Multiplying Fractions: Word Problems and Mistakes | Grade 5 Math | LevelKite

Some stories give you two fractions, and one of them is a part of a part. Here is one.

Half of a pan of cornbread is left. You eat one third of what is left. How much of the whole pan do you eat?

Picture the leftover half. Cut it into 3 equal pieces and eat 1 piece. The other half of the pan would make 3 more pieces of the same size. So 6 of these pieces fill the pan, and each piece is $\frac{1}{6}$ of the pan. You eat $\frac{1}{6}$ of the pan.

You just found $\frac{1}{3}$ of $\frac{1}{2}$ . That is what multiplying these two fractions means: $\frac{1}{3} \times \frac{1}{2}$ tells you to cut $\frac{1}{2}$ into 3 equal parts and take 1 of them. So the story matches this multiplication sentence:

\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}

Every fraction is a part of something. That something is called its whole. The two fractions in the cornbread story do not have the same whole. To see this, ask about each one: a fraction of what? The table shows the answers.

The cornbread story in words and in numbers

In the story	In numbers	A fraction of what?
Half of the pan is left.	$\frac{1}{2}$	The whole pan
You eat one third of what is left.	$\frac{1}{3}$	The leftover half
The part of the pan that you eat	$\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$	The whole pan

Key concept

A part of a part means multiply

Some stories have two fractions with two different wholes.

One fraction is a part of a whole. Call this the first part.
The other fraction is a part of the first part.

When a story is built this way, multiply the two fractions. The product (the answer when you multiply) tells how much you have as a fraction of the whole, not as a fraction of the first part.

Worked example

A part of a garden

Vegetables grow on $\frac{4}{5}$ of a garden. Tomatoes grow on $\frac{2}{3}$ of the vegetable part. What fraction of the whole garden grows tomatoes?

Ask about each fraction: a fraction of what? Four fifths is a fraction of the whole garden. Two thirds is a fraction of the vegetable part, not of the whole garden. The question asks about the whole garden.
Tomatoes are a part of a part: two thirds of the four fifths. A part of a part means multiply, so write the multiplication.
$\frac{2}{3} \times \frac{4}{5}$
See why with pieces. Think of the garden as 5 equal strips. Vegetables fill 4 of them. Cut every strip into 3 equal pieces, so the whole garden has 15 equal pieces. Taking two thirds of the vegetable part is the same as taking 2 of the 3 pieces in each of the 4 vegetable strips. That makes 8 tomato pieces.
$3 \times 5 = 15 \text{ pieces in the garden} \qquad 2 \times 4 = 8 \text{ tomato pieces}$
Multiply the numerators (the top numbers) and multiply the denominators (the bottom numbers). The 8 on top counts the tomato pieces. The 15 on the bottom counts the equal pieces in the whole garden. So the product is a fraction of the whole garden.
$\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$

Answer

Tomatoes grow on $\frac{8}{15}$ of the whole garden.

Worked example

When the whole is 1 mile

Rosa walks $\frac{3}{5}$ of a trail. The whole trail is $\frac{3}{4}$ of a mile long. How far does Rosa walk, in miles?

Ask about each fraction: a fraction of what? Three fourths is a fraction of 1 mile. It tells how long the trail is. Three fifths is a fraction of the trail. So the trail is the first part, and Rosa walks a part of that part. The story names the fractions in the other order this time, but the wholes still show you what to do.
Rosa walks three fifths of three fourths of a mile, so multiply the two fractions. Multiply the numerators, then multiply the denominators.
$\frac{3}{5} \times \frac{3}{4} = \frac{3 \times 3}{5 \times 4} = \frac{9}{20}$
Name the whole for the product, then check the size. The question asks for miles, so nine twentieths is a fraction of 1 mile. Rosa walks only part of the trail, because three fifths is less than 1. So she must walk less than three fourths of a mile. Three fourths equals fifteen twentieths, and nine twentieths is less than that.
$\frac{9}{20} < \frac{15}{20} = \frac{3}{4}$

Answer

Rosa walks $\frac{9}{20}$ of a mile.

Now you will check someone else's work. Two questions help you find a mistake.

Does the size make sense? Every fraction in this lesson is greater than 0 and less than 1. When both fractions are like that, a part of a part is smaller than the part you started with. So the product must be smaller than each of the two fractions. An answer that is not smaller than both has a mistake in it.
Is each step true? To multiply two fractions, you multiply the numerators and multiply the denominators. Check the top numbers and the bottom numbers one at a time.

The size check cannot catch every mistake, because a wrong answer can still be small. So always ask the second question too.

Worked example

Find and fix a mistake

A painter has painted $\frac{3}{5}$ of a wall. Blue paint covers $\frac{1}{2}$ of the painted part. A student wants the fraction of the whole wall that is blue and writes $\frac{1}{2} \times \frac{3}{5} = \frac{5}{10} \times \frac{6}{10} = \frac{30}{10}$ . Find the mistake and correct it.

Check the size first. Blue covers only one half of the painted part, and one half is less than 1. So the blue part must be smaller than the painted part, which is three fifths of the wall. But 30 tenths is 3 whole walls, because 10 tenths make 1 whole. The answer is much too big, so the work has a mistake.
$\frac{30}{10} = 3$
Check the first step. The student renamed both fractions with a common denominator, which is one denominator shared by both fractions. One half equals five tenths, and three fifths equals six tenths, so this step is true. Multiplying does not need it, but it is not the mistake.
$\frac{1}{2} = \frac{5}{10} \qquad \frac{3}{5} = \frac{6}{10}$
Check the second step. The student multiplied the numerators: 5 times 6 is 30. But the student kept the denominator 10. Keeping the denominator is a step for adding fractions with the same denominator, where the pieces stay the same size. Taking a part of a part cuts the pieces smaller, so you must multiply the denominators too. That gives hundredths, not tenths.
$\frac{5}{10} \times \frac{6}{10} = \frac{5 \times 6}{10 \times 10} = \frac{30}{100}$
Correct the work the short way. Multiply the fractions just as the story gives them: numerators together and denominators together. The result names the same amount as 30 hundredths, because dividing both 30 and 100 by 10 gives 3 and 10.
$\frac{1}{2} \times \frac{3}{5} = \frac{1 \times 3}{2 \times 5} = \frac{3}{10}$

Answer

The student kept the common denominator $10$ instead of multiplying the denominators. The correct product is $\frac{3}{10}$ , so $\frac{3}{10}$ of the whole wall is blue.

Watch for this mistake: adding the numerators. A student solves the garden problem and writes $\frac{2}{3} \times \frac{4}{5} = \frac{6}{15}$ . The student multiplied the denominators, $3 \times 5 = 15$ , but added the numerators, $2 + 4 = 6$ .

This fails because the numerator of the product counts the small pieces you take. Tomatoes take 2 pieces in each of 4 strips. That is 4 groups of 2, so you multiply: $2 \times 4 = 8$ . Adding 2 and 4 does not count those pieces. The correct product is $\frac{8}{15}$ .

Notice that $\frac{6}{15}$ is smaller than both fractions: $\frac{2}{3}$ equals $\frac{10}{15}$ and $\frac{4}{5}$ equals $\frac{12}{15}$ . So the size check does not catch this mistake. Checking each step does.

What to remember: when a story gives you a part of a part, multiply the two fractions. Ask a fraction of what? about each fraction and about your answer. The product is a fraction of the whole that the first part came from.

When you check work, remember that multiplying is not adding. To multiply fractions, you do not need a common denominator. You do not keep the denominator, and you do not add the numerators. Multiply the numerators, multiply the denominators, and ask whether the size of the answer makes sense.