Adding and subtracting with unlike denominators

1/2 + 1/3 = 5/6. The answer only becomes visible after both halves and thirds are cut into sixths.

Different pieces cannot be counted together

In the last unit the pieces were the same size, so adding meant counting. Now try 1/2 + 1/3. One piece is a half and the other is a third. You cannot say you have two of anything, because the two pieces are different sizes. Before the pieces can be counted together they have to be cut into a size that fits both.

Sixths work. A half is 3 sixths and a third is 2 sixths, so 1/2 + 1/3 = 3/6 + 2/6 = 5/6. The number 6 is a common denominator: a number that both denominators divide into. Once both fractions are written in sixths, the problem is an ordinary like-denominator sum.

Finding a common denominator

Any common multiple of the two denominators can be the common denominator. The product of the denominators always works: for 3/4 + 5/6 you could use 4 × 6 = 24. The least common denominator (LCD), the smallest common multiple, keeps the numbers smaller. To find it, list multiples of the larger denominator until one is divisible by the smaller. Multiples of 6 are 6, 12, 18 and so on; 12 is the first that 4 divides, so the LCD of 4 and 6 is 12.

The four steps

First, find a common denominator. Second, rewrite each fraction as an equivalent fraction with that denominator, multiplying top and bottom by the same number. Third, add or subtract the numerators. Fourth, simplify the result. For 3/4 + 5/6 with LCD 12: 3/4 = 9/12 and 5/6 = 10/12, so the sum is 19/12. Nothing divides both 19 and 12, so 19/12 is final, a little more than 1 and a half.

Subtraction follows the same steps. 7/8 − 1/6: multiples of 8 are 8, 16, 24, and 24 is the first one 6 divides. 7/8 = 21/24 and 1/6 = 4/24, so the difference is 17/24. With three fractions, find one denominator for all of them. 2/3 + 1/4 + 1/6 uses 12: 8/12 + 3/12 + 2/12 = 13/12.

A shortcut for two fractions

Using the product of the denominators gives a one-line formula: a/b + c/d = (a × d + c × b)/(b × d). For 3/5 − 1/4: (3 × 4 − 1 × 5)/(5 × 4) = (12 − 5)/20 = 7/20. The formula always gives a correct answer, but when the denominators share a factor the result needs simplifying. 1/4 + 1/6 by the formula is 10/24, which simplifies to 5/12, the same answer the LCD gives directly.

Check with an estimate

A rough estimate catches most mistakes. 1/2 + 1/3 must be more than 1/2 and less than 1, and 5/6 is. The tempting wrong answer, adding tops and bottoms to get 2/5, is less than 1/2, which is impossible when one of the two parts is already 1/2. That wrong method produces the mediant from unit 2, a number between the two fractions, never their sum.

Example. A recipe uses 1/2 cup of oats and 2/3 cup of flour. How much dry mix is that? The LCD of 2 and 3 is 6: 3/6 + 4/6 = 7/6 cups, a little more than one cup.