Multiply Mixed Numbers: Word Problems and Mistakes | Grade 5 Math | LevelKite

A mixed number is a whole number and a fraction written together, like $2\frac{3}{4}$ . You already know how to multiply mixed numbers. One way is to write each one as an improper fraction and then multiply. An improper fraction has a top number (the numerator) that is equal to or greater than its bottom number (the denominator), like $\frac{11}{4}$ .

In this lesson you use that skill for two new jobs:

Solver. You read a story problem, decide what to multiply, and check that your answer makes sense.
Checker. You look at finished work, find the mistake, and fix it.

A story problem does not tell you to multiply. You have to spot it. Multiplication fits a story in three common ways:

Equal groups. Every group is the same size. The number of groups can be a whole number or a mixed number, like $2\frac{1}{2}$ batches.
Part of an amount. You take a fraction of something.
Area. Area is the amount of flat space inside a shape. It is measured in square units, such as square feet. For a rectangle, the area is the length times the width.

When you spot one of these, write the multiplication with numbers and symbols. The table does this for four stories. You already know how to work out each one.

From a story to a multiplication

The story	What it means	Multiplication to do
One jug holds $1\frac{3}{4}$ liters. You fill $5$ jugs. How many liters is that?	$5$ groups of $1\frac{3}{4}$ liters	$5 \times 1\frac{3}{4}$
One batch uses $1\frac{1}{3}$ cups of flour. You make $2\frac{1}{2}$ batches. How many cups do you use?	$2\frac{1}{2}$ groups of $1\frac{1}{3}$ cups	$2\frac{1}{2} \times 1\frac{1}{3}$
A board is $5\frac{1}{3}$ feet long. You paint $\frac{3}{4}$ of it. How many feet do you paint?	$\frac{3}{4}$ of $5\frac{1}{3}$ feet	$\frac{3}{4} \times 5\frac{1}{3}$
A poster is $3\frac{1}{2}$ feet long and $2\frac{1}{4}$ feet wide. What is its area?	Length times width	$3\frac{1}{2} \times 2\frac{1}{4}$

Before you multiply, make an estimate. An estimate is a quick answer that is close to the exact one. It tells you about how big the real answer should be.

To estimate, round each mixed number to the nearest whole number. Rounding means swapping a number for a nearby number that is easier to use. Look at the fraction part:

Is it less than $\frac{1}{2}$ ? Keep the whole number. So $4\frac{1}{5}$ rounds to $4$ .
Is it $\frac{1}{2}$ or more? Go up to the next whole number. So $1\frac{2}{3}$ rounds to $2$ .

If a number is already a whole number, or is a fraction less than $1$ , leave it alone.

Then multiply in your head. For $1\frac{2}{3} \times 4\frac{1}{5}$ , the estimate is $2 \times 4 = 8$ .

The exact product, which is the answer to a multiplication, is $\frac{5}{3} \times \frac{21}{5} = \frac{105}{15} = 7$ . It does not equal the estimate, but it is near it. That is what you want to see.

Key concept

Estimate, multiply, compare

An answer to a mixed number problem has to make sense. Three moves tell you if it does:

Estimate. Round each mixed number to the nearest whole number and multiply.
Multiply. Find the exact product.
Compare. The exact product should be near the estimate.

If the two are far apart, that is a sign of a mistake. Go back and check the steps.

Worked example

Equal groups with a whole number of groups

One friendship bracelet uses $2\frac{3}{4}$ feet of cord. You make $4$ bracelets. How many feet of cord do you need?

Spot the multiplication. Every bracelet uses the same amount of cord, so the story has 4 equal groups. Each group is 2 and 3 fourths feet.
$4 \times 2\frac{3}{4}$
Estimate. Three fourths is more than one half, so 2 and 3 fourths rounds up to 3. The 4 is already a whole number, so it stays. You should need about 12 feet.
$4 \times 3 = 12$
Multiply. Write the mixed number as an improper fraction: 2 wholes are 8 fourths, and 3 more fourths make 11 fourths. Then 4 groups of 11 fourths are 44 fourths. Every 4 fourths make 1 whole, so 44 fourths make 11 wholes.
$4 \times 2\frac{3}{4} = 4 \times \frac{11}{4} = \frac{44}{4} = 11$
Compare. The exact answer, 11, is near the estimate, 12. It is a little less, and that makes sense, because you rounded 2 and 3 fourths up.

Answer

You need $11$ feet of cord to make $4$ bracelets.

Worked example

Equal groups with a mixed number of groups

One lap of a bike path is $2\frac{1}{4}$ miles long. You ride $3\frac{2}{3}$ laps. How many miles do you ride?

