reverse percentages homework

Reverse Percentages: Worksheets with Answers

Whether you want a homework, some cover work, or a lovely bit of extra practise, this is the place for you. And best of all they all (well, most!) come with answers.

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How To Calculate Reverse Percentages

Calculating reverse percentages is a really useful skill to have that you’ll use time and time again. It basically means working backwards to figure out the original cost or figure of something.

Our guide will show you how to break reverse percentages down to make overcoming the challenge as easy as possible.

What are reverse percentages?

Reverse percentages can help you to work out the original price or value of something, after it’s had a percentage discount or increase applied.

You’re most likely to come across reverse percentages in numerical reasoning tests , as they examine your general aptitude for numbers, data and mathematical principles.

However, you may also find it useful to know how to do reverse percentages in other aptitude tests, such as mechanical reasoning .

Key points to remember

With reverse percentages, it’s always easiest to work from what the value of 100% is.

A really simple way to remember how to find the percentage of a value (e.g. 20% of £100) is to follow this simple equation: value ÷ 100 × percentage.

If you’re struggling with a particularly complex reverse percentage, it can help to break down a value into its 1%, 10% or 25% form. These are normally easier to calculate mentally, and can provide an alternative starting point.

Although calculators make things easier a lot of the time, you might not always have one with you. Learning how to mentally reverse percentages is an important skill that you’ll be surprised how often you use.

Step 1: Start at 100% and minus the figure you've been given

The first thing you need to do is start at 100% and minus the figure you’ve been given.

So, if you’ve been asked to find out what the original price of an item is in a sale where everything is 20% off, you’d do: 100 - 20 = 80.

This means that you’re only paying 80% of the original price in a 20% off sale.

Step 2: Find the value of 1%

If you know that the sale item you’re purchasing is £160, you know that £160 is 80% of the original cost.

Next, you’ll need to divide the sale price by the percentage of the original price to find the value of 1%: 160 ÷ 80 = 2.

Step 3: Multiply 1% by 100 to get the value of 100%

Then, multiply 2 (or 1%) by 100 to get the value of 100% (or the original price). So, 2 x 100 = 200.

This means that the original price of the sale item was £200, and in a 20% off sale it’s now £160.

Example questions

Sarah buys a scarf in the sale. It’s marked as 75% off and now costs £20. How does Sarah work out the original cost of the scarf?

• Firstly, you need to remember that you’re working backwards from 100%.

So 100% - 75% = 25%, which means the sale price of £20 is 25% of the original price.

Next, divide the sale price by the percentage of the original price to find the value of 1%: 20 ÷ 25 = 0.80

Finally, to work out 100% take the value of 1% and multiply it by 100: 0.80 x 100 = £80. This makes the original value of the scarf £80.

There’s been a 10% rise in prices in Mark’s local coffee shop. If an Americano now costs Mark £2.75, how much did it cost before the increase?

As always, we start by looking at 100%. In this case, we now need to look at 100% + 10% = 110% and in terms of the price of the coffee itself, 110% is £2.75.

Now that we know what 110% is, we need to work out what 1% is. To do this you’d do: 2.75 ÷ 110 = 0.025

Then, multiply 0.025 (or 1%) by 100. So 0.025 x 100 = 2.50

And that’s your answer. Mark’s Americano would have cost £2.50 before the 10% price increase at the coffee shop.

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Reverse Percentages Calculator

Welcome to the our Reverse Percentages Calculator. Here you will our reverse percentage calculator which will help you to find the original value before a percentage increase or decrease.

Our calculators will not only find the original values, but also show you all the working out along the way!

For full functionality of this site it is necessary to enable JavaScript.

Here are the instructions how to enable JavaScript in your web browser .

Which Reverse Percentage Calculator do I need?

Reverse Percentage Calculator 1

• I want to find the original value when I know the final value and the percentage of the original value that made it,

Example: 32% of a number is 150. What's the number?

Reverse Percentage Calculator 2

• I want to find the original value when I know the final value and the percentage increase or decrease from the original value.

Example: After a 12% decrease, the price of a car is $15,000. How much did the car originally cost?

Reverse Percentages Calculator 1

Find Original Value

This calculator will help you to find the original value , when you have been given the final value and the percentage of the original value that made it.

Example: if 70% of a number is 210 then it will find the original value using the steps below.

