Some people find percentages difficult. I have long thought of this as more a reading problem 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.

Try doing these and only then reading the discussion lower down the page. Too much research shows we learn well from mistakes, so please do that; have a go and then see where I disagree. I predict already that most of where we differ is to do with communication more than maths. That’s my failure to persuade you to read the question the way I intended quite as much as it is yours for not recognising what to do. The little piece that is new mathematical understanding is almost certainly covered in the advice in blue above.

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

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

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

**4 For q3: How much VAT do I add? What is the new total?**

**5 What is my selling price for the money spent above? How much of that will be VAT? Work with a 140% addition for overheads and so on.**

**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?**

**7 In reality, 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?**

**8 From essay 248: We estimate that 60,932 stores have closed since 2012, a fall of -15.7% to end-2017. So what do you think the number of stores was at the end of 2017?**

**Discussion, techniques:**

**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 the money answer that 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. Sometimes that increase will be more like 150% - it dpends on your overheads. 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 subsequently you can drop your overhead figure to say 30%, this is a subsequent decision.

Do not confuse that paragraph with what you pay VAT on. The rules are clear; you recover the VAT *you* have paid and you charge VAT on what you *sell*. That is what **V**alue **A**dded **T**ax is all about. I was writing about what you choose as your add-on for overheads and profit, in particular which number you decide to work from.

**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. Note we’re talking about *selling* price, now, not what you paid for the stuff.

One way of looking at this is to take what you spent, the incl. VAT figure of £1410, and add 140%. Which means you want to know what 240% of £1410 is. £1410 x 240% / 100% = £3384. If you sold all your stock at that mark-up your gross take would be £1974, VAT included.

How much of this is VAT? Where is the 100% now? It moved; £3384 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 £3384 x 17.5% / 117.5% = £504. 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 240/100 = £2880. The VAT to add to this is 17.5%, but on the new figure, so the VAT to add is 17.5% of £2880 or £2880 x 17.5% / 100% = £2880 x 0.175 = £504.

And £2880 + £504 = £3384. 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 140% plus VAT; 20 x 50 x 240%/100% x 117.5% / 100%

= £2400 X 1.175 = £2820 or if you prefer, at £141 ex-VAT each, £165.68 including VAT and rounding the last number upwards. Except you probably priced them at a number more like £164.95 or £159.99. In some environments you’d work with ex-Vat prices, in this case perhaps just under £140 or £150 and then add the VAT.

However, we’ll work with the £165.68.

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 (working from £165.68) that would be £149.11, £132.54, £115.98 and £99.41. 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£165.68 = £497.04 so the total income for all 24 can be written as £2820 + £497.04 0r as 23 x 50 x 2.4 x 1.175, or as £2820 x23/20. All of which = £3243.00. I repeat, your customers are paying the VAT so if you drop prices 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 you can only 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 £150, £130, £115 and £100, 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 £3384? £2820? £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 £3243. That says you made a (gross) profit of the difference, £1833. As a percentage, then the 100% here is what you spent, and the profit is £3243/ £1410 x 100% = 230%, which is a profit of 130% and I didn’t need to round any numbers off. 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 £1833 / £1410 x 100% = 130%.

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 (£165.68 - £141) = £567.64. Except that £567.54 might be the argument, working on the total money in; 23 x £165.68 x .175/1.175 = 567.54.

Indeed, What you really do is add up what you actually ‘took’ as income, which might be 20@£165.68 + £(150 + 130+ 115 + 100) = 3808.60, then work out how much of that was VAT [3808.6 x .175/1.175 = £567.24] And then you deduct what you already *spent* in VAT: What you spent at the point of purchase was £1410 – £1200 = £210. So you have to send off the difference to the VAT man, £567.24 - 210.00 = £357.24.

So your profit should be working with the VAT numbers removed, but often people who sell at VAT-included prices don’t bother, calling VAT just another business expense (and why not?). Businesses that work with VAT-EXcluded prices tend to declare their profit with no VAT at any point. In this example, the 3243 collected needs the VAT removing, 3243/1.175 = £2760 and then we take off that the original ex-VAT £1200 to show £1560 gross profit. I repeat that this is the money that pays for the rent, the business rates, heating, lighting, staffing, cleaning, loans - all the things that the business incurs as costs. The tiny amount left over is the net profit. Across a year the total net gain is expressed as a percentage of the total investment and is seen as return on investment. Whether than is calculated beforre or after you pay yourself is one of those pesky management decisions.

Note here how good your records need to be to show that your VAT recovered was not 24 x £(165.68-141)= £592.32. Many businesses would have an argument (one they would lose) and end up paying the larger figure, i.e. £382.32 sent to the taxman, not £357. [and 64p, 54p or 24p]. You might well argue that the work involved is (far) too much for the difference in money. This is why you buy some software, but that doesn’t excuse you needing to understand what is going on and you cetrtainly need to be able to test wther the software is producing numbers you agree with. Even more pertinently, that the Vatman agrees with.

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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.

**8 From essay 248: We estimate that 60,932 stores have closed since 2012, a fall of -15.7% to end-2017. So what do you think the number of stores was at the end of 2017?**

If the 2012 figure was 100% and we’re now at 84.3%, then 60932/0.157 is the 2012 figure and 84.3% of that is 327,000. More than 3sf precision is inappropriate.

DJS

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