 Recently I’ve been quite a fan of LEON. By using your membership card: If you spent 30 pounds, you can get any item for free. A meal deal counts as an item which could cost up to 10 pounds! In effect you get a 33% off discount.

Today, I discovered that there is also a 25% off student discount. However that student discount seems to have made the membership worse. I’d need to eat there more often in order to spend 30 pounds and the value of the free item seems has dropped due to the student discount as well. Can I mathematically show that the membership seems to have gotten “worse”?

## Abstraction

Let’s say you are at a shop that sells only one type of pen. The price of a pen is usually $$c$$, but there’s an ongoing discount right now so each pen only costs $$kc$$ for some $$0 < k < 1$$. There’s also a policy that once you have spent $$m$$, you can get a pen for free.

As such, assuming that you buy a lot of pens, you could calculate that on average spending $$m$$ would get you one free pen plus $$m/kc$$ pens. As such the average price per pen would be

$\frac{m}{\frac{m}{kc}+1}$

Alternatively you can think that $$1$$ unit of currency could get you the following amount of pens.

$\frac{1}{kc} + \frac{1}{m}$

## Back to LEON

For sake of simplicity, assume that I only ever spend 7 pounds to get a wrap every time I visit LEON. Whenever I get a free item from the membership, I would only use it to get the same 7 pound wrap. As such we could use our previous formula by letting $$c=7$$. By tabulating and substituting various values we get

k\m nil 30 50 100
100% 7 5.68
19%
6.14
12%
6.54
7%
90% 6.3 5.21
17%
5.60
11%
5.93
6%
75% 5.25 4.47
15%
4.75
10%
4.98
5%

The nil column is the price of a wrap without the membership deal. The percentages in the table cells are the discounts brought by the membership card combined with the student discounts compared to only counting student discounts.

As it turns out, having the student discount didn’t make the membership deal that much worse!

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