What is the most expensive item I could buy with £50?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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 $X$:

$$r' = -100\frac{r}{100+r}\approx -r + \left(\frac{r}{10}\right)^2\,.$$

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.

Answer 6

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...

Answer 7

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.

Answer 8

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.

Answer 9

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.