Multiplying Fractions

Multiplying fractions is the easy one. Tops times tops, bottoms times bottoms. 2/3 × 4/5 = 8/15. No common denominator needed.

You can cancel before multiplying to keep numbers manageable. In 3/4 × 8/9, the 3 and 9 share a 3, and the 4 and 8 share a 4. Cancel to get 1/1 × 2/3 = 2/3.

Why does multiplying make things smaller? Because "×" here means "of." 1/2 × 1/3 is half of a third, which is a sixth.

The definition is (a/b)·(c/d) = (ac)/(bd), which follows from associativity and commutativity of multiplication once you write each fraction as a·b⁻¹.

Cancelling early is legitimate because the product is one single fraction before it is one single computation: any factor in either numerator may be reduced against any factor in either denominator. Formally you are simplifying (ac)/(bd) before evaluating it, which is why cross-cancelling between different fractions is allowed here but not when adding.

The "gets smaller" intuition only holds for factors between 0 and 1. Multiplying by an improper fraction such as 5/3 increases the value, and multiplying by a negative fraction reverses sign. The precise statement is that multiplying by k scales by k, and scaling by a number less than 1 shrinks.