Adding and Subtracting Fractions

Start with the easy case, where both fractions are already cut into the same size pieces. Three eighths plus two eighths. Look at the bars: three shaded pieces pushed together with two shaded pieces gives five shaded pieces, and every piece is still an eighth. So the answer is five eighths. The rule is simply to add the numerators and keep the denominator. Subtracting is the same idea backwards, so seven ninths take away two ninths leaves five ninths. The one thing you must never do is add the denominators. The bottom number is not a count, it is a name. It tells you how big each piece is, and pushing pieces together does not change their size.

Getting the right pieces is only half the job, because a fraction should be written in its simplest form. Take five sixths minus one sixth. Remove one shaded piece and four sixths are left. Now look at the bar again and group the sixths in twos. There are three of these bigger pieces, and two of them are shaded, so four sixths is the same amount as two thirds. Nothing about the shading changed, only where you drew the cutting lines. To simplify without a picture, divide the numerator and the denominator by the same number until nothing divides them both. Four and six both divide by two, giving two thirds.

Now the harder case, where the pieces are different sizes. One half plus one quarter. You cannot count them together, because a half and a quarter are not the same thing. The fix is to recut one bar so both are chopped the same way. Four is already a multiple of two, so cut each half down the middle. Every half becomes two quarters, which means one half is the same as two quarters. Now both fractions are in quarters, and you are back to the easy case: two quarters plus one quarter is three quarters. Whatever you multiply the denominator by, multiply the numerator by the same amount.

Sometimes neither denominator is a multiple of the other, so both fractions have to change. Two thirds plus one quarter. Count in threes and count in fours, and the first number that turns up in both lists is twelve, so twelve is your common denominator. Cut each square into twelve cells. Two thirds is two rows of the three, which is eight cells, so two thirds is eight twelfths. One quarter is one column of the four, which is three cells, so one quarter is three twelfths. Now add: eight twelfths plus three twelfths is eleven twelfths. Whatever you multiply the bottom by, multiply the top by the same number.

Subtraction follows the very same plan. Three quarters minus two fifths. Twenty is the first number in both the four times table and the five times table, so recut everything into twentieths. Three quarters is fifteen twentieths, two fifths is eight twentieths, and fifteen take away eight leaves seven twentieths. Mixed numbers are just as friendly if you handle the wholes and the parts separately. Two and one quarter plus one and one half: the wholes give three, the parts give one quarter plus two quarters, which is three quarters, so the answer is three and three quarters. Whatever the question, the plan is the same four steps every time.