8.6Reverse Percentage

The value you are given is the one after the change, and you work back to the original. Set the original at $100\%$ and go through $1\%$ :

Say what percentage of the original you were given. After a $25\%$ increase, the new price is $125\%$ of the original.
Divide by that percentage to get $1\%$ of the original.
Multiply by $100$ to get the original itself.

After a $\displaystyle 25\%$ increase in price, a pair of sports shoes now costs $\displaystyle \$175$ . Find the original price.

Show solution▾

The original price is

100\%

, so after a

25\%

increase the new price is

125\%

\displaystyle 125\% \text{ of original}

\displaystyle = \$175

\displaystyle 1\% \text{ of original}

\displaystyle = \dfrac{175}{125}

\displaystyle 100\% \text{ of original}

\displaystyle = \dfrac{175}{125} \times 100

\displaystyle = \$140

The example below. The

\$490

is the

70\%

part, so one part is

490 \div 70

and the whole is one hundred of those.

During a sale, a phone was sold at a $\displaystyle 30\%$ discount for $\displaystyle \$490$ . Find the original marked price.

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The marked price is

100\%

, so after a

30\%

discount the sale price is

70\%

\displaystyle 70\% \text{ of marked price}

\displaystyle = \$490

\displaystyle 1\% \text{ of marked price}

\displaystyle = \dfrac{490}{70}

\displaystyle 100\% \text{ of marked price}

\displaystyle = \dfrac{490}{70} \times 100

\displaystyle = \$700

Now you try

After an increase of $25\%$ , a salary is $\$3\,125$ . Find the original salary.

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Let the original salary be

x

\displaystyle x \times \left(1 + \dfrac{25}{100}\right)

\displaystyle = 3125

\displaystyle x \times 1.25

\displaystyle = 3125

\displaystyle x

\displaystyle = \dfrac{3125}{1.25}

\displaystyle x

\displaystyle = \$2\,500

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