Multiplying Fractions

Three quarters of a square

Let a square stand for one whole. Cut it into 4 equal columns, and color 3 of them. The colored part is $\frac{3}{4}$3/4 of the square.

The square is cut into 4 equal columns, and 3 of them are colored: $\frac{3}{4}$3/4.

Two thirds of that

Now find $\frac{2}{3}$2/3 of the $\frac{3}{4}$3/4. Cut the same square the other way, into 3 equal rows, and color 2 of the rows. The two sets of cuts together divide the square into 4 × 3 = 12 equal small squares.

The 3 columns are already shaded. Drag to shade 2 of the 3 rows, and watch the small squares that are shaded twice.

Where the two agree

The small squares colored twice are $\frac{2}{3}$2/3 of the $\frac{3}{4}$3/4: they are inside the 3 colored columns, and they are 2 of every 3 small squares there. There are 3 × 2 = 6 of them, out of 4 × 3 = 12 small squares in the whole. So $\frac{2}{3}$2/3 of $\frac{3}{4}$3/4 is $\frac{6}{12}$6/12, which simplifies to $\frac{1}{2}$1/2.

Finding a fraction of a fraction is multiplying, so $\frac{2}{3}$2/3 of $\frac{3}{4}$3/4 is written $\frac{2}{3}\times \frac{3}{4}$2/3 × 3/4. Multiply the numerators to count the small squares colored twice, and multiply the denominators to count the small squares in the whole: $\frac{2}{3}\times \frac{3}{4}=\frac{\left(2\times 3\right)}{\left(3\times 4\right)}=\frac{6}{12}=\frac{1}{2}$2/3 × 3/4 = (2 × 3)/(3 × 4) = 6/12 = 1/2.

The columns and the rows overlap in 6 of the 12 small squares, so $\frac{2}{3}$2/3 of $\frac{3}{4}$3/4 is $\frac{6}{12}$6/12.

Multiply straight across

This works for any two fractions, and no common denominator is needed. For $\frac{2}{3}\times \frac{4}{5}$2/3 × 4/5, cut a square into 5 columns and color 4 of them, then into 3 rows and color 2 of them. The square now holds 5 × 3 = 15 small squares, and 4 × 2 = 8 of them are colored twice. So $\frac{2}{3}\times \frac{4}{5}=\frac{8}{15}$2/3 × 4/5 = 8/15.

The answer is smaller than both fractions, because it is a part of a part: $\frac{8}{15}$8/15 is less than $\frac{2}{3}$2/3, and less than $\frac{4}{5}$4/5.

4 of the 5 columns and 2 of the 3 rows overlap in 8 of the 15 small squares: $\frac{2}{3}\times \frac{4}{5}=\frac{8}{15}$2/3 × 4/5 = 8/15.

Simplify first when you can

When a numerator and a denominator share a factor, you can divide both by it before multiplying. In $\frac{2}{3}\times \frac{3}{4}$2/3 × 3/4 there is a 3 in the numerator and a 3 in the denominator, so divide both by 3: $\frac{2}{3}\times \frac{3}{4}=\frac{2}{4}=\frac{1}{2}$2/3 × 3/4 = 2/4 = 1/2. The numbers stay small, and the answer is already simplified.

The usual mistakes

Adding instead of multiplying. $\frac{2}{3}\times \frac{3}{4}$2/3 × 3/4 is not $\frac{\left(2+3\right)}{\left(3+4\right)}=\frac{5}{7}$(2 + 3)/(3 + 4) = 5/7. A fraction of a fraction multiplies both the numerators and the denominators.

Multiplying the numerators but adding the denominators. $\frac{2}{3}\times \frac{3}{4}$2/3 × 3/4 is not $\frac{6}{7}$6/7. The two sets of cuts make 3 × 4 = 12 small squares, so the denominator is 12.

Making a common denominator first. That is needed to add or subtract fractions, not to multiply them.