Calculating Percentages
A percentage (%) is a fraction expressed as a part of one hundred, instead of any other denominator. The word comes from the Latin per cent, meaning ‘out of one hundred’.
A half, therefore, is 50%, because 50 is half of 100.
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The advantage of working with percentages is that they are relatively straightforward to calculate because, unlike fractions, you are always working with a base of 100.
This page explains how to calculate percentages, and provides some simple percentage calculators for you to use.
Percentages vs. Fractions
Percentages are actually fractions, with a denominator (the number below the line) of 100.
What this means in practice is that you don’t need to worry about whether the numerator (top) divides by the denominator (bottom), or whether it is reduced to its lowest form, as you do when you are working with fractions.
It is simply a matter of working out how many hundredths you have, and then expressing it as a decimal if necessary.
Three types of expression
There are three types of percentage calculations:
- What is x% of y?
- What is x as a percentage of y?
- If x is y percent, what is the whole?
We will look at each of these in turn, providing examples of how to calculate them for you to practise, using the percentage calculators on the page if you wish.
Examples
What is 10% of 50?
There are two ways to approach this. The first says “I know that 10 is one-tenth of 100. I will therefore divide 50 by 10. The answer is 5”.
This is fine when the numbers are relatively simple. But suppose the numbers are more complicated.
What is 22% of 46?
Now it’s much less straightforward. You can’t just work out what 22% is, expressed as a fraction, and anyway, it’s not a simple fraction.
Instead, you have to divide 46 into 100 equal parts, and work out what 22 of them would be when added together.
So:
46 ÷ 100 = 0.46 [remember, when you divide by 100, you move the decimal point two places to the left].
0.46 × 22 = 10.12
Answer: 22% of 46 is 10.12.
The same rules apply to questions which are written as word problems.
You are buying some paint and the shop’s prices do not include VAT [sales tax]. You want to know how much you will be paying in tax. VAT is charged at 20%. The paint costs £15 per pot, and you will need three pots.
The total price of the paint is £15 × 3 = £45.
Effectively, you are being asked ‘What is 20% of £45?’
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45 ÷ 100 = 0.45
0.45 × 20 = £9.
Answer: The total tax payable on the transaction will be £9.
Examples
Think of this as turning a fraction into a percentage. Your fraction is x/y, and your percentage is [unknown, here] A/100.
x/y = A/100
The easiest way to do these, is to move the fraction around. If you multiply both sides by 100, you get A (your unknown) = 100x divided by y. Just plug in the numbers and out will come the answer. Some examples may make this even clearer.
What is 10 as a percentage of 50?
Using the formula that we have just worked out, x is 10 and y is 50. The calculation is therefore:
100 × 10 = 1000 1000 ÷ 50 = 20.
Answer: 10 is 20% of 50.
This method also works with word-based problems.
You have been quoted commission of $7.50 on the sale of a table. The sale price is $150. Another company has quoted you commission of 4.5%. You want to know which is better value.
This is asking you ‘What is $7.50 as a percent of $150?’.
Using the formula, therefore, x is 7.5 and y is 150.
7.5 × 100 = 750 750 ÷ 150 = 5.
Answer: The commission of $7.50 is 5% on the sale price. The commission of 4.5% is therefore better value for you as a customer.
Examples
Again, you can think of this as a fraction, but in a slightly different form.
Here, x and y are on opposite sides of the equation.
x/A = y/100
Manipulating this equation, once again, you get A = 100x ÷ y.
If 10 is 45%, what is the total?
x = 10, and y = 45. 100 × 10 = 1000. 1000 ÷ 45 = 22.22
Here 22.22 is the total.
More Complicated Examples
It’s worth thinking about some more complicated, ‘real world’ examples using percentages.
Interest rates are almost always given in percentages, which means that mortgages and credit cards rely heavily on them. Understanding how to calculate them could save you a lot of time and hassle (and money).
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You will probably need to use several steps, and several calculations, to get the answer.
Mortgage Interest Calculations
You want to take out an interest-only mortgage of £215,000, at a fixed interest rate of 1.5% per annum for the first two years, payable in monthly instalments, and then you will move onto the bank’s standard variable rate, currently 2.75%. You want to know how much you will have to pay each month in interest for the first two years.
The yearly interest payment is 1.5% of £215,000.
215,000 ÷ 100 = 2,150 2,150 × 1.5 = 3,225
That gives you the annual interest, but you are going to pay it in monthly instalments. That means that each year’s payment has to be divided by 12 (in practice, your mortgage company will probably do it by day, so that it will alter slightly each month, but this should be close enough for budgeting purposes).
3,225 ÷ 12 = £268.75
The monthly interest payment will be £268.75
Now suppose that you want to know how much interest you will have to pay over the lifetime of the mortgage, 25 years.
For the first two years, the interest rate is 1.5%, and you already know that the annual payment is £3,225. The total for the first two years is therefore £3,225 × 2 = £6,450.
After that, you don’t actually know what the interest rate will be, because the bank’s standard variable rate changes. But right now, it’s 2.75%, so you can use that to calculate it for comparison purposes.
2,150 × 2.75 = £5,912.50 per year
You will be paying this for 23 years (25 years minus the first two), so the total you will have to pay on that interest rate is £5,912 × 23 = £135,987.50.
Altogether, over the 25 years, you will pay the bank £135,987.50 + £6,450 = £142,437.50.
No wonder banks are happy to lend money for mortgages. This is also why it is worthwhile paying off your mortgage early if you can do so.
Now you can take a further step, and calculate what the banks call the annual equivalent rate, that is, the average rate per year over the whole lifetime of the loan.
The average payable per year is the total divided by the number of years, in this case £142,437.50 ÷ 25 = £5,697.50.
Now the question is ‘What is £5,697.50 as a percent of £215,000?’.
Plug this into the formula A = 100x ÷ y. x is £5,697.50, and y is £215,000.
100 × 5,697.50 = 569,750 569,750 ÷ 215,000 = 2.65%
The annual equivalent rate is 2.65%.
Understanding Mortgages
Comparing Like-for-Like
Being able to calculate percentages in several different ways means that you can compare like with like.
You will therefore be able to understand and compare interest rates calculated daily, monthly and yearly. You can also see how to use simple percentage calculators in several different steps to work out complex problems.
You are, in fact, well on the way to mastering an essential skill that will ensure that you understand all your financial commitments.
Understanding Interest
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