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2 Fractions
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AustraliaVIC
Level 7

2.04 Fractions of a quantity

Lesson
Worksheet
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Lesson

Fractions of a quantity

Fractions describe parts of a whole, but they can also describe parts of a quantity.

Find \dfrac{1}{12} of 36.

This image shows a square divided into 36 smaller squares

Let's start by drawing a grid of 36.

This image shows 12 vertical rectangles that contain 3 squares each.

To find one twelfth, we split this grid into 12 equal parts.

Looking at the pieces, each piece has 3 squares. So \dfrac{1}{12} of 36 is 3.

We can also work this out using arithmetic. We know that \dfrac{1}{12} of 36 can be written using multiplication, \dfrac{1}{12} \times 36.

This is the same as 1 \times \dfrac{36}{12} and \dfrac{1 \times 36}{12}. The third expression is the most useful.

First, if we evaluate the multiplication in the numerator we get \dfrac{36}{12}. Then we can cancel the greatest common factor from the numerator and denominator. In this case it is 12. This gives us \dfrac{3}{1} which is the same as 3.

We can check this answer by multiplying back. 12 \times 3 = 36, so we know that 3 is \dfrac{1}{12} of 36.

Examples

Example 1

Evaluate \dfrac25\times35.

Worked Solution
Create a strategy

Multiply the numerator by the whole number.

Apply the idea
\displaystyle \dfrac25\times35\displaystyle =\displaystyle \dfrac{2\times35}{5}Multiply the numerator by the whole number
\displaystyle =\displaystyle \dfrac{70}{5}Evaluate
\displaystyle =\displaystyle 14Simplify

Example 2

Find 3 groups of \dfrac45.

Worked Solution
Create a strategy

The phrase "groups of" means multiplication.

Apply the idea
\displaystyle 3\times\dfrac45\displaystyle =\displaystyle \dfrac{3\times4}{5}Multiply the numerator by the whole number
\displaystyle =\displaystyle \dfrac{12}{5}Evaluate

Example 3

Find \dfrac{5}{7} of 5 weeks in days.

Worked Solution
Create a strategy

The word "of" means multiplication.

Use the fact that 1 \text{ week } = 7 \text{ days}.

Apply the idea

Convert 5 weeks to days:

\displaystyle 5 \text{ weeks}\displaystyle =\displaystyle 5\times7 \text{ days}Mutiply 5 by 7
\displaystyle =\displaystyle 35 \text{ days}Evaluate

So we have:

\displaystyle \dfrac{5}{7}\times 5 \text{ weeks}\displaystyle =\displaystyle \dfrac{5}{7}\times 35 \text{ days}Substitute the number of days
\displaystyle =\displaystyle \dfrac{5\times35}{7}Multiply the numerator by the whole number
\displaystyle =\displaystyle \dfrac{175}{7}Evaluate
\displaystyle =\displaystyle 25 \text{ days}Simplify

So \dfrac{5}{7} of 5 weeks is 25 days.

Idea summary

Finding the fraction of a quantity is the same as multiplying a whole number by a fraction.

To multiply a whole number by a fraction, multiply the whole number by the numerator.

It is often easier to cancel common factors in the numerator and denominator before evaluating the multiplication.

Outcomes

VCMNA245

Express one quantity as a fraction of another, with and without the use of digital technologies

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