• Type 70 for the percentage.
• Type 210 as the Final Value.
• Click 'Find Original Value'.
• Answer: 300. So 70% of 300 = 210.

Reverse Percentages Calculator 2

Find Original Number

This calculator will help you to find the original number, when you have been given the percentage increase or decrease and the final number.

Example: if a toy in a sale marked 20% off costs $210, what is the original price?

• Type 20 for the percentage.
• Select 'Percentage Decrease' because the items is 20% less in the sale.
• Type 210 into the Final Number box.
• Click 'Find Original Number'.
• Answer: 262.5. So the original price is $262.50

Example: Tyger jumps 320 cm which is 15% further than Sally. How far did Sally jump to the nearest cm?

• Type 15 for the percentage.
• Select 'Percentage Increase' as the jump was 15% more.
• Type 320 into the Final Number box.
• Answer: 278.26. So Sally jumped 278 cm to the nearest cm

What is a Reverse Percentage?

Reverse percentages are also called inverse percentages.

When you have a reverse percentage it means that you are given a final amount and a percentage (which could be a straightforward percentage or a percentage increase or decrease).

With reverse percentages you need to find the original value by working backwards from the final value and the given percentage.

Reverse Percentage Calculators Video

We have created a short video to show you all about reverse percentages.

In the video, you will see:

• what a reverse percentage is
• how to spot a reverse percentage problem
• which calculator you should use to solve the problem
• a worked example showing how to solve the problem without a calculator

How to find Reverse Percentages

Read on below if you want to find out how to solve reverse percentage questions without the calculators.

Reverse percentages are used when the percentage and the final number are given, and the original number needs to be found.

Finding the Reverse Percentage of a number in 3 easy steps.

Step 1) Get the percentage of the original number.

If the percentage is an increase then add it to 100, if it is a decrease then subtract it from 100.

• Example: if the percentage is an 18% increase, then the percentage is 100 + 18 = 118%
• Example: if the percentage is a 37% decrease, then the percentage is 100 - 37 = 63%

Step 2) Find 1% of the missing number by dividing the final number by the percentage from Step 1)

Step 3) Find 100% of the missing number by multiplying the result from Step 2) by 100.

Examples of Reverse Percentages

Example 1) 35% of a number is 320. what is the number.

Step 1) The percentage of the original value is 35%

Our percentage equation is 35% of ? = 320

Step 2) So 1% of ? = 320 ÷ 35 = 9.1429 (to 4dp).

Steo 3) 100% of ? = 9.1429 x 100 = 914.29 (to 2dp)

Answer: the original value was 914.29 to 2dp.

You can use calculator 1 to solve this problem.

Example 2) A car is reduced by 20% in price to $40,000. What was the original price?

Step 1) The percentage of the original number is 100% - 20% = 80%

Our percentage equation is 80% of ? = $40,000

Step 2) So 1% of ? = $40,000 ÷ 80 = $500

Step 3) 100% of ? = $500 x 100 = $50,000

Answer: the original car cost $50,000

You can use calculator 2 to solve this problem.

Example 3) Sally invests money in some shares. Five years later, she sells them for $7200 at a profit of 25% of their original value. How much did she spend on the shares?

Step 1) The percentage of the original number is 100% + 25% = 125%

Our percentage equation is 125% of ? = $7200

Step 2) So 1% of ? = $7200 ÷ 125 = $57.60

Step 3) 100% of ? = $57.60 x 100 = $5760

Answer: the amount she spent on shares was $5760.

Example 4) Tyger buys a mountain bike. Three years later, he sells it for $675 which is 45% of its original value. How much did he pay for the bike originally?

Step 1) The percentage of the original number is 45%

Our percentage equation is 45% of ? = $675

Step 2) So 1% of ? = $675 ÷ 45 = $15

Step 3) 100% of ? = $15 x 100 = $1500

Answer: the amount he paid for the bike was $1500.

More Recommended Math Resources

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Online Percentage Practice Zone

Our online percentage practice zone gives you a chance to practice finding percentages of a range of numbers.

You can choose your level of difficulty and test yourself with immediate feedback!

• Online Percentage Practice
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This is the calculator to use if you want to find a percentage of a number.

Simple choose your number and the percentage and the calculator will do the rest.

Find the Percentage Increase Calculator

If you need to find the percentage increase/decrease between 2 numbers, then this calculator is the one you need.

Simple input the original number, then the new number and the calculator will tell you the percentage increase or decrease.

