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.

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

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

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

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

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: $100% = textcolor{purple}{£200}$

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

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)

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]

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

$£13.50div0.75=£18$

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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]

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

$£15,550div1.05=£14,761.90$

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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]

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]

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

$46,235div 0.85 =54,394$

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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]

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

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

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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]

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:

$44% = pounds10,120$

then

$1% = £10,120 div 44 = pounds230$

$1% = pounds230$

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 $0.44$ :

$pounds10,120 div 0.44 = pounds23,000$

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Specification Points Covered

Ratio, proportion and rates of change – 9. define percentage as ‘number of parts per hundred’; interpret percentages and percentage changes as a fraction or a decimal, and interpret these multiplicatively; express one quantity as a percentage of another; compare two quantities using percentages; work with percentages greater than $100%$ ; solve problems involving percentage change, including percentage increase/decrease and original value problems, and simple interest including in financial mathematics