Reverse Percentages

 

What are Reverse Percentages?

Reverse percentages are used to find an original amount after a known percentage change. You work backward using the given percentage to calculate the value before the increase or decrease.

 

Working with Reverse Percentages

Reverse Percentage Increase

When a final value is given after a percentage increase, the original value can be found using this approach:

1. If a value has been increased by a percentage, say $20\%$, the final amount is $120\%$ of the original.
2. Divide the final value by the percentage as a decimal (for example, $120\%$ becomes $1.2$) to find the original amount.

Formula

If the percentage increase is $x\%$: $$\text{Original Value} = \frac{\text{Final Value}}{1 + \frac{x}{100}}$$

 

Example: Finding the Original Price after an Increase

The price of a bike after a $25\%$ increase is $\pounds 250$. What was the original price?

1. Set up the equation:

   • Since the price increased by $25\%$, the final value represent $125\%$ of the original.
   • So we divide $\pounds 250$ by $1.25$
2. Calculation: $$\text{Original Price} = \frac{250}{1.25} = 200$$

So, the original price was $\pounds 200$

 

 

 

Reverse Percentage Decrease

When a final value is given after a percentage decrease, the original value can be found by:

1. Determining the remaining percentage after the decrease.
2. Dividing the final value by this percentage in decimal form to get the original amount.

Formula

If the percentage decrease is $x\%$: $$\text{Original Value} = \frac{\text{Final Value}}{1 - \frac{x}{100}}$$

 

Example: Finding the Original Price after a Decrease

The sale price of a TV after a $20\%$ discount is $\pounds 160$. What was the original price?

1. Set up the equation:

   • After a $20\%$ discount, the final price represents $80\%$ of the original.
   • Divide $\pounds 160$ by $0.8$
2. Calculation: $$\text{Original Price} = \frac{160}{0.8} = 200$$

So, the original price was $\pounds 200$