Some people find percentages difficult. I have long thought of this as a reading problem more than one of reasoning. What follows here is based upon the very many conversations in lessons with people who find this a difficult topic.Many of the people who ‘can’t do percentages’ are the same people who will say things like ‘it’s got numbers in it; I can’t do numbers’. Yet these same people have no problem divining the emotional content of a poem, which seems to me to be infinitely harder.

• The percentage symbol means “out of 100”

• The fundamental question to answer is “What is 100%?”

• Check your result for sense.

1. Your pay of £22,000 per year is to go up by 3%. What is the increase in money terms?

It seems to me obvious that the £22,000 IS 100%. Then 3% of 100% is 3/100 x £22000 = £660.

Check your result for sense. Is £660 a credible size? would £66 or £6600 be better?

You want a small number as a result. 3% of £22000 should be fairly small. £660 is the right size.

2. 17.5% VAT is included in a price of £258.50. What is the price without VAT?

Not so easy. VAT has been added, so the £258.50 figure is 117.5%, 100% plus VAT. The price without VAT is the 100% you want to know. So the number to find is smaller than £258.50, but not by an awful lot. We have two percentages and one price. If we multiply by one % and divide by the other, the % element will cancel out, so the solution will be the result of £258.50 x 100/117.5 = £220.

Sensible? Well, it is a nice round answer, which you’ld expect from an early example. We could check it by adding back 17.5% of £220 to see what we get: £220 x 117.5/100 = 258.50. Correct.

People who write test questions like using VAT for two reasons. (i) It is one of the more likely events where you would want the skill in later life (ii) it uses ‘nasty’ numbers so you probably need a calculator.

If you are working with VAT and running a shop of some sort, then you will be claiming back some VAT from the taxman. You buy products to sell at one figure, you sell these same things for, hopefully, a little more money. You claim back the VAT you paid in your purchasing. So what you often want to know is how much of what you paid out was VAT. Some suppliers will tell you prices ‘ex VAT’ and ‘incl VAT’ [ex VAT excludes it, incl VAT includes it, so is the bigger number; some suppliers will make their products ‘look’ cheaper by only quoting ex VAT figures, which then sometimes catches you out at the point of paying for an order. So if you’re not the end user, you really need to know about this.

3. Your total order comes to £1200 ex VAT. How much will you pay?

What is the VAT charged on? £1200. So what is the 100%? Again, £1200. So how much VAT do I add on? 17.5%. So what number do we want to know? 117.5%, the new bigger figure. So we need to work out £1200 x 117.5/100 = £1200 x 1.175 = £1410.

Notice that I leave in the money symbol. With students who find this hard I might also leave in the % symbol, so writing £1200 x 117.5% / 100% = £1200 x 1.175 = £1410. That allows me to point out that the % signs ‘cancel out’, making sure I have a money answer as I want.

Many shopkeepers would rather know what the VAT was to be added; they will do two calculations only because they actually have two questions:

4 How much VAT do I add? What is the new total?

17.5% of £1200 is 1200 x 17.5%/100% = £1200x0.175 = £210. £1200 plus £210 = £1410.

So this way the £210 which will eventually be recovered is identified. I don’t think this helps particularly, since that assumes you succeed in selling all of what you buy. I have seen a large fraction of classes insist on doing this two-part calculation; when I push them into an answer as to why, the honest response is that this is what they understand. They’re saying that the Incl VAt figure is somehow ‘too hard’.

VAT is charged by you the shopkeeper on your selling price. It is fairly typical to add 40% to your purchase price (your cost) to cover your overheads of staff, rent, heat & light, advertising, etc. Which number (with or without VAT) you choose to start from is something you would make a ‘management decision’ about. I’d suggest that you use the bigger one, since you have to pay the VAT initially anyway. If you discover you can drop your overhead figure to say 30%, this is a subsequent decision.

5 What is my selling price for the money spent above? How much of that will be VAT?

This is an extra-difficult problem because there is at least one intermediate answer, depending on how you approach the problem.

One way of looking at this is to take what you spent, the incl. VAT figure of £1410, and add 40%. Which means you want to know what 140% of £1410 is. £1410 x 140% / 100% = £1974. If you sold all your stock at that mark-up your gross take woudl be £1974, VAT included.

How much of this is VAT? Where is the 100% now? It moved; £1974 is the ‘incl VAT’ number, so must be 100+17.5% = 117.5%. So the VAT included is 17.5% of the 117.5% (and is classed as a ‘hard’ problem). So the VAT included is £1974 x 17.5% / 117.5% = £294. Unfair? Your contribution to the VAT system is that you collect the tax on what you sell, but recover the tax you actually paid (at the time you bought your stock).

