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I was set the following question during the discrete mathematics module of my degree and despite my instructor explaining his working to me I still disagree with the answer he says is correct.

Can someone please help me either understand where my mistake is or help me prove that my instructor's answer is incorrect?

It is the Christmas festive season again and your manager is very pleased with your performance and gives you a £50 Amazon gift card as a Christmas bonus. Value added tax (VAT) or sales tax is currently 20%. Determine the price of the most expensive taxable item you can buy with the gift card. Show your working and not just the answer.

It's a pretty horribly worded question! My gut feeling was £50, as all UK retail prices are already inclusive of VAT.

My Answer:

Let the total price inclusive of VAT $=x=$ £50

Let the rate of VAT $=y=0.2\ (20\%)$

Let the price exclusive of VAT $= z$

$x = z + zy$

$50 = z + 0.2z$

$50 = 1.2z$

$50 / 1.2 = z$

$z = 41.666...$

Instructor's Answer:

Let the price of the most expensive taxable item be $= x$

Let the 20% VAT on £50 $= y = (20/100)*50 = 10$

Our equation can be written as:

$50 = x + y$

$x = 50-y$

$x = 50-10$

$x = 40$

Update from instructor:

It looks like you and I are going to have some interesting discussions during the course of this module. I see where you are “going wrong” for want of a better phrase. You are assuming that the £50 includes VAT, but that is the wrong assumption. Sometimes easy to make that automatic jump or connection to real life scenario, but this question has nothing to do with the actual UK VAT laws. Maybe it could have been phrased differently, but the £50 is SUBJECT to a 20% VAT which implies that VAT is not included and has to be deducted from the £50. Forget the UK law for now and you will see why it’s £40. The point really is not even about how much the item is, but it is about rearranging an equation and solving for x.

In my opinion there are so many errors in his logic that it's not worth me pushing this point any further as he will be my teacher for the next 3 months anyway.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – Daniel Fischer Oct 28 '16 at 17:25
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Your answer looks right. The instructor made the mistake of assuming the taxable part would be 20% of the total, which is a common mistake with percentages, to forget about what percentages mean. You can see that $40$ is right out, by simply taking $40+.2*40=48$, and that's not 50.

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    $\begingroup$ I think the answer 41.6666... looks naively right. But you can go higher by exploring the rounding rules. $\endgroup$ – SusanW Oct 13 '16 at 11:29
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    $\begingroup$ @SusanW if the instructor can't even get this much right, I'm pretty sure they're not looking for the with-rounding-rules answer. $\endgroup$ – Walt Oct 13 '16 at 21:47
  • $\begingroup$ @Walt But ... but ... yeah, you're right :-) $\endgroup$ – SusanW Oct 13 '16 at 22:29
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    $\begingroup$ To me the error here seems to be "if I added 20% to get here, then I have to remove 20% to get back to where I was", which is also a common mistake with percentages. $\endgroup$ – Jack M Oct 15 '16 at 14:05
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Official UK gov calculation for working out the pre tax value of an item when given the post tax value is

To work out a price excluding the standard rate of VAT (20%) divide the price including VAT by 1.2.

source: https://www.gov.uk/vat-businesses/inclusive-exclusive-prices

50/1.2 = 41.666...

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    $\begingroup$ This is correct because adding 20% is equivalent to multiplying by 1.2, so dividing by 1.2 is the inverse operation. $\endgroup$ – paolo Oct 12 '16 at 15:55
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    $\begingroup$ It's almost correct - you could have a higher net value that still rounded to the same (whole pennies) VAT. $\endgroup$ – SusanW Oct 13 '16 at 11:09
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    $\begingroup$ @SusanW - not in accounting, no. Prices are not real numbers. $\endgroup$ – Davor Oct 13 '16 at 17:05
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    $\begingroup$ @Davor No, but they are certainly rational numbers: 29 beans for 85p, and that price can be represented to any degree of precision you choose, as long as your counterparties agree. There's nothing wrong with valuing something at £41.674 net of VAT, getting the gross by taking 41.674*1.2 = 50.0088, rounding down to £50 for sale, and arriving at £8.3348 VAT, which you then pay as £8.33 to HMRC. (Or taking 50-41.674=8.326 and rounding it up to £8.33) So £41.674 is a valid higher net value. $\endgroup$ – SusanW Oct 13 '16 at 19:38
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    $\begingroup$ @SusanW: £50.0088 rounds to £50.01. Anything strictly less than $\frac{£50.005}{1.2}=£41.6708\overline{\mbox{3}}$ should round properly. Still, I doubt the question intended to call VAT rounding procedure into play. $\endgroup$ – MichaelS Oct 14 '16 at 0:24
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You're pretty clearly correct, unless you've misunderstood how VAT is applied (I'm not familiar with UK finances, so I can't speak to that).

