How to write one number as a percentage of another

Writing one number as a percentage of another means finding how big a share the first number is of the second, and then giving that share in hundredths. The usual hurdle is putting the numbers the right way up in the fraction, so that is where we start.

When do I use this?

Whenever a question asks "what percentage is $x$ of $y$": 20 marks out of 60, 45 of the 60 people in a room, the part of a price that is VAT. This is the second of the four types of percentage problem in what is a percentage?. Where the first type asks for a percentage of a number, this one runs the other way: we have the two numbers, and we want the percentage.

The procedure

Let's say we have to find how many per cent 20 is of 60.

1. Write the share as a fraction. We must find $\tfrac{20}{60}$, so we set up exactly this fraction, because it tells us how much 20 is of 60. The number we are asking about, 20, goes on top. The whole, 60, goes on the bottom.
2. Multiply by 100 to get it in per cent. The fraction is the share as a plain number. Per cent means hundredths, so we multiply by 100:

$$\frac{20}{60} \times 100 \approx 33.3$$

So 20 is about 33.3% of 60. That makes sense, because 20 is a third of 60, and a third is 33.3%.

More generally we can say: how big a percentage is $x$ of $y$?

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

Worked examples

Example 1: 20 as a percentage of 60. Set up $\tfrac{20}{60}$ and multiply by 100. As we saw, $\tfrac{20}{60} \times 100 \approx 33.3$, so 20 is about 33.3% of 60.

Example 2: 200 as a percentage of 500. Something has gone up from 500 to 700, and we want the rise in per cent. The rise is $700 - 500 = 200$, so we ask how many per cent 200 is of 500:

$$\frac{200}{500} \times 100 = 40$$

So the rise is 40%. This is exactly how a percentage increase is worked out, and it gets its own section in the guide about percentage increase and decrease.

Example 3: 45 as a percentage of 60. Same method. 45 out of 60 marks is

$$\frac{45}{60} \times 100 = 75$$

so 45 out of 60 is 75%. If you spot that $\tfrac{45}{60}$ simplifies to $\tfrac{3}{4}$ first, you can see the 75% straight away.

Common mistakes

• Putting the numbers the wrong way up. The number you are asking about goes on top, and the whole goes on the bottom. "20 as a percentage of 60" is $\tfrac{20}{60}$. If you write $\tfrac{60}{20}$ instead, you get 300%, which is asking a different question.
• Forgetting the 100. The fraction $\tfrac{20}{60}$ on its own is a share, roughly 0.333. It only becomes a percentage once you multiply by 100. The fraction says how much, the 100 puts it in hundredths.
• Treating it as a new kind of problem. It is still just a fraction. Set up the fraction that says "this much of that", and the rest is one multiplication.

Related

• What is a percentage?
• How to find a percentage of an amount
• Percentage increase and decrease with a multiplier