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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.

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.

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.

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.

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.

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

My name is Jonathan Robinson and I 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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