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Converting Percentage Calculator

If you need to convert a percentage to a fraction - either a decimal fraction or a fraction in its simplest form, then use this calculator.

You can also use the fraction to percentage calculator for converting any fraction into a percentage.

This calculator takes a percentage and converts it at the click of a button.

• Convert Percent to Fraction

• Converting Fractions to Percentages

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Online mathematics book, search radford mathematics, how to calculate reverse percentages, (reverse percentage increases & decreases).

We're faced with a Reverse Percentage problem as soon as we are given the value of a number following either a percentage increase or a percentage decrease and we're asked to find what the number was, its initial value , before the increase (or decrease) took place .

Here's a typical reverse percentage question :

Due to high demand, a pair of shoes' price was increased by \(20\%\) and they now cost \(\$ \ 96\). How much did the shoes cost before their price was increased?

Reverse Percentage Increase or Decrease?

The example, above, is a typical reverse percentage increase question . Had the price of the pair of shoes been decreased by \(20\% \) it would have been a reverse percentage increase question .

In the following two tutorials we learn how to solve such problems. Watch them before trying to work through the questions further down.

Tutorial 1: Reverse Percentage Increase

In the following tutorial we work through an example in which we solve a reverse percentage increase problem.

Tutorial 2: Reverse Percentage Decrease

In the following tutorial we work through an example in which we solve a reverse percentage decrease problem.

• Following a \(20\%\) increase in price, a pair of shoes costs \(\$ \ 96\). How much did the shoes cost before their price was increased?
• Following a \(25\% \) decrease in price, Clara's gym class cost \(\$ \ 90\). How much did her gym class used to cost?
• A few years ago Cathy decided to invest in property and bought an apartment. She has been very fortunate and its value has gone up by \(24\%\) and is now worth \(\$ \ 198\ 400\). How much was Cathy's apartment worth when she bought it?
• A clothes shop puts everything on sale with a \(30\%\) discount. After looking around for a while, Charlotte buys a dress for \(\$ \ 84\). How much would she have had to pay if there hadn't been a discount?
• Next month, the price of Benjamin's monthly bus ticket will increase by \(6\%\) and will cost \(\$ \ 42.40\). How much does Benjamin's ticket currently cost?
• Following a low sugar diet, as well as regular exercise, John reduced his weight by \(8\% \) and now weighs \(72\) kg. Rounding your answer to the nearest kg, how much did John weigh before his diet?
• Last fall, a highly selective university offered \(6\% \) of its applicants a place. Given that the university accepted \(540\) students, find how many students applied.
• From grade 8 to grade 9, the number of hours spent studying at home is expected to increase by \(30\%\) to \(14\) hours a week. Rounding your answer to the nearest hour, find how many hours per week students are expected to study in grade 8.

Exercise 1 - Answer Key

• If we call the initial price \(P\), we find: \[P \times 1.2 = 96\] which leads to \(P = \frac{96}{1.2} = 80\) the shoes cost \(\$ \ 80\).
• If we call the initial price \(P\), we find: \[P \times 0.75 = 90\] which leads to \(P = \frac{90}{0.75} = 120\) Clara's gym classes used to cost \(\$ \ 120\).
• If we call the initial value \(V\), we find: \[V \times 1.24 = 198 \ 400\] which leads to \(V = \frac{198 \ 400}{1.24} = 160\ 000\) Cathy's apartment was initially worth \(\$ \ 160\ 000\).
• If we call the dress' initial price \(P\), we find: \[P \times 0.7 = 84\] which leads to \(P = \frac{84}{0.7} = 120\) Charlotte would have had to pay \(\$ \ 40.00\).
• If we call the ticket's current (initial) price \(P\), we find: \[P \times 1.06 = 42.4\] which leads to \(P = \frac{42.4}{1.06} = 40\) Benjamin's ticket currently costs \(\$ \ 40.00\).
• If we call John's initial weight \(W\), we find: \[W\times 0.92 = 72\] which leads to \(W = \frac{72}{0.92} = 84.783\) which, rounded to the nearest kg, corresponds to \(85\)kg.
• If we call the number of applicants \(A\), we find: \[A \times 0.06 = 540\] which leads to \(A = \frac{540}{0.06} = 9000\) last fall \(9\ 000\) students applied to this university.

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Reverse Percentage

The following diagram shows some examples using reverse percentage. Scroll down the page for more examples and solutions on using the reverse percentage.

