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.

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