# EdPlace's Year 9 home learning maths lesson: Reverse Percentages

Looking for short lessons to keep your child engaged and learning? Our experienced team of teachers have created English, maths and science lessons for the home, so your child can learn no matter where they are.  And, as all activities are self-marked, you really can encourage your child to be an independent learner.

Get them started on the lesson below and then jump into our teacher-created activities to practice what they've learnt. We've recommended five to ensure they feel secure in their knowledge - 5-a-day helps keeps the learning loss at bay (or so we think!).

Are they keen to start practising straight away? Head to the bottom of the page to find the activities.

Now...onto the lesson!

Key Stage 3 statutory requirements for maths
Year 9 students should be able to express one quantity as a percentage of another

# Increasing their interest at a quick rate

Reverse percentages are a common problem that will catch students out as there are some misconceptions around this topic. This step by step guide explains how to tackle these types of question and avoid slipping up.

We'd bet that if you follow our step by step approach your child will:

1) Understand what is meant by reverse percentage

2) Describe how to turn a percentage into a ratio

3) Apply their understanding to a typical question with success!

## Step 1 - What is a reverse percentage?

Before we get going, it's helpful to put this topic into a context your child can relate to.  So, what does a reverse percentage mean?  A reverse percentage is a way of doing a percentage problem backwards.  Think about the ever-popular Black Friday Sales and being drawn to the numerous sales signs... we've all been there!  Items in this sale will proudly display a price (final) and a sticker often detailing the percentage that has been deducted.  While these sales often do it for us, to draw us in, a reverse percentage can be used to work out the original price before the item was discounted, should this not be made clear.

## Step 2 - A typical question looks like...

We will begin by looking at a typical “reverse percentage” problem and understanding what to help children avoid doing.

A sale offers 20% off all prices. You pay £160 for a jacket. How much was the jacket before the sale?

A very common mistake that students make is to find 20% and just taking it away from £160. Unfortunately, this does not give us the correct answer. It's important to encourage children to think more carefully about this type of problem.

## Step 3 - Where to begin?

The key to solving this question is to work backwards. When we're given a percentage reduction the first thing we need to work out is the percentage that is left.

The question states: In a sale, it is 20% off all prices.

Working backwards, 100% - 20% = 80%

The easiest way to understand it is this: if prices are 20% off, then a customer is only paying for 80% of the original price.

## Step 4 - Creating a Ratio

Once we've worked out the percentage,  the next thing to do is set it up as a ratio.  In a sale, it's 20% off all prices. The final cost of the jacket is £160.

We know that we're only paying for 80%, so £160 must represent this amount.  We encourage students to write this as the ratio % : £.  We always write into a ratio the information that we have so we could also write this like 80% : £160

Once we have the ratio set up we've almost finished the question.  It's important at this point, to encourage your child to re-visit the wording of the question.  The question asked for the price of the jacket before the sale. In other words, we need to find the full original amount, which we know is 100%. We can use the ratio to find this percentage and the easiest thing to do here is to divide both sides by the percentage to find 1%, then multiply both sides by 100 to find 100%.   Why not get your child to write it out in the following format to ensure they don't miss out any steps in the process?

And there we have it... the jacket cost £200 before the sale.

Bonus work

If your child has grasped this easily, why not push this topic on one stage further?  Some students may recognise that a similar principle could work if we had a percentage increase, and they'd be right!  For example, we might be told that a price has increased by a given percentage and we need to find the price before the increase.

Ticket prices rise by 15% to £46. How much did they cost before the rise?

• An increase of 15% means we are paying for 115%.

• £46 = 115%

• As a ratio…

So, the tickets cost £40 before the price rise.

## Step 5 - Give it a go...

Why not see if your child has really grasped this topic by completing the questions below together?  Or, you could get them to teach you the topic, that will really show how well they've understood it!

(a) In a sale, prices are 25% off. You pay £450 for a computer. How much did it cost before the sale?

(b) In a sale, prices are 10% off. You pay £46.80 for a game. How much did it cost before the sale?

(c) You are given a 5% pay rise at work. You now earn £1,575 per month. How much were you earning before your raise?

(d) Rent increases by 1.5% to £558.25. How much was the rent before the increase?

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(a) £600
£    : %
450 : 75
6    : 1
600  :100

(b) £52
£     : %
46.80 : 90
0.52  : 1
52    : 100

(c) £1,500
£    :  %
1575 : 105
15    :  1
1500  : 100

(d) £550
£        :   %
558.25   :  101.5
5.5       :  1
550      : 100