Our daughter (9) was so excited she hugged me and said &quot;I love maths!&qu...
- 2017-01-18 17:33:03+0000 - Updated: 2017-01-18 17:33:03+0000
Our daughter (9) was so excited she hugged me and said "I love maths!"

One of her homework questions was: "Tom spends £2 on 10p and 20p stamps. He buys three times as many 10p stamps as 20p stamps. How many of each did he buy?"

It's a standard sort of question that usually makes me sigh and reach for some algebra and start rearranging equations. But she'd seen a simpler way, which was this:

If he'd bought just one 20p stamp, and thus three 10p stamps, he would have spent 50p. So he must have done that four times, to make £2. Done.

It's a small but significant insight, and she experienced the joy of seeing it for herself.
Shared with: Public, Timothy Gowers, Neela Hutton
- 2017-01-20 12:40:39+0000
That very nice story provokes in me a wish to analyse what kind of thought process could have led to the insight. I'm not sure I have a the best possible answer, but here's a first stab at it. We are given a relationship between the number of 10p stamps bought and the number of 20p stamps bought. We are also given a relationship between the number of stamps bought of each type and the total price. It's easy to calculate the total price given the number of stamps bought of each type, and it's also easy to calculate the total price given the number of stamps bought of just one type. That's particularly easy if you're given just the number of 20p stamps, since then you can multiply by 3 rather than dividing by 3 to get the number of the other type.

So now we've established that from the number of 20p stamps we can easily calculate first the number of 10p stamps and then the total price. Unfortunately, we're asked to reverse this: we're given the price and asked to calculate the number of stamps of each kind (but it will be sufficient to work out the number of 20p stamps).

However, and this seems to be the key point, if we just go ahead and do things the wrong way round but in general, then we find that the relationship between the number of 20p stamps and the total price is actually very simple: they are in a ratio of 1 to 50. And if we now look at the total price, we find that we do after all have something that's easy to invert, since dividing £2 by 50 is easy.
- 2017-01-20 14:58:47+0000 - Updated: 2017-01-20 15:13:10+0000
, yes, it's like she started from a brute force approach (try a few values until one works) and then spotted a shortcut in the thought process.

Maybe she remembers - ?

I'll ask but I think her teachers are encouraging the approach of 'try a few values' which I think is very healthy, precisely because it gives room for the kids to look for shortcuts. Which reminds me of the story of Gauss being told to add the numbers from 1 to 100 - maybe that's a good lesson to set!

For adults perhaps this question about stamps is a bit of misdirection, because we've seen similar ones where it's not so simple. e.g. "Tony has 11 more nickels (5c) than quarters (25c). How many coins does he have if the total value of his coins is \$2.65?"

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