Another way to do this is to work with the ex VAT figures. You bought ex VAT at £1200 , so the ex VAT selling price is £1200 x 140/100 = £1680. The VAT to add to this is 17.5%, but on the new figure, so the VAT to add is 17.5% of £1680 or £1680 x 17.5% / 100% = £1680 x 0.175 = £294.

And £1680 + £294 = £1974. So this checks with your other numbers.

6. In practice, you probably bought a box of things for your original £1200. Let’s assume it was two dozen items each of £50. Let’s hope you succeeded in selling twenty at your full price and then the last four sold at less, say with reductions of 10%, 20% 30% and 40%. I’m picking easy numbers, but in practice you’d be choosing sale numbers that fit with your perception of what the customer will spend. We’ll come back to that.

What was your total income from these 24 items?

You sell the first 20 items at £50 plus 40% plus VAT; 20 x 50 x 140%/100% x 117.5% / 100%

= £1400 X 1.175 = £1645 or if you prefer, at £70 ex-VAT each, £82,25 including VAT.

The last four, you can take the discount off either figure, before or after VAT. But the 100% for the discount is where you start from, so these four sell at 90%, 80%, 70% and 60% of the previous sale figure. With the VAT included that would be £74.03, £65.80, £57,57 and £49.35. We could work these out separately and add them up (as a shopkeeper might), or we could add up the percentages (90+80+70+60 = 300%) and simply say that we sell the last four for the previous figures for three. Which would mean that the income on the last four is 3x£82.25 = £246.75 so the total income for all 24 can be written as £1645 + £246.75 0r as 23 x 50 x 1.4 x 1.175, or as £1645 x23/20. All of which = £1891.75. I repeat, your customers are paying the VAT so if you drop prices so the VAT drops in proportion. You can’t sell something without the VAT (well you can, but it is costing you personally). You send a load of money off to the taxman and he sends you your bit (in practice, you’ll send off the difference of course, but can imagine how very sharp the taxman is over numbers being right).

7 In practice of course, these four reduced prices would be something more like £75, £65, £55 and £50, or a penny or a pound less than each figure. One might simply say that of any 24, recovering the value of 23 is good going. How much profit did you make on these two dozen items?

How much did you spend? Was it £1974? £1645? £1410? £1200 ?Just “One of those other numbers up the page”? Well you spent £1410 on the original bill to your supplier. You seem to have recovered £1891.75. That says you made a profit of the difference, £481.75. As a percentage, then the 100% here is what you spent, and the profit is £481.75 / £1410 x 100% = 34.2%, call that 34% profit. Note how this time we want a % answer, so we make the money £ cancel out. We could do a simpler calculation by saying the recovery is £1891.75 / £1410 x 100% = 134.2%, so 34% profit.

But that forgets the VAT recovered. The VAT man receives the difference between what you charged as VAT and what you paid as VAT. The ‘charged’ figure was the same as 23 items at your full price, 23 x (£82.25 - £70) = £281.75. Not the £294, because you didn’t sell them all at full price. What you spent at the point of purchase was £1410 – £1200 = £210. So you have to send off the difference to the VAT man, £71.75.

Note here how good your records need to be to show that your VAT recovered was not £294. Many businesses would have an argument (one they would lose) and end up paying the larger figure, i.e. £84 sent to the taxman. You might argue that the work involved is too much for the difference in money.

How does this affect your profit? Surely this VAT payment is a ‘cost’, collecting tax for the nation, so it reduces your gross profit from £481.75 to £410, which reduces the profit percentage to

£410 / £1410 x 100% = 29.1%

At this point you might want to do all the figures over again, without VAT, and then deduct the cost of VAT from the end result. Spent £1200. Recovered 23x£50x1.4 = 23x£70 = £1610, a profit of £410, less the £71.75 VAT sent, £338.25 money gain. Making the % profit £338.25 / £1200 x 100% = 28.2%

How can these two numbers for percentage profit be different? Is one more right than the other? The answer to that depends on what you think the 100% is. Was it £1200 or £1410? The £1410 includes VAT, so is probably the figure with some error built into it.

Try looking at this differently. VAT is 17.5% and the mark-up is 40%, combining to 47% (you work that out). Sales are good if 23 out of 24 are made at the target price, 95.83%. Generally the VAT return will be 7%, but at this point you should be wondering 7% of what, quite? It is at this point that the whole model fails, because one is no longer clear which number is the right one to work with. What businesses do is work in the numbers without VAT, because then they are looking at the effect of the sales process. The collection of VAT is then taken as an aspect of the business overhead and then both the money exchange and the costs of doing that become part of the overhead and can be costed within that figure. Small businesses find this onerous, large businesses need a department to deal with this. Both are costs upon the business.

DJS

40% x 1.175 = 47% 40% x 0.175 = 7% 0.175 = 7/40 for those who like fractions.