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Reverse Percentages

Here we will learn about reverse percentages including how to work backwards to find an original amount given a percentage of that amount or a percentage increase/decrease.

There are also reverse percentages worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are reverse percentages?

Reverse percentages (or inverse percentages) means working backwards to find an original amount, given a percentage of that amount.

• We can do this using a calculator by taking the percentage we have been given, dividing to find 1% and then multiplying by 100 to find 100% .
• We can also do this without a calculator by using factors of the percentage we have been given.
• Sometimes we are given a percentage of an amount and we need to work out what the original value was.

We need to remember that the original amount is 100% of the value.

How to use reverse percentages given a percentage of an amount (calculator method)

In order to find the original amount given a percentage of the amount (using a calculator):

Write down the percentage and put it equal to the amount you have been given.

• Divide both sides by the percentage. (e.g. if you have 80% , divide both by 80 ). This will give you 1% .
• Multiply both sides by 100 . This will give you 100% .

Explain how to find the original amount given a percentage of the amount in 3 steps

Reverse percentages worksheet

Get your free reverse percentages worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Reverse percentage examples (calculator method)

Example 1: calculator.

45% of a number is 36 . Find the original number.

Put the percentage equal to the amount.

2 Divide both sides by the percentage to find 1% .

In this case the percentage is 45%, so divide by 45.

3 Multiply by 100 to find 100% .

The original number was 80 .

Example 2: calculator

150% of a number is 690 . Calculate the original number.

Divide both sides by the percentage to find 1%. In this case the percentage is 150%, so divide by 150.

Multiply by 100 to find 100%.

The original number was 460 .

How to use reverse percentages given a percentage of an amount (non-calculator method)

In a situation where we do not have a calculator, we can often simplify the problem by using common factors. Rather than finding 1% , which might involve a difficult division, we could find 10%, 25%, or any other percentage which is a factor of 100% .

• Identify a common factor of the percentage and 100% (a number which goes in to both).
• Use division to find that percentage of your amount.
•  Use multiplication to find 100% .

Reverse percentages examples (non-calculator method)

Example 3: non-calculator.

70% of an amount is 56 . Find the original amount.

Identify a common factor of 70% and 100%.

Factors of 70: 1 , 2 , 5 , 7, 10 , 14, 35, 70

Factors of 100: 1 , 2 , 4, 5 , 10 , 20, 25, 50, 100

Here 1, 2, 5 and 10 are all factors of both 70 and 100 . We could use any of these but select the one which results in the easiest calculations (usually the highest common factor). In this case we are going to use 10% .

As 10% is a factor of both 70% and 100%, we need to find 10% of our amount. To do this we will divide by 7 because 70% ÷ 7 = 10%.

As we now have 10%, we need to multiply by 10 to find 100%.

The original amount was 80 .

Note: In this example, and every example, the method of finding % would also work. The reason that we found 10% here rather than 1% is that 56 ÷ 7 is easier to work out without a calculator than 56 ÷ 70 .

Example 4: non-calculator

125% of a number is 350 . Find the original value.

Identify a common factor of 125% and 100%.

Factors of 125: 1, 5 , 25 , 125

Factors of 100: 1, 2, 4, 5 , 10, 20, 25 , 50, 100

Here we are going to use 25% .

As 25 is a factor of both 125% and 100%, we need to find 25% of our amount. To do this we divide by 5 because 125% ÷ 5 = 25%.

As we now have 25%, we need to multiply by 4 to find 100%.

The original amount was 280.

How to use reverse percentages given a percentage increase/decrease

Sometimes, instead of being told a percentage of the amount, we are told what percentage increase or decrease has occurred. The only difference here compared to what we have already looked at is that we first need to identify what percentage of the original amount we now have.

• Identify what percentage of the original amount you now have. If it has been increased by a percentage, add that percentage onto 100% . If it has been decreased by a percentage, subtract that percentage from 100% .
• Use either the calculator or non-calculator method to find 100% .

Reverse percentages examples (percentage increase/decrease)

Example 5: percentage increase, calculator.

The number of fans attending a football match this week was 12% more than last week. If 728 people attended the match this week, how many attended last week?

This is a percentage increase of 12%.

100% + 12% = 112%

This is a calculator question, so use the method of finding 1%.

The number of fans last week was 650.

Example 6: percentage decrease, calculator

The value of a car has decreased by 16.5% in the last year. If its value now is £5845 , find its original price.