To demonstrate that your instructor is incorrect, simply show that there's a more expensive item that can be bought. In your case, 41 is nicely in between; bigger than his answer, but smaller than yours. 20% of 41 is 8.2. 41 + 8.2 = 49.2 < 50, so you can buy something for 41.

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    $\begingroup$ VAT is just an ordinary sales tax so we've all correctly understood it. The rate is 20% so, if an item is worth £40, then £8 in VAT is due and the price paid by the consumer is £48. The only difference from the sales taxes you're used to in the US is that the price on the shelf includes the tax and usually doesn't even mention it: the customer would just see £48 as the price in this case. $\endgroup$ – David Richerby Oct 12 '16 at 11:06
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    $\begingroup$ The fundamental difference between VAT and sales tax is that business are not exempt from VAT when making purchases, and get the VAT refunded by the state. This simplifies things for sellers as they charge VAT to everyone, whether business or consumer. $\endgroup$ – bdsl Oct 12 '16 at 14:01
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    $\begingroup$ @bdsl That's a great point, though it doesn't affect the question. I should probably have prefixed by comment with "For the purposes of this question, ..." $\endgroup$ – David Richerby Oct 13 '16 at 11:14
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Just because it's for a discrete maths course, and no-one else seems to have pointed it out, the correct answer should actually be £41.67 since this is the largest amount that will give a total not greater than 50, when multiplied by 1.2 (i.e. added 20% tax) and rounded to the nearest number of pennies.

And as everyone else has already pointed out, your method of dividing by (1 + 0.2) is correct, and your instructor's method of multiplying by (1 - 0.2) is wrong.

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    $\begingroup$ Not sure the net amount needs to be a whole number of pennies. The sale price does, and the total VAT amount submitted does, but the net amount doesn't. $\endgroup$ – SusanW Oct 13 '16 at 11:05
  • $\begingroup$ @ Ergwun : I did mention £41.67 in my original answer. Please see the comments that follow my answer after which I decided to do the edit and make £41.66 just to escape the clutches of the taxmen!! $\endgroup$ – naveen dankal Oct 14 '16 at 14:13
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    $\begingroup$ @naveendankal Yeah, I think you had it right before the edit. I wanted to emphasise the point about rounding though since it comes from a discrete maths course which will presumably care about such issues. $\endgroup$ – Ergwun Oct 15 '16 at 3:03
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The issue isn't the math, it's your instructors understanding of how VAT is calculated; is it 20% pre tax or post tax? In the UK it is 20% of pre-tax price (hence "value added") (i.e. 50/1.2, not 50x.8) - You are correct.

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    $\begingroup$ I think the derivation of "value added" is a little different. It's based on the idea that product goes through several steps in a supply chain, and typically each one charges more than the last. The tax is paid at every step on the 'value added'. So if a retailer buys something for £1 and sells it for £2 they send the state £0.20 in tax. (both the £1 and £2 are exclusive of vat) $\endgroup$ – bdsl Oct 12 '16 at 14:03
  • $\begingroup$ @bdsl nope, if a retailer buys something for £1, then the seller will owe the state £0.20. When the retailer sells it for £2, he then owe the state £0.40. If UK laws are like Belgium, which I'm most doubtful of, retailers also have right to "deduct" VAT PAID from their tax declaration, so in this case it would be like retailer didn't actually pay the VAT, the end user paying it all. $\endgroup$ – Laurent S. Oct 12 '16 at 15:02
  • $\begingroup$ @Laurent In the UK (and I think the EU in general) buisnesses producing VATable goods/services (including zero rated but not including exempt) can reclaim the VAT they paid on their inputs. For transactions between VAT-registered buisnesses in different EU countries there is a special procedure, the seller zero-rates the supply, the buyer "pays" VAT through a reverse charge procedure but can usually claim it back immediately. $\endgroup$ – Peter Green Oct 12 '16 at 15:40
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    $\begingroup$ Generally the concept of VAT works like that in every country that uses the VAT scheme (instead of something like the sales tax) $\endgroup$ – SztupY Oct 12 '16 at 18:37
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    $\begingroup$ The net outcome of VAT is always as @bdsl describes — pretty much by definition. The implementation varies — but the way that Laurent and Peter Green describe is the most common way in Europe. $\endgroup$ – Peter LeFanu Lumsdaine Oct 13 '16 at 10:28
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Once again, your answer is correct to me. You can do a quick check with the following formula (and its approximation for small values) that converts the percentage change $r$ from price $X$ to $Y$ in a percentage change $r'$ from $Y$ to $X$:

