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

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.

Solution 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.

Solution 2:

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...

Solution 3:

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.