Spot the multiplication. Every lap is the same length, so the laps are equal groups. You have 3 and 2 thirds groups, and each group is 2 and 1 fourth miles.
$3\frac{2}{3} \times 2\frac{1}{4}$
Estimate. Two thirds is more than one half, so 3 and 2 thirds rounds up to 4. One fourth is less than one half, so 2 and 1 fourth rounds down to 2. You should ride about 8 miles.
$4 \times 2 = 8$
Write each mixed number as an improper fraction. 3 wholes are 9 thirds, and 2 more thirds make 11 thirds. 2 wholes are 8 fourths, and 1 more fourth makes 9 fourths.
$3\frac{2}{3} = \frac{11}{3} \qquad 2\frac{1}{4} = \frac{9}{4}$
Multiply the numerators to get 99, and multiply the denominators to get 12. To write 99 twelfths as a mixed number, divide 99 by 12. You get 8 wholes with 3 twelfths left over, and 3 twelfths is the same as 1 fourth.
$\frac{11}{3} \times \frac{9}{4} = \frac{99}{12} = 8\frac{3}{12} = 8\frac{1}{4}$
Compare. The exact answer, 8 and 1 fourth, is very near the estimate, 8. The answer makes sense.

Answer

You ride $8\frac{1}{4}$ miles.

Now switch jobs and be the checker. Start with the estimate. If an answer is far from its estimate, that is a sign of a mistake.

The estimate does not show you where the mistake is, and a small slip can hide from it. So you also check the steps. Two mistakes happen often when people multiply mixed numbers. Here is the first one.

The four pieces of the product

The piece in words	The piece in symbols	Found by the mistaken way?
$2$ groups of $3$	$2 \times 3 = 6$	Yes
$2$ groups of $\frac{3}{4}$	$2 \times \frac{3}{4} = 1\frac{1}{2}$	No
$\frac{2}{3}$ of $3$	$\frac{2}{3} \times 3 = 2$	No
$\frac{2}{3}$ of $\frac{3}{4}$	$\frac{2}{3} \times \frac{3}{4} = \frac{1}{2}$	Yes

Add all four pieces to get the real product: $6 + 1\frac{1}{2} + 2 + \frac{1}{2} = 10$ . The mistaken answer, $6\frac{1}{2}$ , left out the $1\frac{1}{2}$ and the $2$ .

Improper fractions give the same product. Each improper fraction holds its whole mixed number, so no piece gets left out: $\frac{8}{3} \times \frac{15}{4} = \frac{120}{12} = 10$ . And $10$ is near the estimate of $12$ . It is a little less because both numbers were rounded up.

Improper fractions only help if you write them correctly. That is where the second common mistake hides. Watch for it in the next example.

Worked example

Find the mistake and fix it

A garden bed is a rectangle. It is $6\frac{1}{4}$ meters long and $3\frac{1}{5}$ meters wide. Ava finds its area like this: $6\frac{1}{4} \times 3\frac{1}{5} = \frac{7}{4} \times \frac{16}{5} = \frac{112}{20} = 5\frac{3}{5}$ square meters. Find Ava's mistake and fix it.

Ava is right to multiply, because the area of a rectangle is its length times its width. Now estimate. One fourth and one fifth are both less than one half, so the length rounds to 6 and the width rounds to 3. The area should be about 18 square meters. Ava's answer is less than 6 square meters. That is far from 18, so look for a mistake.
$6 \times 3 = 18$
Check each improper fraction. For 3 and 1 fifth: 3 wholes are 15 fifths, and 1 more fifth makes 16 fifths. Ava's 16 fifths is right. For 6 and 1 fourth: 6 wholes are 24 fourths, and 1 more fourth makes 25 fourths. Ava wrote 7 fourths, so this is the mistake.
$6\frac{1}{4} = \frac{24}{4} + \frac{1}{4} = \frac{25}{4}$
See why it happened. Ava added the whole number 6 to the numerator 1 and got 7. That counts the 6 wholes as only 6 fourths. But each whole is worth 4 fourths, so 6 wholes are 24 fourths. You can also tell by size: 7 fourths is less than 2, so it cannot equal a number that is more than 6.
$\frac{7}{4} = 1\frac{3}{4}$
Fix it. Multiply again with 25 fourths in place of 7 fourths. 25 times 16 is 400, and 4 times 5 is 20. Then 400 divided by 20 is 20.
$\frac{25}{4} \times \frac{16}{5} = \frac{400}{20} = 20$
Compare. The corrected answer, 20, is near the estimate, 18. It is a little more, which makes sense, because both numbers were rounded down.

Answer

Ava wrote $6\frac{1}{4}$ as $\frac{7}{4}$ , but it is $\frac{25}{4}$ . The correct area of the garden bed is $20$ square meters.

Keep this idea: a product of mixed numbers has to make sense. Estimate by rounding each mixed number to the nearest whole number. Multiply to find the exact product. Then compare the two.

If an answer is far from its estimate, hunt for the mistake. Did every part of one number multiply every part of the other? Is each improper fraction right? To check one, count the pieces in the wholes first, then add the numerator.