# Reverse Percentages

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## Reverse Percentages

Reverse percentages are where we are given a value or an amount that has increased or decreased by a certain percent, and then we have to use this information to calculate the original amount.

Make sure you are happy with the following topics before continuing.

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## Method 1: Reverse Percentages 1% to 100%

• Write the amount in the question as a percentage of the original value
• i.e. write an increase of $10\%$ as $110\%$ and a decrease of $10\%$ as $90\%$
• Divide the amount by this value to find $1\%$ of the original value
• Multiply by $100$

Example:

Felicity buys a dress in a sale. It is reduced by $\textcolor{red}{40\%}$ down to a price of $\textcolor{blue}{£54}$. Work out the original price of the dress.

Step 1: We need to work out the cost of the dress as a percentage of its original value. We know it was $\textcolor{red}{40\%}$ off so:

$100\% - \textcolor{red}{40\%} = 60\%$

Step 2: Divide the cost by $60\%$ to find $1\%$ of the original value

\begin{aligned} (\div \, 60) \, \, \, \, \, 60\% &= \textcolor{blue}{£54} \,\,\,\,\, (\div \, 60) \\ 1\% &= £0.90 \end{aligned}

Step 3: Multiply by $100$ to get $100\%$ and the original value

\begin{aligned} (\times \, 100) \, \, \, \, \, 1\% &= £0.90 \,\,\,\,\, (\times \, 100) \\ 100\% &= \textcolor{purple}{£90} \end{aligned}

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## Method 2: Reverse Percentages using decimals

• Write the amount in the question as a percentage of the original value
• i.e. write an increase of $10\%$ as $110\%$ and a decrease of $10\%$ as $90\%$
• Convert this percentage to a decimal or a fraction.
• Divide the amount by this value to find the original value

Example:

Felicity buys another dress in a sale. It is reduced by $\textcolor{red}{30\%}$ down to a price of $\textcolor{blue}{£42}$. Work out the original price of the dress.

Step 1: We need to work out the cost of the dress as a percentage of its original value. We know it was $\textcolor{red}{30\%}$ off so:

$100\% - \textcolor{red}{30\%} = 70\%$

Step 2: Convert this percentage to a decimal or fraction $70\% = 0.7$

Step 3: Divide the sale value by this decimal equivalent to get the original value

$£42 \div 0.7 = \textcolor{purple}{£60}$

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## Example 1: Reverse Percentage

A TV cost $\textcolor{limegreen}{£160}$ in a $\textcolor{blue}{20\%}$ off sale. Calculate the original price of the TV before the sale.

[1 mark]

Step 1: $80\% = \textcolor{limegreen}{£160}$

Step 2: Divide both sides by $80$ to find $1\% = £2$

Step 3: Multiply by $100$ to find $100\%$, i.e. the original amount: $100\% = \textcolor{purple}{£200}$

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## Example 2: Reverse Percentage

A gym’s membership increased in price by $\textcolor{red}{15\%}$ to $\textcolor{orange}{£23}$ per month. What was the cost of the membership before the increase?

[1 mark]

Step 1: $115\% = \textcolor{orange}{£23}$

Step 2: Divide both sides by $115$ to find $1\% = £0.20$

Step 3: Multiply by $100$ to find $100\%$, i.e. the original amount:  $100\% = \textcolor{purple}{£20}$ per month

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## Example 3: Reverse Percentage

Rogelio’s new record for the $100$m sprint is $\textcolor{red}{10.8}$ seconds. This is $\textcolor{limegreen}{5.8\%}$ faster than his previous record. What was his previous record (to $2$ dp)?

[1 mark]

Step 1: $94.2 \% = \textcolor{red}{10.8}$ seconds

Step 2: Divide both sides by $94.2$ to find $1\% = 0.11464$

Step 3: Multiply by $100$ to find $100\%$, i.e. the original amount: $100\% = \textcolor{purple}{11.46}$ seconds ($2$ dp)

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## Reverse Percentages Example Questions

Question 1: The price of a t-shirt is reduced by $25\%$ to $£13.50$ during a sale. How much did it originally cost?

[1 mark]

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To find the original price of the t-shirt we can divide the new cost by $0.75$:

$£13.50\div0.75=£18$

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Question 2: The price of a car has increased by $5%$ to $£15,500$. What was the cost of the car prior to this increase?

[1 mark]

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To find the original cost of the car we can divide the new cost by $1.05$:

$£15,550\div1.05=£14,761.90$

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Question 3: A sports recovery bar contains $15.6$ g of protein.  $18\%$ of the recovery bar is protein. Work out the total mass of the bar.

[1 mark]

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To find the total mass of the bar we can divide the mass of protein by $0.18$:

$15.6$g $\div \, 0.18 = 86.67$g

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Question 4: The crowd attendance at a recent football game is $46,235$. This is $85\%$ capacity of the ground. What is the total capacity of the stadium?

[1 mark]

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To find the total capacity of the stadium we can divide the recent attendance by $0.85$:

$46,235\div 0.85 =54,394$

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Question 5: The population of London is $9,300,000$. This represents $14\%$ of the population of the United Kingdom. Work out the total population of the U.K.

[1 mark]

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To find the total population of the U.K. we can divide the population of London by $0.14$:

$9,300,000\div 0.14 =66,428,571$

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Question 6: Petra buys a car. After $2$ years of her owning it, it is now worth $£10,120$, which constitutes a $56\%$ decrease from its original value. What was its original value?

[2 marks]

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If a car has decreased in value by $56\%$, this means that the car is now worth $44\%$ of what it was worth before ($100\% - 56\% = 44\%$).

This means that the current value of $£10,120$ represents $44\%$ of the original price.

The original price is the $100\%$ amount, so we need to work out what $100\%$ represents if $44\% = £10,120$.  The easiest way to do this is to work out what $1\%$ is:

If

$44\% = \pounds10,120$

then

$1\% = £10,120 \div 44 = \pounds230$

If

$1\% = \pounds230$

then

$100\% = £230 \times100 = \pounds23,000$

So, the original price of Petra’s car was $£23,000$.

Another way we can view this question is by working what we would multiply the original amount by in order to work out the new value.  If we are calculating a $56\%$ decrease, we would multiply the original amount by $0.44$ to work out the new value.  Therefore, if we know the new amount and wish to work out the original amount, then we can simply divide the new amount by $0.44$:

$\pounds10,120 \div 0.44 = \pounds23,000$

Gold Standard Education

## Reverse Percentages Worksheet and Example Questions

### (NEW) Reverse percentages Exam Style Questions

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