## Tuesday, November 25, 2008

### A lesson in percentages

Subsequent to GMTV's inability to calculate VAT I found someone still struggling here's the conversation which I'll follow with what is hopefully some simple explanations

"How do you calculate VAT? Say I had something that I bought for £100 how do I do it?"
"If you want the base price divide by 1.175"
"About £85. Now how do I work out what it would be if it was 15%?"
"Um divide by 1.15, but what are you trying to do?"
"£87. So there's only £2 difference"
"What are you trying to do?
"Work out the price of something when the VAT is 15%"
"Well you've got to start with the base price that doesn't change so divide by 1.175"
"£85"
"Then multiply by 1.15"
"£98"
"So that's how much it will be at 15%"

Explanation time

"per cent" means divide by 100 so 15% can be written as 15/100. So to determine what 15% of something is you multiply it by 15/100 that is you can multiply by 15 then divide by 100 or divide by 100 then multiply by 15 it doesn't matter which.

So 100% of something just means 100/100 which you should spot is just 1 so to determine what 100% of something is you just multiply it by 1 - duh!

So now for the tricky part taking something and adding 15% to it. I'll go through it step by step-

First off we want to find 15% of our amount which we'll call x so as above that's

15x /100 or x/100 * 15, but the former is easier to write.

Now we want to add it on to 100% of the amount x, so that's

100x/100 + 15x/100

Now when you have the same denominator (the bit below the line) you can simply add the numerator (the bit above the line) to reach

115x/100

So we can just multiply by 115 then divide by 100 or 115/100; as with the 100/100 equals 1 we can rewrite this as a decimal 1.15.

For 17.5% that's 17.5/100 or 0.175 to add it to the whole 1 it makes 1.175.

That works in reverse too so for something you've bought at £9.99 inc VAT@17.5% you can work out what the original price would be by simply dividing by 1.175. To find out what it would be with VAT@15% take your new figure and multiply by 1.15.

That's for adding on percentages that already exist. How do you subtract with say an item at £24.99 with 22% off; what's the price?

Well it's just as you've already done just with a minus figure. Take the full amount and subtract the percentage

100x/100 - 22x/100 or 78/100x

78/100 is the same as 0.78 so simply take the £24.99 and multiply it by 0.78 to get £19.49.

Important note

What happens if you take 22% from £24.99 to get £19.49 then add 22% back on? You get £23.78. Why haven't we got the full £24.99 back? Because we're adding on 22% of £19.49 and not 22% of £24.99. Unless we know that £19.49 is the reduced price and how much it was reduced by we can't get back to the original amount.

Extra Credit

If we do know that £19.49 represents a 22% reduction then we can use the above formula to realise that this price is 78% of the original (remember that's what per cent means and lookee there at the 78/100 we used). So to get back to the original price we can invert it, that is instead of multiplying by 78 and dividing by 100 we can multiply by 100 then divide by 78.

100x/78 or 100*19.49 / 78

Result: £24.99.

Extra Extra Credit

So that's all fine and dandy but what if you've got your £24.99 and want to work out just the VAT@17.5%? Simple - work out the base price (divide by 1.175) and then get the VAT (multiply by 0.175)

24.99/1.175 * 0.175

Result: £3.72

The two most important things to remember are that per cent simply means divide by 100 and that if you subtract or add 15% to something you can't simply add or subtract 15% to get back where you started from.

Anonymous said...

So, the super duper extra credit question: why doesn't everyone know this already? It's pretty simple arithmetic. I was at primary school when people were bemoaning that the introduction of electronic calculators meant kids didn't learn any mental arithmetic, and I learned about percentages then. (Actually, I learned about them from my parents, who ran a business, long before I did in school. But they were covered in school.) So how is it that the people who weren't 'handicapped' by calculators know nothing?

FlipC said...

Like so much it's only easy if you know what you're trying to do.

Unlike other basic operators percentages do behave oddly hence my NB about +15% != -15%. When it's laid out it's easy (I hope) to understand why this is true, but in this instance the operator function gets in the way of that.

Take the simple equation

100 + 30% = 130

Basic algebra teaches us that

100 = 130 - 30%

Which is of course wrong. Write out the full equation

100 + (30/100)*100 = 130, and
100 = 130 - (30/100)*100

which is correct and hopefully the error becomes apparent.

Still it's scary that I can overhear comments along the lines of '10% off? How much is that then?'