Writing and Simplifying Ratios

A ratio compares two or more amounts. If a bag has 6 red counters and 9 blue counters, the ratio of red counters to blue counters is 6:9. The colon is read as "to".

The order matters. Red to blue is 6:9, but blue to red is 9:6. Write the numbers in the same order as the words in the question.

In the exam you are usually asked to "Write down the ratio of" one amount to another and "Give your ratio in its simplest form", for 2 marks. You may also be asked to write a ratio in the form 1:n or n:1, or to compare amounts given in different units.

Equivalent ratios describe the same comparison. You make one by multiplying or dividing every number in the ratio by the same number. So 6:9, 2:3 and 20:30 are all equivalent.

A ratio is in its simplest form when no whole number bigger than 1 divides into every number in it. To get there in one step, divide every number by their highest common factor (HCF).

You can also divide in stages, but do not stop too soon. 30:42 halves to 15:21, but both numbers still divide by 3, so the simplest form is 5:7

Equivalent ratios also give a missing amount. If red : blue =3:5 and there are 12 red counters, the 3 has been multiplied by 4. Multiply the 5 by 4 too, so there are 20 blue counters.

Before you simplify, both amounts must be in the same units. Change the larger unit into the smaller one, so you get whole numbers. The units then cancel, and the answer has no units in it.

• 1\text{ m}=100\text{ cm}
• 1\text{ cm}=10\text{ mm}
• 1\text{ kg}=1000\text{ g}
• 1\text{ litre}=1000\text{ ml}
• 1\text{ hour}=60\text{ minutes}

If a ratio has decimals, multiply every number by 10 (or 100) to make whole numbers, then simplify. For example, 0.6:1.5=6:15=2:5

A ratio can compare three amounts. Simplify it in the same way: divide all three numbers by the same number.

It is only in its simplest form when no whole number bigger than 1 divides into all three. For example, 6:10:15 cannot be simplified. 2 divides into 6 and 10, but not into 15.

Some questions ask for a ratio in the form 1:n. The first number must be 1, so divide both numbers by the first number. The other number can be a decimal.

For the form n:1, the second number must be 1, so divide both numbers by the second number.

For example, 4:10 in the form 1:n is 1:2.5, because 10\div4=2.5

If red : blue =2:3, think of 5 equal parts: 2 parts are red and 3 parts are blue.

So the fraction of the counters that are red is \frac{2}{5}, not \frac{2}{3}. The 3 is only the blue counters. Add the parts to find the whole.

This works the other way too. If \frac{3}{10} of the drinks sold are tea and the rest are coffee, then 3 parts out of 10 are tea and 7 parts are coffee. So tea : coffee =3:7

A percentage works the same way, because it is out of 100. If 30\% of the drinks are tea, then 70\% are coffee, and 30:70=3:7

• Not simplifying fully. Keep dividing until no whole number bigger than 1 divides into every number.
• Writing the ratio the wrong way round. Put the numbers in the same order as the words in the question.
• Writing a fraction instead of a ratio. Use a colon, as in 3:5, when the question asks for a ratio.
• Dividing the numbers by different amounts. Divide every number in the ratio by the same number.
• Using the total instead of the rest. If 30\% are tea, coffee is 70\%, so use 30:70, not 30:100.
• Not giving the form asked for. In the form 1:n the first number must be 1, even if n is a decimal.