# 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.

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.

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

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

Let the price exclusive of VAT $= z$

$x = z + zy$

$50 = z + 0.2z$

$50 = 1.2z$

$50 / 1.2 = z$

$z = 41.666...$

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.

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.