# How do you find the percent change given the original number is 60 and new number is 45?

Feb 19, 2017

-26%

#### Explanation:

Reduction requires subtraction so the percentage is negative as well

$\left(\text{change")/("original value")-> ("2nd number - 1st number")/("original value}\right) = \frac{- 15}{60}$

$\textcolor{b l u e}{\text{Shortcut method}}$

Most people would show this as: color(purple)(-15/60xx100% = -25%)

This can hide from people what is actually happening
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$\textcolor{b l u e}{\text{First principle method with full explanation - so it is long}}$

Starting point: you need to end up with: $\frac{\text{some number}}{100}$

Very important point: the symbol % is like a unit of measurement that is worth $\frac{1}{100}$

So: ("some number")/100 -> "some number "xx1/100

But $\frac{1}{100}$ is the same as % so we have:

("some number")/100 -> "some number "xx%

which is the same as: "some number"%

$\textcolor{b r o w n}{\text{The calculations}}$
To change the denominator of 60 into 100 multiply by $\frac{100}{60}$

$\textcolor{g r e e n}{- \frac{15}{60} \textcolor{red}{\times 1} \text{ "=" "-15/60color(red)(xx(color(white)(.)100/60color(white)(.))/(100/60))" "=" } - \frac{26}{100}}$

Notice that the numerator of $\textcolor{g r e e n}{15} \textcolor{red}{\times \frac{100}{60}} \to \frac{15}{60} \times 100$ Which is exactly the same as in the shortcut method.

but $\textcolor{g r e e n}{- \frac{26}{100}}$ is the same as $\text{ } \textcolor{g r e e n}{- 26 \times \frac{1}{100}}$

but 1/100 -> % giving:

-26% as in the shortcut method