Multiply a Mixed Number by a Whole Number | Grade 5 Math | LevelKite

You can already multiply fractions. Now you will multiply a mixed number by a whole number.

A mixed number is a whole number written together with a fraction that is less than 1, like $1\frac{1}{3}$ . It means $1 + \frac{1}{3}$ . The $1$ is its whole number part, and the $\frac{1}{3}$ is its fraction part.

Multiplying by a whole number means making that many equal groups. So $2 \times 1\frac{1}{3}$ means 2 groups of $1\frac{1}{3}$ . You may also see it written as $1\frac{1}{3} \times 2$ . When you multiply, switching the order of the two numbers does not change the answer.

Look at one group first. In the model below, each bar is 1 whole cut into 3 equal parts, so each part is 1 third. One whole bar is shaded, and 1 third of the second bar is shaded. That shows $1\frac{1}{3}$ . Count every shaded part and you get 4 thirds.

One group of

1\frac{1}{3}

: 1 whole bar and 1 third of the second bar are shaded, which is 4 thirds in all.

Now make 2 groups. Each group has 1 whole and 1 third, so there are 2 groups of each part.

2 groups of 1 whole make 2 wholes: $2 \times 1 = 2$ .
2 groups of 1 third make 2 thirds: $2 \times \frac{1}{3} = \frac{2}{3}$ .

The model below puts the two groups together. The first two bars are fully shaded. They are the 2 wholes, one from each group. The third bar has 2 thirds shaded, one from each group. Altogether, 2 wholes and 2 thirds are shaded, so $2 \times 1\frac{1}{3} = 2\frac{2}{3}$ . Count every shaded part and you get 8 thirds, which is 2 times the 4 thirds in one group.

Two groups of

1\frac{1}{3}

: 2 whole bars and 2 thirds of the third bar are shaded, so

2 \times 1\frac{1}{3} = 2\frac{2}{3}

.

Key concept

Every group has both parts

A whole number times a mixed number means that many equal groups of the mixed number. Each group has a whole number part and a fraction part. So multiply the whole number part by the number of groups, multiply the fraction part by the number of groups, and then add the two results. The number sentence below shows this for the two groups in the bar model.

2 \times 1\frac{1}{3} = (2 \times 1) + \left(2 \times \frac{1}{3}\right) = 2 + \frac{2}{3} = 2\frac{2}{3}

It helps to estimate before you multiply. An estimate is a quick answer that tells you about how big the exact answer should be. After you find the exact answer, you can compare it with your estimate to see whether it makes sense.

The numbers you multiply are the factors, and the answer is the product. To estimate a product of mixed numbers, round each factor to the nearest whole number. That means you swap it for the whole number closest to it. Then multiply the rounded factors.

To round a mixed number, look at its fraction part.

Less than $\frac{1}{2}$ : the mixed number is closer to its whole number part. Round down by dropping the fraction part.
Greater than $\frac{1}{2}$ : the mixed number is closer to the next whole number, which is 1 more than its whole number part. Round up to that number.
Exactly $\frac{1}{2}$ : the mixed number is halfway between those two whole numbers. The usual rule is to round up.

For example, $9\frac{5}{8}$ rounds up to 10, because $\frac{5}{8}$ is greater than $\frac{4}{8}$ , and $\frac{4}{8}$ equals $\frac{1}{2}$ . A factor that is already a whole number stays the same.

The table shows four estimates. In the last row, both factors are mixed numbers, so you round both.

Estimates made by rounding each factor

Product to estimate	First factor	Second factor	Estimate
$7 \times 3\frac{1}{8}$	$7$ stays $7$	$3\frac{1}{8}$ rounds down to $3$	$7 \times 3 = 21$
$3 \times 9\frac{5}{8}$	$3$ stays $3$	$9\frac{5}{8}$ rounds up to $10$	$3 \times 10 = 30$
$2 \times 6\frac{1}{2}$	$2$ stays $2$	$6\frac{1}{2}$ is halfway, so it rounds up to $7$	$2 \times 7 = 14$
$2\frac{5}{6} \times 4\frac{1}{5}$	$2\frac{5}{6}$ rounds up to $3$	$4\frac{1}{5}$ rounds down to $4$	$3 \times 4 = 12$

Worked example

Multiply each part, then add

Estimate, then find $4 \times 2\frac{1}{5}$ by multiplying the whole number part and the fraction part separately.

