Re: Giant (IMM) 16% Discount

Hmm…

We just went to Giant @ IMM to spend one of our $8 voucher yesterday. We’ve confirmed with the staff that only one $8 voucher can be used for every $50 purchase.

Let’s see how this affects things.

First off, this is a reponse to Dear2’s earlier post. Strictly speaking, the $8 voucher is more of a rebate than a discount. A discount is something that you receive upfront before paying. A rebate is something that you receive (and can be used for a later purchase) after paying the original amount.

So, strictly speaking again, the percentage savings should be calculated as \frac{8}{50+50}=8\%, although not as high as 16% mentioned earlier, it is still a very significant amount!

Of course, we can take a look at the more general case to see how much savings can we actually achieve if we keep returning to Giant @ IMM to spend $58 each time to continually accumulate $8 vouchers.

Let n \subset \mathbb{Z}^+ be the number of times we visit Giant @ IMM. For the special case n=1, we make no savings at all, so we shall ignore it.

We know that we should spend $50 on the first visit to earn the first voucher. On subsequent visits, we should spend $58 so that we can use the first voucher and then continue to earn one more voucher. We should repeat this until the last visit where we should only spend $50 simply to use our last voucher.

Hence, we can define the percentage of savings as

\frac{8}{50+50},n=2

\frac{8+8}{50+58+50},n=3

\frac{8+8+8}{50+58+58+50},n=4

\frac{8+8+8+8}{50+58+58+58+50},n=5

\vdots

\frac{8(n-1)}{100+58(n-2)},n=n

Simplifying, we get

\displaystyle \frac{4n-4}{29n-8}

In the limit that we visit Giant @ IMM a large enough number of times

\displaystyle \lim_{n \to \infty} \frac{4n-4}{29n-8} = \frac{4-\frac{4}{n}}{29-\frac{8}{n}} = \frac{4}{29} \approx 13.79\%

In order to better visualize this relationship, we plot the graph of the amount of savings against the number of visits from 0 to 100 visits in Fig. 1.

Fig. 1. Graph of Savings against Number of visits

We can observe that the amount of savings rises rapidly during the first few visits before again rapidly tapering off towards the horizontal asymptotic value of 0.1379.

Of course, we’re not going to visit Giant @ IMM that many times since the promotion is going to end in 30 more days (i.e. 31 Oct 2010). Practically, visiting 13 times (including first and last) is enough to put the savings at 13%, which isn’t too bad at all.

Take note that this is only the theoretical limit. Practically, it isn’t quite possible to purchase *exactly* $50 or $58 every single time. So, the practical savings is likely smaller than 13%.

Solved. :D

-Dear1

p.s. This post is just poking fun at how much an engineer Dear1 is! I don’t usually do such things when we go shopping. :P

\mathbb{Z}
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