Find a Fraction of a Number | Grade 5 Math | LevelKite

You already know how to multiply a fraction by a whole number. Now you will use that skill to find a fraction of a number, such as $\frac{2}{3}$ of $12$ .

Start with what the fraction says. The denominator is the bottom number. It tells how many equal parts to split the number into. The numerator is the top number. It tells how many of those parts to take.

So $\frac{2}{3}$ of $12$ means: split $12$ into $3$ equal parts, then take $2$ of the parts.

Picture $12$ counters in $3$ equal rows. Each row has $4$ counters. Take $2$ of the rows and you have $8$ counters. So $\frac{2}{3}$ of $12$ is $8$ .

The table shows this idea three ways: with counters, in words, and as a number sentence, which is math written with numbers and symbols.

Two thirds of 12, shown three ways

With counters	In words	Number sentence
Put $12$ counters in $3$ equal rows. Each row has $4$ counters.	Split $12$ into $3$ equal parts. Each part is one third of $12$ .	$12 \div 3 = 4$
Take $2$ of the rows. You have $8$ counters.	Take $2$ of the parts. That is two thirds of $12$ .	$2 \times 4 = 8$

Now link this to multiplying. Finding a fraction of a number means multiplying the fraction by the number. So $\frac{2}{3}$ of $12$ is $\frac{2}{3} \times 12$ .

Here is why the two match. With the counters, you divided $12$ by the denominator, $3$ . Then you multiplied by the numerator, $2$ . When you multiply $\frac{2}{3} \times 12$ the way you already know, you multiply $12$ by the numerator first and get $\frac{24}{3}$ . Then you divide $24$ by the denominator, $3$ .

These are the same two jobs in a different order, and either order gives the same answer. Dividing first takes one third of $12$ two times: $4 + 4$ . Multiplying first makes two copies of $12$ , which is $24$ , and then takes one third of the $24$ . That is one third of each copy, so it is $4 + 4$ again. Both ways give $8$ .

\frac{2}{3} \text{ of } 12 = \frac{2}{3} \times 12 = \frac{2 \times 12}{3} = \frac{24}{3} = 8

Key concept

A fraction of a number means multiply

To find a fraction of a number, split the number into equal parts and take some of them. The denominator tells how many equal parts. The numerator tells how many parts to take. This is the same as multiplying the fraction by the number.

In the formula, $\frac{a}{b}$ is the fraction and $n$ is the number. The letters stand for whole numbers, and $b$ is not $0$ .

You can divide by $b$ first or multiply by $a$ first. Both orders give the same answer. Dividing first is the easy way when $n$ divided by $b$ is a whole number. When it is not, multiply first.

\frac{a}{b} \text{ of } n = \frac{a}{b} \times n = \frac{a \times n}{b}

Worked example

Split, then take

Find $\frac{5}{8}$ of $16$ .

The denominator is 8, so split 16 into 8 equal parts. Divide to find the size of one part. Each part is one eighth of 16.
$16 \div 8 = 2$
The numerator is 5, so take 5 of the parts. Each part is 2, so 5 parts make 5 times 2.
$5 \times 2 = 10$
Check by multiplying the fraction by the number. Multiply 16 by the numerator, then divide by the denominator. The answer matches, because these are the same two jobs in a different order.
$\frac{5}{8} \times 16 = \frac{5 \times 16}{8} = \frac{80}{8} = 10$

Answer

$\frac{5}{8}$ of $16$ is $10$ .

Word problems in this lesson multiply a fraction by a whole number. In them, the fraction does one of two jobs. To decide which, ask: what does the whole number count?

The fraction is the size of each group. The whole number counts equal groups, and the fraction tells how big each group is. Multiply the number of groups by the size of one group.
The fraction is the part taken. The whole number is the whole amount, and the fraction tells what part of that amount you take. Multiply the fraction by the whole amount.

Both jobs lead to multiplying, but the answers mean different things. The table shows the same numbers doing each job.

The same numbers, two different jobs

Story	Job of the fraction	Number sentence	What the answer means
There are $8$ bottles. Each bottle holds $\frac{3}{4}$ liter of juice. How much juice is there in all?	Size of each group. The $8$ counts the bottles.	$8 \times \frac{3}{4} = \frac{24}{4} = 6$	$6$ liters of juice in all. That is $8$ groups of $\frac{3}{4}$ liter.
A tank can hold $8$ liters of water. It is $\frac{3}{4}$ full. How much water is in the tank?	Part taken. The $8$ liters is the whole amount.	$\frac{3}{4} \times 8 = \frac{24}{4} = 6$	$6$ liters of water. That is a part of the $8$ liters.

Worked example

Decide the job, then multiply

A trail is $4$ kilometers long. Maya has walked $\frac{3}{5}$ of the trail. How many kilometers has she walked?

Decide the job of the fraction. The story does not have 4 equal groups. The 4 kilometers is the whole trail, and Maya walked a part of it. So the fraction is the part taken.
Write the multiplication. A fraction of a number means the fraction times the number, so three fifths of 4 is three fifths times 4.
$\frac{3}{5} \times 4$
Multiply. Here 4 divided by 5 is not a whole number, so multiply first. Multiply 4 by the numerator. Keep the denominator, because the parts are still fifths.
$\frac{3}{5} \times 4 = \frac{3 \times 4}{5} = \frac{12}{5}$
Check the size. Since 10 fifths make 2 wholes, 12 fifths is 2 wholes and 2 fifths more. That is less than 4, which makes sense because Maya walked only part of the trail.
$\frac{12}{5} = 2\frac{2}{5}$

Answer

Maya has walked $\frac{12}{5}$ kilometers, which is $2\frac{2}{5}$ kilometers.

Worked example

Find and fix a mistake

Leo has $20$ stickers. He gives away $\frac{7}{10}$ of them. To find how many, Leo writes $\frac{7}{10} \times 20 = \frac{7 \times 20}{10 \times 20} = \frac{140}{200}$ . Find Leo's mistake and correct it.

Test the answer against the story. One tenth of 20 stickers is 2 stickers, so seven tenths must be more than 2 stickers. Leo's numerator is smaller than his denominator, so his answer is less than 1 sticker. It cannot be right.
$\frac{140}{200} < 1$
Find the mistake. Leo multiplied the denominator by 20 as well as the numerator. Multiplying both the top and the bottom by 20 only renames the fraction, so his answer is still seven tenths.
$\frac{140}{200} = \frac{7}{10}$
Correct the work. Multiply only the numerator by 20. The parts are still tenths, so the denominator stays 10.
$\frac{7}{10} \times 20 = \frac{7 \times 20}{10} = \frac{140}{10} = 14$
Confirm by splitting and taking. Split 20 into 10 equal parts, then take 7 of the parts. Both ways give 14.
$20 \div 10 = 2, \quad 7 \times 2 = 14$

Answer

Leo multiplied the denominator by $20$ as well as the numerator. The correct work is $\frac{7}{10} \times 20 = \frac{140}{10} = 14$ , so Leo gives away $14$ stickers.

The big idea: a fraction of a number is the fraction times the number. Split the number into as many equal parts as the denominator says. Then take as many parts as the numerator says.

In a word problem, ask what the whole number counts. If it counts equal groups, the fraction is the size of each group. If it is the whole amount, the fraction is the part taken.

When you multiply a fraction by a whole number, the whole number multiplies only the numerator. The denominator stays the same in that step, because the size of the parts does not change. Use this to find and fix mistakes.