Republic of Mathematics blog

Save 30%, be 30% better off?

Posted by: Gary Ernest Davis on: February 22, 2011

A travel advertisement running, currently in the US, says you can save 30% on travel costs, and so be 30% better off.

Is that right?

How about 50% off?

To make it easier to think about, let’s think how much we would save if we got 50% off the price of something.

Suppose some item we wanted cost $100, and the salesperson said we would get 50% off.

This means we would pay only $50 when we thought we might have to pay $100.

So we could actually get 2 items for the $100 we originally intended to spend.

That is, we could get the 1 item we wanted, plus an extra 1 for the $100.

This, to me, means we are 100% better off.

What about 1/3 off?

Now let’s hone in on that 30% off.

Suppose we are offered 33% off the price of an item.

That’s pretty close to \frac{1}{3} off, so let’s pretend it is, in fact, \frac{1}{3} off the price.

Let’s suppose the item was going to cost $90.

Now, with the discount of \frac{1}{3} it will only cost $60.

That means we have saved half the cost of getting another one.

In other words, if we were prepared to spend $180 we would not get just 2 items, as we would have thought before we heard about the \frac{1}{3} discount: we would actually get \frac{180}{60}=3 items.

So whereas we thought we would get 2 items for $180 we will actually get 3 items.

This means we are one-half – or 50% – better off than without the discount.

So a \frac{1}{3} discount makes us 50% better off.

Because \frac{1}{3} is close to 30%, we should suspect that a 30% discount is probably going to make us closer to 50% better off, than 30% better off.

So how much better off are we after a 30% discount?

What if an item costs $100 and we are given a 30% discount?

This means we only pay $70 for the item.

So if we were to spend $700 we would get 10 items after the discount.

Bu before the discount $700 would have only bought us 7 items, at $100 each.

So we are \frac{3}{7} \approx 0.43 = 43\% better off after the 30% discount.

Even better than the advertisement stated.

It’s not often advertisers misrepresent the truth in your favor!

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