For example, 1/three, one/6 and 1/9 do not make 'dandy' percentages (they are 33.33333%, 16.66666% and 11.11111%).
Working out Percentages of a Whole
So far we accept looked at the basics of percentages, and how to add or subtract a percentage from a whole.
Sometimes information technology is useful to be able to work out the percentages of a whole when you are given the numbers concerned.
For case, let's suppose that an organisation employs ix managers, 12 administrators, 5 accountants, 3 human resource professionals, 7 cleaners and iv catering staff. What percentage of each blazon of staff does it apply?
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Start by working out the whole.
In this case, y'all exercise not know the 'whole', or the total number of staff in the organisation. The get-go footstep is therefore to add the different types of staff.
9 managers + 12 administrators + v accountants + 3 Hour professionals + 7 cleaners + 4 catering staff = 40 members of staff.
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Piece of work out the proportion (or fraction) of staff in each category.
We know the number of staff in each category, but we need to convert that to a fraction of the whole, expressed as a decimal. The calculation nosotros need to do is:
Staff in Category ÷ Whole (Encounter our segmentation page for assist with segmentation sums or use a figurer)
We tin can use managers as an instance:
9 managers ÷ forty = 0.225
In this case it tin can be helpful if, instead of thinking of the division symbol '÷' as meaning 'divided past', nosotros can substitute the words 'out of'. We use this often in the context of test results, for example 8/10 or 'viii out of 10' correct answers. So we summate the 'number of managers out of the whole staff'. When nosotros use words to describe the adding, information technology can help it to brand more sense.
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Convert the fraction of the whole into a percent
0.225 is the fraction of staff that are managers, expressed equally a decimal. To convert this number to a percentage, we need to multiply it past 100. Multiplying by 100 is the same equally dividing by a hundred except you lot move the numbers the other style on the place values calibration. So 0.225 becomes 22.five.
In other words, 22.v% of the organisation'southward employees are managers.
We then practise the same ii calculations for each other category.
- 12 administrators ÷ 40 = 0.iii. 0.3 × 100 = 30%.
- 5 accountants ÷ twoscore = 0.125. 0.125 × 100 = 12.5%.
- three HR professionals ÷ 40 = 0.075. 0.075 × 100 = 7.v%.
- 7 cleaners ÷ 40 = 0.175. 0.175 × 100 = 17.5%.
- 4 catering staff ÷ twoscore = 0.1. 0.1 × 100 = 10%.
Height TIP! Bank check you have a total of 100%
When you accept finished calculating your percentages, it is a good idea to add them together to make sure that they equal 100%. If they don't, then bank check your calculations.
In summary, we tin can say that the organisation is fabricated up of:
| Roles | Number of Staff | % of Staff |
| Managers | nine | 22.5% |
| Administrators | 12 | 30% |
| Accountants | v | 12.5% |
| Hr professionals | iii | 7.v% |
| Cleaners | vii | 17.5% |
| Catering staff | iv | 10% |
| Full | 40 | 100% |
It tin can exist useful to show percent data representing a whole on a pie nautical chart. You can quickly see the proportions of categories of staff in the example.
For more on pie charts and other types of graphs and charts see our folio: Graphs and Charts.
Points to remember
- Percentages are a style to describe parts of a whole.
- They are a bit like decimals, except that the whole is always split into 100, instead of tenths, hundredths, thousandths and so on of a unit.
- Percentages are designed to make calculations easier.
Further Reading from Skills You Need
Proportion
Part of The Skills You Need Guide to Numeracy
This eBook covers proportion looking at numbers as parts of other numbers, as parts of a larger whole, or in relation to other numbers. The volume covers fractions and decimals, ratio and percentages with worked examples for you to try and develop your skills.
Whether you want to brush up on your basics, or help your children with their learning, this is the book for you.
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