A Fraction of a Fraction | Grade 5 Math | LevelKite

You have already multiplied a fraction by a whole number. For example, $\frac{2}{3} \times 6$ means $\frac{2}{3}$ of 6: cut 6 into 3 equal parts and take 2 of them. That makes 4.

A fraction times a fraction works the same way. The only change is that the amount you start with is a fraction too. Read $\frac{2}{3} \times \frac{4}{5}$ as $\frac{2}{3}$ of $\frac{4}{5}$ .

The second fraction, $\frac{4}{5}$ , is the amount you start with.
The denominator (bottom number) of the first fraction is 3. It tells you to cut $\frac{4}{5}$ into 3 equal pieces.
The numerator (top number) of the first fraction is 2. It tells you to take 2 of those pieces.

What you get is a part of a part: a piece of an amount that is already a piece of one whole. Try it with the bar below.

The bar is $\frac{4}{5}$ of one whole. Cut it into 3 equal pieces and take 2 of them. Then try unequal cuts, or take a different number of pieces, and see what changes.

Press the groove under the bar to cut it, and slide a cut to move it. Tap a piece to take it.

Given45

4/5 of 1 whole

On the bar now

The given amount is one piece. It is not cut yet.

Nothing is taken yet.

Notice: With 3 equal pieces, each piece is one third of the bar, not one third of the whole. Two pieces make $\frac{2}{3}$ of $\frac{4}{5}$ , and all 3 make the whole bar. With unequal pieces, 2 of the 3 pieces may not be $\frac{2}{3}$ of the bar.

If you cut the bar into 3 equal pieces and took 2, you found $\frac{2}{3}$ of $\frac{4}{5}$ . That amount is the product of $\frac{2}{3}$ and $\frac{4}{5}$ . A product is the answer to a multiplication.

But how much of one whole is it? A third of the bar is smaller than a third of the whole, because the bar is less than one whole. So the 2 pieces are not $\frac{2}{3}$ of the whole. To name the product, you need to see the bar inside the whole, cut into equal parts.

A grid model lets you see this. It is a drawing of one whole, cut into equal columns (strips that run up and down) and equal rows (strips that run across).

The shaded columns show the amount you start with, like the bar did.
The rows cut those columns into equal pieces. The shaded rows show the pieces you take.
The overlap, where the shaded columns and the shaded rows cross, is the part of a part.

First, see how it works with a simpler product.

One whole in 2 rows and 3 columns: the shaded column is

\frac{1}{3}

, and the overlap is half of that column.

Worked example

Half of one third

Use the grid above to find $\frac{1}{2} \times \frac{1}{3}$ as a fraction of the whole.

Read it as one half of one third. The grid has 3 equal columns, so each column is one third of the whole. The 1 shaded column is the amount you start with.
$\frac{1}{2} \times \frac{1}{3} \text{ means } \frac{1}{2} \text{ of } \frac{1}{3}$
The denominator 2 says to cut that column into 2 equal pieces. The 2 equal rows do this. They run across the whole grid, not just the column, so every small part is the same size. The numerator 1 says to take 1 piece: the overlap with the 1 shaded row. So the overlap is half of the column.
Now name the overlap as a fraction of the whole. The 2 rows and 3 columns cut the whole into 6 equal parts, and the overlap is 1 of them.
$2 \times 3 = 6, \text{ so } \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$

Answer

Half of one third is $\frac{1}{6}$ of the whole, so $\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$ .

One whole in 3 rows and 5 columns: the 4 shaded columns are

\frac{4}{5}

, and the overlap is 2 of their 3 equal pieces.

Worked example

Two thirds of four fifths

Use the grid above to find $\frac{2}{3} \times \frac{4}{5}$ as a fraction of the whole. This is the same part of a part as on the bar.

Read it as two thirds of four fifths. Four of the 5 equal columns are shaded, so they show four fifths of the whole.
$\frac{2}{3} \text{ of } \frac{4}{5}$
The 3 equal rows cut the shaded columns into 3 equal pieces. The overlap with the 2 shaded rows is 2 of those pieces, so it is two thirds of the four fifths.
Count the equal parts in the whole: 3 rows of 5 parts. The rows are equal and the columns are equal, so all 15 parts are the same size. Each part is one fifteenth of the whole.
$3 \times 5 = 15$
Count the parts in the overlap: 2 rows of 4 parts. The overlap is 8 of the 15 equal parts.
$2 \times 4 = 8, \text{ so } \frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$

Answer

Two thirds of four fifths is $\frac{8}{15}$ of the whole, so $\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$ .

Key concept

A part of a part, named in the whole

A fraction times a fraction means the first fraction of the second. To name the product as a fraction of one whole, think of the grid model:

The denominators give the number of equal rows and the number of equal columns. Multiply them to count the equal parts in the whole. That is the denominator of the product.
The numerators give the number of shaded rows and the number of shaded columns. Multiply them to count the parts in the overlap. That is the numerator of the product.

So to multiply two fractions, multiply the numerators and multiply the denominators. The grids here show fractions less than 1, but this rule works for any two fractions.

\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}

Worked example

Three fourths of five eighths

Find $\frac{3}{4} \times \frac{5}{8}$ .

Read it as three fourths of five eighths: cut the five eighths into 4 equal pieces and take 3 of them.
$\frac{3}{4} \text{ of } \frac{5}{8}$
Multiply the denominators to count the equal parts in the whole. To make the 4 equal pieces, you cut every eighth of the whole into 4 equal parts.
$4 \times 8 = 32$
Multiply the numerators to count the parts you take. Each of the 4 pieces has 1 part from each of the 5 eighths, and you take 3 pieces.
$3 \times 5 = 15$
Write the parts you take over the equal parts in the whole.
$\frac{3}{4} \times \frac{5}{8} = \frac{3 \times 5}{4 \times 8} = \frac{15}{32}$

Answer

Three fourths of five eighths is $\frac{15}{32}$ of the whole, so $\frac{3}{4} \times \frac{5}{8} = \frac{15}{32}$ .

Takeaway: A fraction times a fraction is a part of a part. Cut the amount you start with into equal pieces, take some of them, and name the result as a fraction of the same whole. In that whole, the denominators multiply to count all the equal parts, and the numerators multiply to count the parts you take.