Chapter 2 · Fraction review

See a Fraction of a Fraction with Area Models

See what a fraction of a fraction means, then use an area model to find the product by counting the overlap as a part of the whole.

10 min read

With whole numbers, 3×53 \times 5 means 3 groups of 5. The first number tells how many groups to take. When the first number is a fraction, it tells what fraction of a group to take.

So 12×34\frac{1}{2} \times \frac{3}{4} means 12\frac{1}{2} of 34\frac{3}{4}. Start with the amount 34\frac{3}{4}. Cut it into 2 equal parts. Take 1 of those parts.

The numbers you multiply are called factors, and the result is called the product. When both factors are fractions, the product is a fraction of a fraction.

The whole is the amount that stands for 11. Here, 34\frac{3}{4} means 3 of 4 equal parts of the whole, and the 12\frac{1}{2} tells you to take half of that amount. Your goal is to name the result as a part of the whole. First, try it on a bar.

Cut the bar, which is 34\frac{3}{4} of a whole, into 2 equal pieces and take 1 piece to show 12\frac{1}{2} of 34\frac{3}{4}. Then try a cut that makes unequal pieces, 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.

Given34

3/4 of 1 whole

Keyboard: on the cutting groove, Left and Right move along the bar, Enter or Space makes or takes away a cut, and Shift with Left or Right slides a cut. Each piece is a button: Enter or Space takes it or puts it back.

On the bar now

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

Nothing is taken yet.

Notice: Whatever you take comes from the bar, which is 34\frac{3}{4} of a whole, so 1 of 2 equal pieces is 12\frac{1}{2} of 34\frac{3}{4} and not 12\frac{1}{2} of a whole. If the 2 pieces are unequal, neither piece is 12\frac{1}{2} of the bar.

On the bar, 1 of 2 equal pieces is half of the bar. But the bar is 34\frac{3}{4} of a whole, so half of the bar is not half of a whole. What part of the whole is it?

An area model shows the answer. An area model is a rectangle that stands for 1 whole. Equal columns cut it one way, and equal rows cut it the other way. The boxes they make are called cells, and every cell is the same size.

A band is a group of whole columns or whole rows that is marked on the grid. The part of the grid that lies in both bands is called the overlap.

Look at the grid for 12\frac{1}{2} of 34\frac{3}{4}. One band covers 3 of the 4 columns. It is 34\frac{3}{4} of the whole, the same amount as the bar. The other band covers 1 of the 2 rows, so it is 12\frac{1}{2} of the whole. The overlap is the part of the first band that the second band also covers.

One whole in 2 rows and 4 columns: one band covers 3 columns (34\frac{3}{4}) and the other covers 1 row (12\frac{1}{2}).

Worked example

Half of three fourths

Use an area model to find 12×34\frac{1}{2} \times \frac{3}{4}.

  1. Read the product as one half of three fourths. Three fourths of the whole is the amount you start with, and you take half of that amount.

    12×34 means 12 of 34\frac{1}{2} \times \frac{3}{4} \text{ means } \frac{1}{2} \text{ of } \frac{3}{4}

  2. Show three fourths of the whole. Fourths are 4 equal parts, so cut the rectangle into 4 equal columns and shade 3 of them. This band is the amount you start with.

  3. Take half of that band. Halves are 2 equal parts, so cut the rectangle into 2 equal rows and shade 1 row. The row line cuts every column into 2 equal cells, so the shaded row holds half of each column. That means it holds half of the band of 3 columns. The overlap is that half, so the overlap is one half of three fourths.

  4. Count the cells in the whole. There are 2 rows with 4 cells in each row. The rows are equal and the columns are equal, so every cell is the same size and each cell is one eighth of the whole.

    2×4=8 cells2 \times 4 = 8 \text{ cells}

  5. Count the cells in the overlap. It is 1 row with 3 cells in it.

    1×3=3 cells1 \times 3 = 3 \text{ cells}

  6. Name the overlap as a part of the whole. It has 3 of the 8 equal cells, so it is three eighths of the whole.

    12×34=38\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}

Answer

The product 12×34\frac{1}{2} \times \frac{3}{4} is 38\frac{3}{8}. Half of three fourths of the whole is three eighths of the whole, so 1 of 2 equal pieces of the bar is 38\frac{3}{8} of a whole.

The overlap is 1 row of 3 cells, and the whole has 8 equal cells, so 12\frac{1}{2} of 34\frac{3}{4} is 38\frac{3}{8} of the whole.

Key concept

The overlap is the product

To find a fraction of a fraction, use one rectangle as the whole. Show one fraction as a band of columns and the other as a band of rows. The overlap of the two bands shows the product.

Name the overlap as a part of the whole: the number of cells in the overlap out of the number of cells in the whole. This works when every cell is the same size and both fractions are between 00 and 11, so each band fits inside one whole.

product=cells in the overlapcells in the whole\text{product} = \frac{\text{cells in the overlap}}{\text{cells in the whole}}
One whole in 3 rows and 5 columns: one band covers 4 columns (45\frac{4}{5}), the other covers 2 rows (23\frac{2}{3}), and they overlap in 2 rows of 4 cells.

Worked example

Two thirds of four fifths

Use an area model to find 23×45\frac{2}{3} \times \frac{4}{5}.

  1. Read the product as two thirds of four fifths. Show four fifths first. Fifths are 5 equal parts, so cut the whole into 5 equal columns and shade 4 of them.

    23×45 means 23 of 45\frac{2}{3} \times \frac{4}{5} \text{ means } \frac{2}{3} \text{ of } \frac{4}{5}

  2. Take two thirds of that band. Thirds are 3 equal parts, so cut the whole into 3 equal rows and shade 2 of them. The row lines cut the band of 4 columns into 3 equal parts, and the overlap is 2 of those parts.

  3. Count the equal cells in the whole: 3 rows with 5 cells in each row.

    3×5=15 cells3 \times 5 = 15 \text{ cells}

  4. Count the cells in the overlap: 2 rows with 4 cells in each row.

    2×4=8 cells2 \times 4 = 8 \text{ cells}

  5. Name the overlap as a part of the whole: 8 of the 15 equal cells.

    23×45=815\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}

Answer

The product 23×45\frac{2}{3} \times \frac{4}{5} is 815\frac{8}{15}. Two thirds of four fifths of the whole is eight fifteenths of the whole.

Here is the idea to keep. When both fractions are between 00 and 11, a fraction of a fraction is a part of a part of one whole. On an area model, one fraction is a band of columns and the other is a band of rows. Their overlap is the product. Name it as a part of the whole: the cells in the overlap out of all the equal cells in the whole.

Next, you will build on this picture to multiply fractions by whole numbers and to multiply two fractions.

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Multiply Fractions by Whole Numbers and by Fractions

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