This is a percentage decrease of 16.5%.

100% – 16.5% = 83.5 %

83.5% = 5845

Use a calculator to find 1%.

The original price was £7000 .

Example 7: percentage increase, non-calculator

A puppy’s weight has increased by 20% to 4.8kg . What was the puppy’s weight before the increase?

This is a percentage increase of 20%.

100% + 20% = 120%

This is a non-calculator question, so use the common factor method.

Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20 , 24, 30, 40, 60, 120

Factors of 100: 1, 2, 4, 5, 10, 20 , 25, 50, 100

There are several common factors here. It would be easiest to use 10% or 20% . Here we are going to use 20% .

The puppy’s original weight was 4kg .

Example 8: percentage decrease, non-calculator

A television is in a 10% sale. The sale price of the television is £450 . Find the original price of the television.

This is a percentage decrease of 10%.

100% – 10% = 90 %

Factors of 90: 1, 2, 3, 5, 6, 9, 10 , 15, 30, 45, 90

Factors of 100: 1, 2, 4, 5, 10 , 20, 25, 50, 100

Here we are going to use 10% .

The original price of the television was £500 .

Common misconceptions

• Calculating a percentage and adding it on

A common mistake is to work out the percentage of the number and then add it on. E.g.  Given 70% of a number, a common error is to calculate 30% of that number to add on to the 70% . Remember, this does not work as 30% of 70% would not be the same as 30% of the original value.

• Not adding/subtracting from 100% when it is a percentage increase/decrease

A common mistake is to use the actual percentage increase/decrease rather than adding/subtracting from 100% . E.g If you are told it is a 30% decrease, a common error would be to use 30% instead of 70% .

Related lessons

Reverse percentages is part of our series of lessons to support revision on percentages. You may find it helpful to start with the main percentages lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Percentages
• Percentage increase
• Percentage decrease
• Percentage of an amount
• Percentage multipliers
• Percentage change
• One number as a percentage of another
• Percentage profit

Practice reverse percentages questions

You may use a calculator for questions 1, 2 and 5.

1. 36\% of a number is 324 . Find the original number.

Dividing both sides by 36 gives

Multiplying both sides by 100 gives

2. 145\% of a number is 2900 . Find the original number.

Dividing both sides by 145 gives

3. 60\% of a number is 210 . Find the original number.

Dividing both sides by 3 gives

Multiplying both sides by 5 gives

4. 150\% of a number is 33 . Find the original number.

Multiplying both sides by 2 gives

5. The price of a car is reduced by 15\% . The reduced price is £ 6800 . Find the original price.

The price has been reduced by 15\% , so £ 6800 is equal to 85\% of the original price.

Dividing both sides by 85 gives

6. The number of customers who visited a shop today was 10\% higher than the number who visited yesterday. Today 231 customers visited the shop. How many customers visited the shop yesterday?

The number of customers has increased by 10\% , so 231 is equal to 110\% of the number from yesterday.

Dividing both sides by 11 gives

Multiplying both sides by 10 gives

Reverse percentages GCSE questions:

1. 40\% of the children in Rahim’s class walk to school.

12 children walk to school.

How many children are in Rahim’s class?

    40\% = 12

            (1)

    10\% = 3

    100\% = 30

2. In a sale, prices are reduced by 15\% .

A phone is reduced by \pounds 36 .

Find the original price of the phone.

    15\% = \pounds 36

    5\% = \pounds 12

    100\% = \pounds 240

3. Tony receives a pay increase of 12\% .

His new salary is \pounds 31920 per annum.

Calculate how much more money he earns each year following the pay increase.

    112\% = \pounds 31920

    1\% = \pounds 285

    100\% = \pounds 28500

    \pounds 31920 − \pounds 28500 = \pounds 3420

Learning checklist

You have now learned how to:

• Find an amount given a percentage of that amount (calculator and non-calculator)
• Calculate reverse percentages involving percentage increase/decrease

The next lessons are

• Simple interest
• Compound interest
• Fractions, decimals and percentages

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Compound & reverse percentage questions

Subject: Mathematics

Age range: 14-16

Resource type: Worksheet/Activity

Last updated

2 October 2020

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nazish_khaalid

Empty reply does not make any sense for the end user

cornishhalfpint

Brilliant! With answers, thanks<br /> <br />

lynneinjapan

Thanks, glad you like it.

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