Simplifying Ratios

In the last lesson you built equivalent ratios by multiplying both parts. Simplifying is the same idea running backwards: you divide both parts by the same number. Take twelve to eighteen. Divide both by two and you get six to nine. Divide both of those by three and you get two to three. Look down the four bars. Every single one changes colour in exactly the same place, so all four describe the identical mixture — the only thing that changed is how many pieces you chopped it into. That is why simplifying is safe. You have not altered the comparison at all, you have just found the tidiest pair of numbers that says the same thing.

You can simplify in little steps, but there is a faster way. Find the highest common factor of the two numbers and divide by that once. For twenty four to thirty six, the factors of twenty four are one, two, three, four, six, eight, twelve and twenty four, and the factors of thirty six are one, two, three, four, six, nine, twelve, eighteen and thirty six. The highest they share is twelve. Twenty four divided by twelve is two, thirty six divided by twelve is three, so the answer is two to three. Notice the order never changes: forty five to twenty seven simplifies to five to three, not three to five.

How do you know when to stop? A ratio is in simplest form when the only number that divides into both parts is one. Four to nine is finished, because nothing except one goes into both. So are five to seven, three to eight and seven to ten. But six to nine is not finished, because three divides both, giving two to three. Eight to twelve is not finished either — halve it to four to six, and if both numbers are still even you halve again to get two to three. That near miss catches people out constantly. Work through two, then three, then five, then seven, and stop only when nothing divides both parts.

Many ratio questions arrive with units attached, and they need two steps in a fixed order. First convert both amounts to the same unit, then simplify. Simplify five hundred grams to two kilograms like this. Two kilograms is two thousand grams, so you have five hundred to two thousand, and the highest common factor is five hundred, giving one to four. Try forty centimetres to one metre: that becomes forty to one hundred, divide both by twenty, and you get two to five. Twenty minutes to one hour becomes twenty to sixty, which is one to three. If you simplify before converting you get garbage, because the two numbers were never measuring the same thing.

Two things to finish with. First, checking. If you divided twelve to eighteen by six to get two to three, then multiplying two to three back by six should rebuild the original, and it does: two sixes are twelve, three sixes are eighteen. That takes five seconds and catches almost every slip. Second, ratios can compare three quantities, and they simplify in exactly the same way. Six to nine to twelve all divide by three, giving two to three to four. Ten to twenty to thirty all divide by ten, giving one to two to three. The only extra care needed is that your dividing number must go into every part — two is useless on six to nine to twelve, because two does not divide nine.