Percentage Change, Forwards and in Reverse

You can increase an amount by a percentage in one step. After a fifteen percent increase you have the original one hundred percent plus fifteen percent more, which is one hundred and fifteen percent, or one point one five as a decimal. So eighty dollars increased by fifteen percent is eighty times one point one five, which is ninety-two dollars. Check the long way: fifteen percent of eighty is twelve, and eighty plus twelve is ninety-two. The number you multiply by is called the multiplier.

For a decrease, multiply by what is left. Taking twelve percent off leaves eighty-eight percent, so the multiplier is nought point eight eight. Two hundred and fifty dollars decreased by twelve percent is two hundred and fifty times nought point eight eight, which is two hundred and twenty dollars. Check: twelve percent of two hundred and fifty is thirty, and two hundred and fifty minus thirty is two hundred and twenty. A multiplier bigger than one means an increase; smaller than one means a decrease.

To find the percentage change from one amount to another, first find the change, then divide by the original amount, the one you started from. From forty to fifty-two, the change is twelve. Twelve divided by forty is nought point three, which is a thirty percent increase. Check: forty times one point three is fifty-two. If you divide by fifty-two instead, you get about twenty-three percent, which is wrong, because the change is measured against where you started.

Here is the classic trap. A price goes up by twenty-five percent and is now one hundred and fifty dollars. What was it before? The original was multiplied by one point two five to get one hundred and fifty, so go backwards: divide one hundred and fifty by one point two five. That gives one hundred and twenty dollars. Check: one hundred and twenty times one point two five is one hundred and fifty. The tempting wrong answer is to take twenty-five percent off one hundred and fifty, giving one hundred and twelve dollars fifty. That is wrong, because the twenty-five percent was of the original price, not of the new one.

Reverse problems after a decrease work the same way. A car has lost thirty percent of its value and is now worth fourteen thousand dollars. Losing thirty percent leaves seventy percent, so the original times nought point seven is fourteen thousand. Divide: fourteen thousand divided by nought point seven is twenty thousand dollars. Check: thirty percent of twenty thousand is six thousand, and twenty thousand minus six thousand is fourteen thousand. Adding thirty percent to fourteen thousand gives eighteen thousand two hundred, which is wrong. Quiz and exercises now.