How to find a percentage of an amount

Finding a percentage of an amount means finding a fraction with 100 on the bottom of that amount. The usual hurdle is deciding what to divide by and what to multiply by, so we go through it with 1% as the stepping stone every time.

When do I use this?

Whenever a question says "find $x\%$ of something": 15% of a price, 20% of a class, 40% of a tank. The something is the whole, and the whole is 100%. If you want the background on why that works, it is all in what is a percentage?: per cent means hundredths, so 15% is the fraction $\tfrac{15}{100}$, and we are finding that fraction of the amount.

The procedure

Let's say we had to find 15% of 60.

1. Decide what the whole is. Here 60 is the whole, that is 100%. So we have to find $\tfrac{15}{100}$ of 60.
2. Find 1%. We know that 1% is $\tfrac{1}{100}$, so we can find 1% of 60 by dividing by 100. That gives $\tfrac{60}{100}$, which is 0.6.
3. Multiply by the percentage you want. We want 15%, so we multiply 1% by 15.

Written in one line:

$$\frac{60}{100} \times 15 = 9$$

So we find how much 1% of 60 is, and then we multiply by 15 to find 15% of 60, which is 9.

More generally we can say that $x\%$ of $y$ can be found by

$$\frac{y}{100} \times x$$

The $\tfrac{y}{100}$ is the 1%, and the $\times x$ takes you up to $x\%$.

Worked examples

Example 1: 20% of 60. We find 1% first, which is $\tfrac{60}{100}$, and then multiply by how many per cent we want:

$$\frac{60}{100} \times 20 = 12$$

So 20% of 60 is 12.

Example 2: 20% of £100. This one is the VAT on a £100 item, since VAT in the UK is 20%. With 100 it is pretty straightforward, but what we do is still to find 1% by dividing by 100, and then multiply by 20 to find 20%:

$$\frac{100}{100} \times 20 = 20$$

It turns out that 20% of 100 is 20, so the VAT is £20. (my Danish notes use 25% here, because Danish VAT is 25%. The method is the same.)

Example 3: 15% of 60, the other way round. It does not matter whether you divide first or multiply first, because $\tfrac{60}{100} \times 15$ is the same as $\tfrac{15}{100} \times 60$. We can check that this is right, because we know that $\tfrac{15}{100} \times 60 = \tfrac{900}{100} = 9$, the same answer as before. Finding 1% first is just the tidiest order to do it in, and it keeps the numbers small.

Common mistakes

• "Percentages need their own formula." They do not. 15% of 60 is the fraction $\tfrac{15}{100}$ of 60, and the fraction rules do the rest. If you can multiply a fraction by a number, you can do this.
• Forgetting which number is the whole. The amount you are taking the percentage of is the whole, that is 100%. It is the number that gets divided by 100. In "15% of 60", the 60 is the whole, not the 15.
• Skipping the 1% step. The whole method is: divide by 100 to get 1%, then multiply. If you multiply 60 by 15 without dividing by 100 anywhere, you get 900, which is 100 times too big.

Related

• What is a percentage?
• How to write one number as a percentage of another
• Percentage increase and decrease with a multiplier