Number · Fractions, Decimals and Percentages · Lesson 6

Finding a Fraction of a Quantity

Sharing money, sharing food and sharing work all need fractions of quantities: half of the lorry fare, $\frac{2}{3}$ of the class, $\frac{3}{4}$ of an hour. In this lesson you will learn a reliable way to find any fraction of an amount of money, a number of objects or a measurement, and to read fractions of a group from a graph.

NaCCA B7.1.3.3.2 ⏱ about 45 min

By the end you will be able to

✔Explain why "a fraction of a quantity" means multiplying the fraction by the quantity.
✔Find a fraction of an amount of money or a number of objects.
✔Find the remaining part of a quantity after a fraction has been taken away.
✔Find fractions of measurements (time, mass, length) by first changing to smaller units.
✔Read the fraction of a group from a graph and use it to find a number of people.

Key words

Quantity: An amount that can be counted or measured, such as GH₵60, 132 learners or 2 kg.
Unit fraction: A fraction with numerator 1, such as $\frac{1}{3}$ . It is one of the equal parts.
"Of": In mathematics, "of" means multiply: $\frac{2}{3}$ of 60 = $\frac{2}{3} \times 60$ .
Remainder (what is left): The part of a quantity left after some fraction of it has been used or taken away.

Curriculum reference

Content standard: B7.1.3.3 Demonstrate an understanding of the process of multiplying and dividing positive fractions and apply this in solving problems.

Indicator: Find a fraction of given quantity (i.e. money or given quantity of objects).

Starter

Grandpa's cocoa money

Nana Kwabena sold his cocoa at Goaso and decided to give $\frac{2}{3}$ of GH₵60 to his granddaughter Abena for her school supplies. Abena's little brother says, "That means she gets GH₵2 and GH₵3!" Of course that is not right. How much does Abena really get? Market women, teachers, chiefs and bankers answer questions like this every day — and you are about to learn how.

1. "Of" means multiply

In the last lesson you saw that when a fraction comes first, the × sign is read as "of". Turn that around: whenever a question asks for a fraction of a quantity, you multiply the fraction by the quantity. So $\frac{2}{3}$ of GH₵60 means $\frac{2}{3} \times 60$ .

To understand what is happening, think about the denominator and numerator separately. The denominator (3) tells you to cut the GH₵60 into 3 equal shares, so each share, $\frac{1}{3}$ , is 60 ÷ 3 = GH₵20. The numerator (2) tells you how many of those shares to take: 2 × GH₵20 = GH₵40.

▶ Notice the two moves: the denominator cuts the whole into equal parts, and the numerator chooses how many parts to take.

Two methods that always give the same answer
Method	Working for $\frac{2}{3}$ of GH₵60	Best when…
A: divide, then multiply	60 ÷ 3 = 20, then 20 × 2 = 40	the quantity divides exactly by the denominator
B: multiply as fractions	$\frac{2}{3} \times \frac{60}{1} = \frac{120}{3} = 40$	you want a written method that always works

2. What is left over?

Many questions give you one fraction but ask about the other part. For example: *There are 132 learners in a class. If $\frac{2}{3}$ of the learners are girls, how many boys are in the class?* There are two good ways to answer.

Find the girls, then subtract. Girls = $\frac{2}{3}$ of 132 = (132 ÷ 3) × 2 = 44 × 2 = 88. Boys = 132 − 88 = 44.
Find the fraction left, then use it. The whole class is $\frac{3}{3}$ , so boys are $1 - \frac{2}{3} = \frac{1}{3}$ of the class. $\frac{1}{3}$ of 132 = 44 boys.

Sometimes a fraction is taken from what remains, not from the whole. Suppose Kofi earns GH₵1,800, spends $\frac{1}{4}$ on rent and then $\frac{1}{3}$ of the remainder on food. Rent = $\frac{1}{4}$ of 1,800 = 450, so 1,350 remains. Food = $\frac{1}{3}$ of 1,350 = 450 (not $\frac{1}{3}$ of 1,800 = 600!). Kofi has 1,350 − 450 = GH₵900 left. Read carefully whether the fraction is of the whole or of the rest.

3. Fractions of measurements

Quantities are not only money and people. You may need $\frac{3}{4}$ of an hour, $\frac{2}{5}$ of a kilogram or $\frac{3}{8}$ of 2 metres. The trick is to change the quantity into a smaller unit first, so that it divides exactly by the denominator.

