Sales tax included

Sometimes the wrong answer is more interesting than the right answer. If the wrong approach almost works, it may be fun to understand why.

Suppose you want to figure a price so that the final price including tax has a given value. For example, say you want a T-shirt to sell for $10 after adding 8% sales tax. A common mistake would be to subtract 8% from $10 and sell the shirt for $9.20 plus tax. But this makes the price with tax $9.94. Observe two things: (1) the result was wrong, but (2) it wasn’t far off.

The correct solution would be to divide $10 by 1.08. Then when we add 8%, i.e. multiply by 1.08, we get $10. That says we should price the shirt at $9.26 before tax. That explains (1). But what about (2), that is, why was the result nearly correct? Subtracting a percentage p amounts to multiplying by (1-p). But we should have multiplied by 1/(1+p). We’ve stumbled on the fact that for small p, 1/(1+p) approximately equals 1 – p.

With a little experimentation we might discover a couple more things. First, we notice that multiplying by 1-p always gives a smaller result than multiplying by 1/(1+p), i.e. the common mistake always gets the price too low. Also, with a little experimentation we might notice that the difference between 1/(1+p) and (1-p) gets smaller as p gets smaller. Not only that, making p a little smaller makes the difference between 1/(1+p) and (1-p) a lot smaller. In summary, we noticed that 1/(1+p) – (1-p) is

  1. small,
  2. positive, and
  3. gets small faster than p gets small.

What accounts for these observations? The Taylor series for 1/(1+p) says that for |p| < 1,

1/(1+p) = 1 – p + p2 + p3 – p4 + …

The error in truncating a Taylor series is roughly equal to the first term that was left out, so the error in approximating 1/(1+p) by (1-p) is roughly p2. That explains why the difference between 1/(1+p) and (1-p) is small, positive, and decreases with p. For example, we should expect that cutting p in half reduces the difference by a factor of four.

The Taylor series argument only necessarily holds for p sufficiently small. If we go back and calculate 1/(1+p) – (1-p) directly we find it’s p2/(1+p). This confirms that 1/(1+p) is greater than 1-p for all p larger than -1.

Posted in Math
7 comments on “Sales tax included
  1. Roger Peng says:

    Hmm, I’m not sure this explanation would fly at the local pizza shop here, but it’s worth a shot, I suppose.

  2. John Venier says:

    Yeah, you can see the same thing when folks think about applying successive discounts on sale items. A coupon good for 10% off any purchase, applied to an item which has already been discounted by 20%, results in a 28% discount as opposed to a 30% discount.

    Another great example of surprising percentages was in a reccent issue of Games magazine. I’ll paraphrase from memory:

    A farmer goes to market starting with 1000 kilos of watermelons, which are 99% water. But on the way there, the watermelons lose some water. When he arrives, they are 98% water instead of 99%. How much does his load of watermelons weigh when he arrives at the market? First make a wild guess before working out the exact answer. I was amazed!

  3. Jason Adams says:

    Methinks the farmer got stopped by bandits with a thirst for watermelon juice. Or he had to walk a thousand miles across the desert.

  4. Kevin says:

    So… a little late here, but any solution or references for the watermelon question?

  5. Douglas says:

    At the start:

    1000 kg watermelon
    = 990 kg water + 10 kg non-water

    At end:
    10 kg = 2% (non-water)
    100% = 10 * (100/2)
    = 500kg

  6. Pieter says:

    Kevin, the caveat is that the relative amount of non-water in a watermelon doubled, while non of it was lost, hence the amount of water is actually halved.

    At start, 1% non -water = 10 kilo, at arrival only water is lost, and that 10 kilo is now 2% of the watermelon, so 100% would be 500 kilo. (this of course assumes the 99% refers to the weight and not the volume, but I guess that’s implied…)

  7. John Venier says:

    Kevin,

    Douglas and Pieter got the answer right. Thanks, guys!

    As far as a reference goes, it was in an issue of Games Magazine. They have a web site here. It must have been in a 2008 issue. I believe it was in their “Wild Cards” section, printed on slick color paper instead of the pencil friendly paper they use almost everywhere else. I’ll see if I still have it hanging around. It is usually quite some time before I’m finished with an issue. If not and you need the reference for some reason, I’d imagine the folks at Kappa Publishing (Games’ publisher) would help. By the way, Games is and has been the best puzzle magazine I’ve ever seen. They also have a less-frequently published “Games’ World of Puzzles” serial which is excellent. Anything they make is absolutely top-notch. If I’m not mistaken, even the superhuman and most brilliant Henry Hook used to write for them. If there should be a patron saint of crosswords (especially cryptics) he’d be an excellent choice IMO. If you prefer more logical / arithmetic puzzles check out Conceptis online (from Japan). Extremely good puzzles, and very, very prolific. Plus, they put weekly samples online for free (I think registration is required though). They even send out t-shirts to random puzzle reviewers, but I’ve never got one.