Scaling by simple fractions

In a nutshell

Simple fractions only have a fractional part. Simple fractions can be used to scale other numbers up or down. This helps you to quickly compare different expressions without performing complex calculations.

Fractions relative to 1

By comparing the numerator (the top number of a fraction) and the denominator (the bottom number of a fraction), you can identify whether it is greater than, equal to, or less than $1$ .

Example 1

Identify whether the following fractions: $\dfrac 12, \dfrac 33,\dfrac 54$ , are greater than, equal to, or less than $1$ .

$\begin{aligned} & a) \space \dfrac 12 = \end{aligned}$ The numerator is less than the denominator.

$\underline{\dfrac 12 \lt 1}$

$\begin{aligned} &b) \space \dfrac 33 = \end{aligned}$ The numerator is equal to the denominator.

$\underline{\dfrac 33 = 1}$

$\begin{aligned} &c)\space \dfrac 54 = \end{aligned}$ The numerator is greater than the denominator.

$\underline{\dfrac 54 \gt 1}$

Scaling with simple fractions

By comparing simple fractions, expressions can be sorted from largest to smallest without performing multiplication.

Example 2

Andy and Janet host a bake sale with $40$ cakes each. Andy manages to sell $\dfrac 25$ of his cakes while Janet sells $\dfrac 35$ . Who sold more cakes?

Form expressions for the number of cakes Andy and Janet have sold.

Andy: $40 \times \dfrac{2}{5}$

Janet: $40 \times \dfrac{3}{5}$

Compare the expressions using scaling.

$\dfrac{3}{5}$ is greater than $\dfrac 25$ as the numerator is larger.

Therefore, Janet sold more cakes.

Note: If you wish to multiply a number by a fraction, multiply the number by the numerator and then divide by the denominator.

Example 3

Arrange the following expressions in order of size, from largest to smallest:

$\dfrac 34 \times \dfrac99 \quad\quad\quad\quad \dfrac 47 \times \dfrac34 \quad\quad\quad\quad \dfrac 34\times \dfrac 53$

Rearrange all the expressions to be in terms of a common multiple.

$\dfrac 34 \times \dfrac99 \quad\quad\quad\quad \dfrac 34 \times \dfrac47 \quad\quad\quad\quad \dfrac 34\times \dfrac 53$

Compare the uncommon multiples in each expression by checking whether the fractions are smaller than, greater than or equal to $1$ .

$\dfrac 53 > \dfrac99 > \dfrac47$

Rearrange the expressions.

$\underline{\dfrac 34 \times \dfrac53 \quad\quad\quad\quad \dfrac 34 \times \dfrac99 \quad\quad\quad\quad \dfrac 34\times \dfrac 47}$