Change to a smaller unit, then find the fraction
Question	Smaller unit	Working	Answer
$\frac{3}{4}$ of 1 hour	60 minutes	60 ÷ 4 = 15; 15 × 3 = 45	45 minutes
$\frac{2}{5}$ of 1 kg	1,000 g	1,000 ÷ 5 = 200; 200 × 2 = 400	400 g
$\frac{3}{8}$ of 2 m	200 cm	200 ÷ 8 = 25; 25 × 3 = 75	75 cm
$\frac{5}{6}$ of 1 day	24 hours	24 ÷ 6 = 4; 4 × 5 = 20	20 hours

4. Reading fractions from a graph

A graph can tell you what fraction of a group has a certain property. The bar chart below shows the ages of the 32 learners in a Primary 5 class in Cape Coast. The fraction of learners who are 10 years old is $\frac{number of 10-year-olds}{total number of learners} = \frac{12}{32} = \frac{3}{8}$ .

▶ Add all the bars to get the whole class (4 + 12 + 10 + 6 = 32). Each bar over 32 is that age group's fraction of the class.

Going the other way, if you know that $\frac{5}{16}$ of a class of 32 learners are 11 years old, the number of 11-year-olds is $\frac{5}{16} \times 32 = 10$ , which matches the bar. When a graph has no numbers on it, you can still estimate a fraction — for example a bar that is about a quarter of the total height of all the bars stacked together is roughly $\frac{1}{4}$ of the group.

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

A fraction of money

Find

\frac{2}{3}

of GH₵60.

Show the solution

Step 1"Of" means multiply: $\frac{2}{3} \times 60$ .
Step 2Divide by the denominator: 60 ÷ 3 = 20 (that is $\frac{1}{3}$ ).
Step 3Multiply by the numerator: 20 × 2 = 40.

Answer: GH₵40

Example 2

Finding the other part

There are 132 learners in a class. If

\frac{2}{3}

of the learners are girls, how many boys are in the class?

Show the solution

Step 1Fraction of boys = $1 - \frac{2}{3} = \frac{1}{3}$ .
Step 2Boys = $\frac{1}{3}$ of 132 = 132 ÷ 3 = 44.
Step 3Check: girls = $\frac{2}{3}$ of 132 = 88, and 88 + 44 = 132 ✓

Answer: 44 boys

Example 3

A fraction of a mass

A trader in Techiman has a 2 kg bag of groundnuts. She sells

\frac{3}{5}

of it. How many grams does she sell?

Show the solution

Step 1Change to grams: 2 kg = 2,000 g.
Step 2 $\frac{1}{5}$ of 2,000 g = 2,000 ÷ 5 = 400 g.
Step 3 $\frac{3}{5}$ of 2,000 g = 3 × 400 = 1,200 g.

Answer: 1,200 g

Example 4

A fraction of the remainder

Kofi earns GH₵1,800. He spends

\frac{1}{4}

of it on rent and

\frac{1}{3}

of the remainder on food. How much does he have left?

Show the solution

Step 1Rent = $\frac{1}{4}$ of 1,800 = 450. Remainder = 1,800 − 450 = 1,350.
Step 2Food = $\frac{1}{3}$ of 1,350 = 450.
Step 3Left = 1,350 − 450 = 900.

Answer: GH₵900

Example 5

Using a graph

Use the bar chart of ages in section 4. (i) What fraction of the learners are 10 years old? (ii) How many learners are 11 years old if

\frac{5}{16}

of the 32 learners are 11?

Show the solution

Step 1(i) Total = 4 + 12 + 10 + 6 = 32. Fraction of 10-year-olds = $\frac{12}{32}$ .
Step 2Simplify by dividing top and bottom by 4: $\frac{12}{32} = \frac{3}{8}$ .
Step 3(ii) $\frac{5}{16}$ of 32 = (32 ÷ 16) × 5 = 2 × 5 = 10 learners.

Answer: (i) $\frac{3}{8}$ (ii) 10 learners

Summary

"A fraction of a quantity" means fraction × quantity.
Quick method: divide the quantity by the denominator, then multiply by the numerator.
To find what is left, either subtract from the whole or use 1 − the fraction.
Watch whether a second fraction is of the whole or of the remainder.
For measurements, change to a smaller unit (minutes, grams, centimetres) first.
From a graph: fraction of a group = number in the group ÷ total.

Grandpa's cocoa money

1. "Of" means multiply

2. What is left over?

3. Fractions of measurements

4. Reading fractions from a graph

Worked examples

A fraction of money

Finding the other part

A fraction of a mass

A fraction of the remainder

Using a graph

Summary

Practice: 15 questions