Converting Between Fractions, Decimals and Percentages

Fractions, decimals and percentages are three languages for describing the same thing: part of a whole. Seven of these ten squares are shaded. As a fraction that is seven tenths. As a decimal it is 0.7. As a percentage it is 70 percent. All three describe exactly the same amount — one number wearing three costumes. Per cent is Latin for per hundred, so 70 percent literally means 70 out of 100. You need to convert between them because each form is best at a different job. Fractions stay exact and never round. Decimals are the easiest to compare and the only form your calculator really understands. Percentages are the easiest to picture, which is why shops use them.

To turn a fraction into a decimal, remember that a fraction bar is a division sign in disguise. Three eighths means 3 divided by 8. Set out the long division with three point zero zero zero: 30 divided by 8 is 3 remainder 6, then 60 divided by 8 is 7 remainder 4, then 40 divided by 8 is 5 with nothing left, giving 0.375. There is a shortcut when the denominator fits neatly into 10, 100 or 1000. For seven twentieths, 20 times 5 is 100, so multiply the top by 5 as well to get thirty-five hundredths, which is 0.35 straight off. Once you have the decimal, the percentage is one more step: multiply by 100 and write the percent sign. So three eighths is 0.375, which is 37.5 percent.

Decimals and percentages are the easiest pair to swap between, because it is only ever a factor of 100. To go from a decimal to a percentage, multiply by 100, which moves the decimal point two places right: 0.375 becomes 3.75, then 37.5, so 0.375 is 37.5 percent. To go the other way, divide by 100, which moves the point two places left: 45 percent becomes 4.5, then 0.45. Two places, every single time. The trap is not counting both places on small numbers. 0.08 is 8 percent, not 80 percent, and 7 percent is 0.07, not 0.7. And a decimal above 1 gives a percentage above 100, so 1.25 is 125 percent — perfectly legal when something has grown.

Going from a decimal to a fraction is a place-value job. The number of digits after the point tells you the denominator: one digit means tenths, two means hundredths, three means thousandths. So 0.375 has three digits after the point, making it three hundred and seventy-five thousandths, or 375 over 1000. Then you simplify. The highest common factor of 375 and 1000 is 125, and dividing both gives three eighths. The same routine handles the rest: 0.6 is six tenths, which simplifies to three fifths; 0.45 is 45 hundredths, which becomes nine twentieths; 0.08 is 8 hundredths, which becomes two twenty-fifths. Count those digits carefully, because 0.08 is 8 over 100, not 8 over 10.

Percentages to fractions is the simplest conversion of all, because percent already means over 100. Write 45 percent as 45 over 100, then simplify: the highest common factor of 45 and 100 is 5, giving nine twentieths. That is the whole method. The bigger win in this topic, though, is memorising the common conversions so you never have to work them out. A half is 0.5 and 50 percent. A quarter is 0.25 and 25 percent. Three quarters is 0.75 and 75 percent. A fifth is 0.2 and 20 percent, two fifths is 0.4 and 40 percent, a tenth is 0.1 and 10 percent, and an eighth is 0.125 and 12.5 percent. One third is the awkward one, because 0.333 goes on forever, so 33.3 percent is only an approximation.