Multiplying & Dividing Fractions

Multiplying a fraction by a fraction means taking a part of a part. Half of a half is a quarter: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$. That's why "of" in a problem so often means multiply.

Dividing asks how many of one fraction fit into another — how many $\frac{1}{4}$-cups fit in $\frac{1}{2}$ a cup? The trick is that dividing by a fraction is the same as multiplying by its reciprocal (its flip). Both operations come down to multiplying, so neither needs a common denominator.

Multiply straight across — tops together, bottoms together — then simplify:

$\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}$

Divide by flipping the second fraction, then multiply. To divide by $\frac{1}{2}$, multiply by $\frac{2}{1}$:

$\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2}$

That answer being more than 1 makes sense: more than one half-sized piece fits into three-quarters. "Flip and multiply" is the whole move for division.

Read division as "how many fit?" and the flip-and-multiply will confirm your intuition. Predict before you check.