What a Ratio Is

A ratio is a way of comparing two amounts by counting parts. In this picture there are three red counters and two blue ones, so the ratio of red to blue is three to two, and you write it with a colon: 3 : 2. Notice carefully what the ratio does and does not tell you. It tells you that for every three reds there are two blues. It does not tell you there are only five counters in the world — a much bigger pile with thirty reds and twenty blues has exactly the same 3 : 2 relationship, and so does three hundred and two hundred. That is the whole point of a ratio. It fixes how two amounts compare, not how big they are.

The order you write a ratio in is part of the answer. Red to blue is three to two, but blue to red is two to three, and those are genuinely different statements — look at the two bars and you can see the colours swap over. So read the question to find which quantity is named first, and write that number first. It helps to underline the two words. You say three to two out loud, and the colon is doing exactly the job of the word to. One more habit worth building right now. A ratio has no units written inside it. You sort out the units before you write the numbers, and then you record just the two numbers.

So far you have compared one part with another part. You can also compare a part with the whole. With three red and two blue there are three plus two, which is five parts altogether, so the reds make up three of those five parts. As a fraction, the reds are three fifths of the pile and the blues are two fifths. That switch from parts to a fraction is what lets you handle totals. Suppose a pile really does hold twenty counters in the ratio three to two. There are five parts, so one part is twenty divided by five, which is four counters. That gives twelve red and eight blue, and twelve plus eight is twenty, exactly as it should be.

Equivalent ratios are different pairs of numbers describing exactly the same comparison, and you build them the same way you build equivalent fractions: multiply both parts by the same number. Start with two to three. Double both parts and you get four to six. Triple them and you get six to nine. Multiply by four and you get eight to twelve. Look at the two bars — the colours change over at the identical place, so the mixture really is the same. What you must never do is add the same number to both parts. Adding two to each part of two to three gives four to five, and four to five is a different comparison altogether.

A ratio compares numbers, so before you write one down both amounts have to be measured in the same unit. Comparing fifty centimetres with two metres as fifty to two is nonsense, because you would be comparing centimetres against metres. Convert first. Two metres is two hundred centimetres, so the ratio is fifty to two hundred. It works the same way everywhere. Five hundred grams against three kilograms becomes five hundred to three thousand, forty five minutes against two hours becomes forty five to one hundred and twenty, and two hundred and fifty millilitres against two litres becomes two hundred and fifty to two thousand. Convert to the smaller unit and you will never have to deal with a decimal.