# How do you find the percent change given the original number is 48 and new number is 60?

Feb 15, 2017

41color(white)(.) 2/3%

#### Explanation:

Goes from $48 \to 60$

So this is an increase of $60 - 40 = 20$

We express this as a change in relation to the original price. So the change expressed as a fraction of the original price is :

$\text{ } \textcolor{g r e e n}{\frac{+ 20}{48}}$

$\textcolor{p u r p \le}{\text{Using first principles to explain where the } \times 100}$
$\textcolor{p u r p \le}{\text{in the shortcut method comes from}}$

color(green)(20/48color(red)(xx1)" "=" "20/48color(red)(xx100/100) " "=" "(20color(red)(xx100) )/48color(red)( xx1/100)

But $\frac{1}{100}$ is the same as % giving:

" "=color(green)((20color(red)(xx100) )/48color(red)( %)

Writing this as in the shortcut method we have:

" "=(20/48 xx 100)% = 41.6bar(6)%
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Foot note}}$

The bar over the last 6 indicates that the value 6 repeats for ever.

It is really helpful to remember that $0.333 \overline{3} \text{ is the same as } \frac{1}{3}$

So $0.666 \overline{6}$ is the same as $2 \times \frac{1}{3} = \frac{2}{3}$

Thus: 41.6bar6 = 41 2/3%