$$r' = -100\frac{r}{100+r}\approx -r + \frac{r^2}{100}\,.$$

Take $r=+20$ (percent), you find $r'=-16.6666\ldots$ "per cent", hence from £50 you add the half (you add a negative number), and get $50-8.3333 =41.6667$.

The above approximation (from a Taylor development) is easy to compute by mental calculus, and gives $r'\approx -20+400/100\approx-16$, not too far from the real value, but $20$ percent is not so small. That allows a quick double check of your precise computations.

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Seems your instructor has done the common mistake of applying percentage to the total given amount that is £50. This way he has arrived at the "VAT" that can be applied on £50 and not on the "maximum cost price" of the item that can be bought with £50. So his basic starting assumption that the VAT is £10 is wrong and you are correct in your approach . So, £40 is a wrong answer and £41.67 (rounding up £41.66666...) is correct.

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    $\begingroup$ I see what you're trying to say here but your wording is wrong. He's not arrived at the "Maximum VAT" that can be applied to £50 as there is no maximum or minimum VAT that can be applied. There is one value which would be £50 - £41.6666... as the OP has eluded to. $\endgroup$ – dougajmcdonald Oct 12 '16 at 11:43
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    $\begingroup$ @tomi why round down? The net amount does not need to be a whole number of pennies. $\endgroup$ – SusanW Oct 13 '16 at 11:07
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    $\begingroup$ I also thought about it yesterday when you gave the comment but I thought better be on the "losing end" than on the "receiving" when dealing with taxes!!! Who knows when £0.0002 shows up as a £1.0 in dues!! $\endgroup$ – naveen dankal Oct 13 '16 at 12:28
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    $\begingroup$ @tomi it's ... quite hard, isn't it? :-) Earlier, I wrote a whole answer here where I tried to find the largest net value that would give a VAT amount (£8.335) that would round down (according to VAT rules) to £8.33, so net could be as high as £41.675 - HMRC only really care that they get their share, and I don't think they care if your net+VAT>gross. But then my eyes started bleeding... because the true answer could be "a £30m LearJet - priced at £50!" and I got fuddled by price vs value vs accounting, and I lost faith. Probably for the best... $\endgroup$ – SusanW Oct 13 '16 at 13:19
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    $\begingroup$ I think I should stick to my original answer £41.67. $\endgroup$ – naveen dankal Oct 16 '16 at 7:56
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The correct answer is, of course, 50 pounds. There are items that have a reduced (5%) VAT and even some with no VAT added (food, I think). So you just have to find one of those items.

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    $\begingroup$ Books is an appropriate zero-rated one, seeing as it's an Amazon voucher :-) $\endgroup$ – SusanW Oct 13 '16 at 19:52
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    $\begingroup$ The question specifies "taxable item". And generally, you're supposed to use the tax values given, even if real life has changed since the question was written, or there are other tax values in the real world. $\endgroup$ – MichaelS Oct 14 '16 at 0:09
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This is totally out of the realm of mathematics, but still...

The most expensive item that you can buy with a £50 gift card costs £50. Unless Amazon doesn't actually have any product that costs £50, in which case the most expensive item might cost £49.99 for example.

Now your instructor seems to interpret this as "what could be the most expensive item pre-tax that you can buy for £50". In the UK, most items have 20% VAT. His calculation then is totally bogus. He subtracts 20% from (item price + VAT) and gets £40. But you only pay £8 VAT on a £40 item. As others have calculated, you can buy an item costing £41.67 + £8.33 rounded VAT with your £50 gift card.

But since your instructor is determined to not be corrected, I'll just note that there are plenty of items that are VAT free, like baby wear, children's clothing and children's footwear, books, newspapers, and magazines, and other items with a 5% VAT rate. If he doesn't accept VAT free items as an answer, then you can most definitely buy a children's car set at Amazon for £49.99 including 5% VAT, which would be £47.61 + VAT.

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protected by J. M. is a poor mathematician Oct 13 '16 at 12:52

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