1.3 Going down, going up
In the next activity you will explore whether the order in which you carry out percentage increases and decreases matters, or whether you can apply them either way round. One practical situation in which you might come across this is when shopping.
Activity 5 Going down, going up
You have set your eyes on a new dresser for your room, which is on sale with a 30 per cent discount. But does it matter if the discount is applied first, and then VAT, or if the VAT is applied before the discount? If the cost of the dresser is £100 before the discount and VAT are applied, work out the cost of dresser including both, first starting with the discount and then with the VAT. Take the current value of VAT as 20 per cent.
Discount first, then VAT
Discount on dresser = 30% of £100 = 0.3 × £100 = £30
So price including discount = £100 – £30 = £70
Now add the VAT onto this discounted price.
VAT = 20% of £70 = 0.2 × £70 = £14.00
So the final price to pay = £70 + £14 = £84
VAT first, then discount
VAT on original price = 20% of £100 = 0.2 × £100 = £20
So price including VAT = £100 + £20 = £120
Discount on price = 30% of £120 = 0.3 × £120 = £36
So final price = £120 – £36 = £84
Both prices are the same!
It does not seem to matter whether the discount or the VAT is applied first. But can you be sure that this is always true from exploring one example? Doing more numerical examples would confirm it. However, these could just be lucky choices of numbers. What you need to do is analyse what you can tell about the maths being used by looking at these particular discount and VAT rates.
The original cost of the dresser is 100 per cent of the price – that is, the whole price. You know that the discount is 30 per cent, so the price after the discount will be 100% – 30% = 70% of the original cost. You can find the discounted price by multiplying the cost by 0.7, since 0.7 is 70% as a decimal.
To calculate the price including VAT, you need to find 20 per cent of the price and add this onto the price (100 per cent). To do this in one step, you can work out 100% + 20% of the price, which is the same as 120 per cent of the price (or 1.2 as a decimal).
So if the discount is applied first and then the VAT, it’s 120 per cent of 70 per cent of the original price and can be expressed as 1.2 × 0.7 × original price.
If the VAT is applied first and then the discount, it’s 70 per cent of 120 per cent of the original price, and can be expressed as 0.7 × 1.2 × original price.
In Week 2 you learned that multiplication is commutative, which means it doesn’t matter what order you carry out multiplication operations: the answer will be the same. Hence by looking in more detail at the maths, you’ve seen that this idea will work for any type of combined percentage increase and decrease.
A situation that can be a little trickier to understand is how to undo a percentage change that has been added to or subtracted from a number. Again, you’re going to use discounts and VAT to help with this.