Calculating with fractions

Baking calls for 1/2 a cup of sugar plus another 1/4, and after pizza night a few slices of each pizza are left over: as soon as you combine shares, you are calculating with fractions. The good news: with one single trick, the common denominator, it turns into perfectly ordinary arithmetic.

Expanding and simplifying

Expanding means multiplying numerator and denominator by the same number. That turns 1/2 into 2/4 or 3/6. Simplifying is the way back: you divide numerator and denominator by the same number, turning 4/8 into 1/2. In both cases the value of the fraction stays exactly the same, only the way it is written changes. A fraction is fully simplified when numerator and denominator share no divisor except 1.

Same denominator: just the numerators

If two fractions have the same denominator, their pieces are the same size. Then you simply add or subtract the numerators and keep the denominator: 2/7 + 3/7 = 5/7, and 5/8 − 2/8 = 3/8. Picture it with pizza slices: 2 eighths plus 3 eighths make 5 eighths, and the size of each slice never changes.

Different denominators and multiplying

With different denominators you first expand both fractions to a common denominator. You can always find one by multiplying the two denominators: for 1/2 + 1/3 that gives sixths, so 3/6 + 2/6 = 5/6. Often a smaller denominator works too: for 1/5 + 3/10 the 10 is enough, because 10 is already a multiple of 5. After that everything works as usual. And a fraction times a whole number? You only multiply the numerator: 3 × 2/5 = 6/5, because three lots of 2 fifths make 6 fifths. A result like 6/5 being larger than 1 is perfectly fine: five fifths make one whole, and one fifth is left over.