Calculating a Reverse Percentage

Questions that involve calculating a reverse percentage are difficult for two reasons: students do not always understand they are required to work out the original amount after a percentage change and the written method involves multiple lines of working which, without a clear writing frame, can be confusing.

Here are two examples from exam papers on calculating a reverse percentage.

Question 1

The normal price of a television is reduced by 20% in a sale.

The sale price of the television is £360

Work out the normal price of the television.

The common misconception was to incorrectly add 20% on to the sale price so £432 is seen as an incorrect answer.

Question 2

Anita buys a laptop.

20% VAT is added to the price of the laptop.
Anita then has to pay a total of £400.

What is the price of the laptop with no VAT added?

The most common mistake was to use £400 as 100% instead of 120%, with students working out 20% of 400 and subtracting to get £320.

Some thought that as £400 was 120% they had to find 80% of £400 to get back to the original value.

100% represents the original value.

To overcome these difficulties students, need to understand the original amount, before any percentage change, is represented as 100% .  They also need to have a clear model that draws on prior learning to break down the problem.

Calculating a Percentage Change

Calculating a Reverse Percentage
Reviewing the multiplier to calculate a percentage change.

To address these misconceptions, the lesson starts by reviewing how to calculate an amount after a percentage change using a multiplier.  This is because, to calculate the multiplier students must have started at 100% as the original value.  I ask students to work on whiteboards so I can check this when feeding back.

Using ratios to model reverse percentages

To calculate the original amount after a percentage change I model the percentage and amount using equivalent ratios.  As you can see in this video.

If students needed more practice I use the following Interactive Excel File to randomly generate more questions and solutions, which you can download by clicking on the image.

Calculating a Reverse Percentage

Real Life Reverse Percentage Problems

When the class can model calculating a reverse percentage we move on to solving more worded, real-life problems. These are included in the Interactive Excel File.

Calculating a Reverse Percentage
Real-life problems with calculating a reverse percentage.

Later, as learning progresses students work independently through the questions on the third slide and then through the worksheet.

Calculating a Reverse Percentage

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About Mr Mathematics

My name is Jonathan Robinson and I am passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

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