Estimate. The fraction part, 1 fifth, is less than 1 half, because a fifth of a whole is smaller than a half of it. So the mixed number rounds down to 2. The factor 4 is already a whole number, so it stays 4. The estimate is 8.
$4 \times 2\frac{1}{5} \text{ is about } 4 \times 2 = 8$
Now find the exact product. Multiply the whole number part. Each of the 4 groups has 2 wholes, so there are 4 groups of 2 wholes. That makes 8 wholes.
$4 \times 2 = 8$
Multiply the fraction part. Each of the 4 groups also has 1 fifth, so there are 4 groups of 1 fifth. That makes 4 fifths.
$4 \times \frac{1}{5} = \frac{4}{5}$
Add the two results, because the 4 groups hold 8 wholes and 4 fifths altogether.
$8 + \frac{4}{5} = 8\frac{4}{5}$

Answer

The product is $4 \times 2\frac{1}{5} = 8\frac{4}{5}$ . That is a little more than the estimate of 8, which makes sense because rounding $2\frac{1}{5}$ down to 2 left out the fifths.

Worked example

When the fraction parts make more than 1 whole

For a bake sale, you make 5 batches of muffins. Each batch uses $2\frac{3}{4}$ cups of flour. How many cups of flour do you need in all?

The 5 batches are 5 equal groups, and each group is the flour for one batch. So multiply the mixed number by 5. Estimate first. The fraction part, 3 fourths, is greater than 2 fourths, which equals 1 half. So the mixed number rounds up to 3. The factor 5 stays 5, and the estimate is 15 cups.
$5 \times 2\frac{3}{4} \text{ is about } 5 \times 3 = 15$
Multiply the whole number part. Each of the 5 batches has 2 whole cups, so there are 5 groups of 2 cups. That makes 10 cups.
$5 \times 2 = 10$
Multiply the fraction part. Each batch also has 3 fourths of a cup, so there are 5 groups of 3 fourths. That makes 15 fourths.
$5 \times \frac{3}{4} = \frac{15}{4}$
Rename 15 fourths as a mixed number. Every 4 fourths make 1 whole, so 12 of the fourths make 3 wholes, and 3 fourths are left over.
$\frac{15}{4} = \frac{12}{4} + \frac{3}{4} = 3\frac{3}{4}$
Add the two results. The 10 cups come from the whole number parts, and the rest comes from the fraction parts.
$10 + 3\frac{3}{4} = 13\frac{3}{4}$

Answer

You need $13\frac{3}{4}$ cups of flour, because $5 \times 2\frac{3}{4} = 13\frac{3}{4}$ . That is a little less than the estimate of 15 cups, which makes sense because you rounded $2\frac{3}{4}$ up.

There is a second way to multiply. First write the mixed number as an improper fraction. That is a fraction whose top number, the numerator, is equal to or greater than its bottom number, the denominator. One example is $\frac{4}{3}$ .

Think back to the bar models. One group of $1\frac{1}{3}$ was 4 shaded thirds: 3 thirds for the whole bar and 1 more third. So $1\frac{1}{3} = \frac{4}{3}$ . Two groups were 8 shaded thirds, so $2 \times \frac{4}{3} = \frac{8}{3}$ . Every 3 thirds make 1 whole, so 8 thirds are 2 wholes and 2 thirds: $\frac{8}{3} = 2\frac{2}{3}$ . That is the same product as before.

This works because the mixed number and the improper fraction are the same amount, written in two ways. So both ways of multiplying give the same product. Use the one you find easier.

Worked example

Use an improper fraction

Estimate, then find $3 \times 5\frac{7}{8}$ by writing the mixed number as an improper fraction.

Estimate. The fraction part, 7 eighths, is greater than 4 eighths, which equals 1 half. So the mixed number rounds up to 6. The factor 3 stays 3, and the estimate is 18.
$3 \times 5\frac{7}{8} \text{ is about } 3 \times 6 = 18$
Write the mixed number as an improper fraction. Each whole is 8 eighths, so 5 wholes are 40 eighths. Add the 7 eighths to get 47 eighths.
$5\frac{7}{8} = \frac{40}{8} + \frac{7}{8} = \frac{47}{8}$
Multiply. There are 3 groups of 47 eighths. The parts are still eighths, and 3 times 47 is 141, so that makes 141 eighths.
$3 \times \frac{47}{8} = \frac{141}{8}$
Write the product as a mixed number. Every 8 eighths make 1 whole. 141 divided by 8 is 17 with 5 left over, so 136 of the eighths make 17 wholes, and 5 eighths are left over.
$\frac{141}{8} = \frac{136}{8} + \frac{5}{8} = 17\frac{5}{8}$

Answer

The product is $3 \times 5\frac{7}{8} = 17\frac{5}{8}$ . That is a little less than the estimate of 18, which makes sense because you rounded $5\frac{7}{8}$ up.

Here is the big idea. A whole number times a mixed number means equal groups, and every group has a whole number part and a fraction part. The product counts both parts from every group. You can multiply the parts separately and add the results, or you can write the mixed number as an improper fraction first. Before you start, estimate by rounding each factor to the nearest whole number, so you know about how big the product should be.

Next, you will use improper fractions to multiply a mixed number by a fraction and by another mixed number.