Imagine that our friend bought us a birthday present in the sale, and we want to know how much the item was worth before the sale.

Or we may have missed out on a promotional offer on Black Friday, and want to know how much the price of an item has increased by.

Finding the original amount, before a percentage change took place, is also known as reverse percentages.

The key word here is reverse - remembering this is going to help us throughout this activity.

Let's look at reverse percentages in action now in some examples.

**e.g. A radio sells for £63, after a 40% increase in the cost price.**

**What was the original cost price? **

We know that the radio sells for £63 **after** a 40% increase, so we can write this as:

c × 1.4 = 63 (where 'c' represents the original cost)

If we solve this using normal algebraic rules, we find that:

c = 63 ÷ 1.40 = **£45**

**e.g. A packet of washing powder is advertised as having an extra 20% powder.**

**If it contains 1.32 kg, what amount would be in the original pack?**

p × 1.20 = 1.32 kg

p = 1.32 ÷ 1.20 =** 1.1 kg**

**e.g. A television has been reduced by 8% in a sale.**

**It now costs £586.**

**What was the cost of the television before the reduction in price?**

This time, as we need to find a **reduction** rather than an increase, we need to take 8% **away from 100%** to find our multiplier:

100 - 8 = 92% or × 0.92

Now we need to apply the **inverse** (opposite) operation to an increase, like this:

£586 ÷ 0.92 = **£636.96**

**Let us extend this a little more now...**

You could see questions that don't appear as straightforward as these examples, and may require a different approach.

Here's an example to illustrate what we mean.

**e.g. At a hockey game, there are 60 children present.**

**This is 20% of the total amount of people at the game.**

**How many people are present at the game in total?**

Think about what we know here:

20% = 60 children and 20% × 5 = 100%

So if we multiply 20% by 5, we reach the total amount in this case:

60 × 5 = **300 people**

**e.g. In a shoe factory, 33% of the shoes are leather and the remaining 4355 are suede.**

**How many shoes are leather?**

As we cannot easily turn 33% into 100% (as we did in the previous example), we have to adopt a different approach here.

Think about what we know again.

If 33% of the shoes are leather, the suede shoes **must **make up the remaining 67%, so:

4355 = 67%

To make life easier, we want to find **1%**:

4355 ÷ 67 = 65

If 1% = 65, then...

33% = 65 × 33 = **2145 shoes**

In this activity, we will apply reverse percentages to find starting amounts before a percentage increase or decrease has occurred, using the methods shown above.