Multiplying Fractions

How to multiply two fractions

The rule is short: top × top, bottom × bottom.

So $\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}$. Multiply across, then reduce.

The picture is just as short. A b × d rectangle (here 3 × 4) holds 12 unit cells. Shade a rows for the first fraction, then c columns for the second. The cells that are shaded both ways are the product. In $\frac{2}{3} \times \frac{3}{4}$, that's 6 cells out of 12 — exactly $\frac{6}{12} = \frac{1}{2}$.

Drag the two handles. Try $\frac{1}{2} \times \frac{1}{2}$ — the shaded square is exactly a quarter of the whole. Try $\frac{2}{3} \times \frac{3}{4}$ — the shaded rectangle is half. The shaded area as a fraction of the whole IS the product.

Mixed numbers and improper fractions

So far every shaded rectangle has fit inside a single unit square. That works as long as both factors are between 0 and 1. What about

$1\tfrac{1}{2} \times \tfrac{1}{2}$? The first factor is bigger than 1, so the rectangle won't fit in a single unit square anymore — it spills into the next one.

The arithmetic is the same. First convert the mixed number to an improper fraction: multiply the whole part by the denominator, add the numerator, keep the same denominator.

Read it as "two halves make one whole, plus the extra half, gives three halves." Then multiply across the same way as before:

The picture is the same, just on a bigger canvas. Drag past 1 on either axis and watch the rectangle cross into the next unit square.

Try $1\tfrac{1}{2} \times 1\tfrac{1}{2}$ — both factors are bigger than 1, so the rectangle covers more than one unit square. The product, $\tfrac{9}{4} = 2\tfrac{1}{4}$, is now bigger than each factor.

That's the rule, in one line: multiplying by a number bigger than 1 grows; multiplying by a number smaller than 1 shrinks. Mixed numbers are just numbers bigger than 1 written in two pieces.

Where it shows up in real life

Recipes scale by multiplication. A hot-chocolate recipe calls for $\frac{3}{4}$ cup of cocoa powder. To make $\frac{2}{3}$ of the recipe, the cocoa powder is $\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$ cup.

The scaled cup is smaller than the original — exactly because we multiplied by $\frac{2}{3}$, a number less than 1. Same arithmetic, real ingredients.

Worksheet

These aren't graded. The goal is fluency with the rule and a feel for when the product is smaller than the factors.

Going further

The same area-model picture comes back in algebra. Multiplying two binomials (x + 1)(x + 2) is geometrically the same idea — the area of a rectangle whose sides each split into two terms.

Decimal multiplication is the same idea using base-10 fractions: 0.4 × 0.3 = 0.12 is just $\tfrac{4}{10} \times \tfrac{3}{10} = \tfrac{12}{100}$.