Solving Ratio Problems

Here is the most common ratio question of all. A paint mix is blue to white in the ratio two to five, and you are using eight tins of blue. How much white? Start with the side where you know both numbers, which is the blue: two has become eight, so everything has been multiplied by four. Now do exactly the same to the other side. Five times four is twenty, so you need twenty tins of white. Check it by simplifying backwards: eight to twenty divides by four and gives two to five, which is the recipe you started with. The trap is looking at the difference instead. Two plus six is eight, but five plus six gives eleven, and that is wrong.

There is one idea underneath every ratio problem: work out what a single part is worth. Sand to cement is three to four and you have twenty one kilograms of sand. The sand is three parts, so one part is twenty one divided by three, which is seven kilograms. Now everything unlocks. Cement is four parts, so cement is four times seven, which is twenty eight kilograms. Two parts would be fourteen kilograms and the whole seven parts would be forty nine. Check the answer by simplifying twenty one to twenty eight, which divides by seven and gives three to four, exactly the ratio you were given. Find one part, and the rest is multiplication.

Now the big one: sharing a whole amount in a given ratio. Share one hundred and twenty dollars between Ava and Ben in the ratio two to three. Three moves. First, total the parts: two plus three is five. Second, find one part: one hundred and twenty divided by five is twenty four dollars. Third, rebuild each share: Ava gets two lots of twenty four, which is forty eight dollars, and Ben gets three lots, which is seventy two dollars. Then the check that must never be skipped — forty eight plus seventy two is one hundred and twenty, so the shares put the whole back together. The classic disaster is dividing by one side of the ratio instead of by the total parts.

Sharing questions are easy to get slightly wrong, so build two checks into your routine. Share sixty lollies in the ratio three to seven. Ten parts, so one part is six, and the shares are eighteen and forty two. Check one: the shares must add back to the total, and eighteen plus forty two is sixty. Check two: the shares must simplify back to the original ratio, and eighteen to forty two divides by six to give three to seven. Now watch Ana share eighty dollars in the ratio three to five and write thirty and forty. Check one fails at once, because thirty plus forty is only seventy. The correct shares are thirty and fifty.

The last skill is reading the question properly, because the same ratio can be handed to you in three different ways. Given one side, like four cups of cordial in a one to six mix, one part is four cups and the water is six times four, which is twenty four. Given the total, like eighty dollars shared in three to five, add the parts to get eight, divide to find one part is ten dollars, and rebuild thirty and fifty. Given one share, like the smaller share of a three to five split being twenty seven dollars, three parts equal twenty seven, so one part is nine, the other share is forty five and the total is seventy two. Always find one part first.