# 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.*

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