Ratio and Unit Rates

A ratio compares two amounts and it works in groups. Two red for every three blue means the counters come five at a time: three groups give six red and nine blue, fifteen altogether. The total is always a multiple of five, so you could never have twelve counters in this ratio. And two to three is not the fraction two thirds — red is two of every five, which is two fifths of the total.

To scale a ratio, work out how many groups there are first. Twelve red in the ratio two to three means twelve divided by two, six groups, and six groups need six times three, eighteen blue. Going the other way, fifteen blue is five groups, so there are ten red. The total also follows: six groups is thirty, which matches twelve plus eighteen. Divide to find the groups, then multiply to find the part you want.

To share a total in a ratio, add the parts, divide the total by that number, then multiply each share. Forty dollars in the ratio three to five is eight parts, so each part is five dollars, and the shares are fifteen and twenty-five. Check by adding them back to forty and by making sure the bigger part of the ratio got the bigger share. Dividing the total by three or by five instead of by eight is the usual mistake.

For a rate, go through one. Fifteen dollars buys three toy cars, so one car is five dollars and forty-five dollars buys nine. You can also spot that forty-five is three times fifteen and so buys three times as many, nine again. The same method handles speed and recipes — two hundred and forty kilometres on twenty litres is twelve kilometres per litre, and three hundred grams of flour for four people is seventy-five grams each. Then check the direction: more money must buy more.

Three shapes of question, and all three go through a single unit. A ratio question asks how many groups there are; a sharing question adds the parts before dividing; a rate question finds what one is worth. Work out that unit, multiply back up, and then check the direction and the total. Quiz and exercises now.