# JOST A MON

Dec 8, 2009

## Vitya’s Maths

In Nikolai Nosov’s Vitya Maleev at School and at Home, the rather gormless Vitya has been promoted to the fourth grade despite being utterly at sea with mathematics. He is ashamed of his lack of skill, particularly because his younger sister, a year below him, aces all her subjects and never fails to point this out to him. One day, he is given a homework assignment – a mathematics problem – and in the space of a couple of pages, Nikolai Nosov demonstrates why he is possibly the finest expositor of mathematical pedagogy in children’s literature anywhere. Here is a translation.

…I opened the textbook and began to read the assignment:

In the shop, there were eight saws; axes were three times as many. One team of carpenters bought half the axes and three saws for 84 roubles. The remaining saws and axes were sold to another team for 100 roubles. What are the prices of a saw and an axe?

At first I understood nothing and began to read the problem a second time, and then a third. Presently I realised that whoever set the exercises purposely garbled them up so that the students couldn’t solve them at once. It said: In the shop, there were eight saws; axes were three times as many. Why couldn’t they have said simply that there were twenty-four axes? After all, if there were eight saws and there were three times as many axes, then it’s clear to anyone that there were 24 axes. There’s no need to make a mountain out of a molehill! And then it said: One team of carpenters bought half the axes and three saws for 84 roubles. Why couldn’t they just say bought twelve axes? As though it’s not clear that if there were 24 axes, half that number is 12! And they bought the lot for 84 roubles. Further on, it said that the remaining axes and saws were sold to another team of carpenters for 100 roubles. What do they mean ‘the remaining’? Why can’t they talk like normal human beings? If there were 24 axes and they sold 12, then, clearly, 12 remain? There were in all 8 saws, three were sold to the first team, so obviously the second team got 5. They could have said all this clearly; instead, they twisted and turned just so they could say that kids these days are all numbskulls and can’t solve problems.

I rewrote the problem in my own words so that it appeared simpler, and this is what I got:

In the shop there were 8 saws and 24 axes. One team of carpenters was sold 12 axes and 3 saws for 84 roubles. Another team was sold 12 axes and 5 saws for 100 roubles. What is the price of one saw and one axe?

Having rewritten the problem, I re-read it and saw that it had become somewhat shorter in length, but still I couldn’t figure out how to solve it, because the numbers twisted around in my head and prevented me from thinking straight. I decided to shorten it a bit more so that it had fewer numbers in it. After all, the total number of saws and axes was completely unimportant because, at the end, all of them were sold. I shortened the problem again, and it ended up as follows:

One team was sold 12 axes and 3 saws for 84 roubles. The other team was sold 12 axes and 5 saws for 100 roubles. What is the price of one saw and one axe?

The problem was now shorter and I thought about how to shrink it a bit more. It wasn’t important who bought what. The only important thing was for how much was it all sold. I thought and thought, and I got the following:

12 axes and 3 saws cost 84 roubles. 12 axes and 5 saws cost 100 roubles. What is the price of one saw and one axe?

It couldn’t be shortened any further, and I began to think about how to solve the problem. First I thought that if 12 axes and 3 saws cost 84 roubles, then I needed to put all the axes and saws together and divide 84 into that number. I added 12 axes and 3 saws and obtained 15. Then I tried to divide 84 into 15, but I couldn’t do it because I kept getting a remainder. I realised then that I had made a mistake, and hunted for another approach. I added 12 axes and 5 saws, obtaining 17, and then I tried to divide 100 into 17, but again I ended up with a remainder. So then I put the axes together and the saws together and the roubles together, and divided the roubles for the axes into the saws, but didn’t get anywhere. So I took away the saws from the axes, and threw the money at what remained, to no avail. Then I tried to separate the axes and saws and take away the axes from the roubles, and divided the remainder into the number of saws, and who knows what else I tried. Nothing worked. So then I took the problem over to Vanya Pakhomov.

- Listen, – I said, – Vanya. 12 axes and 3 saws together cost 84 roubles, and 12 axes and 5 saws cost 100 roubles. What is the price of one axe and one saw? How do you think should I solve the problem?

- What do you think? – he said.

- I think I need to add the 12 axes and 3 saws and divide 84 into 15.

- Wait! Why are you adding the axes and saws together?

- Well, I’ll then know how many items there are in all, and if I divide 84 into that number, I’ll know how much each one costs.

- What do you mean ‘each one’? One saw or a one axe?

- One axe, – I say, – or one saw.

- So then you think that they cost the same?

- What, – I say, - they don’t cost the same?

- Of course not. It doesn’t say in the problem that they cost the same. On the contrary, it asks for the cost of one axe and one saw separately. That means that we can’t just add them together.

- Well, – I said, – add them, don’t add them, nothing works!

- Well, – he said, - that’s why it doesn’t work.

- What to do then? – I ask.

- Okay, so look at the problem again, – says Vanya, – What do you see?

- I see, – I say – that 12 axes and 3 saws cost 84 roubles, and 12 axes and 5 saws cost 100 roubles.

- Do you see that in both cases, they sold the same number of axes, but there were extra saws?

- I do

- And do you see that in the second case, it cost 16 roubles more?

- Yes, I see that too. In the first case, it cost 84 roubles; in the second, 100 roubles; 100 minus 84 is 16.

- And why do you think that in the second case it cost 16 roubles more?

- Well, that’s clear to anyone, – I reply. – They bought two extra saws, and so they had to pay an extra 16 roubles.

- Which means that two saws cost 16 roubles?

- Yes, – I agree.

- So then, how much does a saw cost?

- Well, if two cost 16, – I say, - then one would cost 8.

- There you go, – Vanya says. – Now you know how much one saw costs.

- Phooey! – I said. - What a trivial problem! I can’t believe I couldn’t do it by myself!

- Wait, you still need to figure out the price of an axe!

- Well, that’s easy-peasy, isn’t it? – I said. – 12 axes and 3 saws cost 84 roubles. 3 saws cost 24 roubles. 84 minus 24 is 60. That is, 12 axes cost 60 roubles. So one axe – 60 divided by 12, equals 5 roubles.

I went home and I was very upset that I hadn’t solved the problem by myself. But I decided that next time I would definitely figure out the assignment on my own. Even if it took me five hours, I would do it.

Space Bar said...

this is very lovely and a bit of a keeper!

Fëanor said...

Thanks! Math rules, SB, I always say.

Abha Iyengar said...

omigod. had me in fits. maths is simple simon if u know how. but then u have to be simple or simon and i take it u r not. neither am i. thanks for this. if anyone saw me reading this and giggling away, they would never think it was maths that made me so.
this was so good.

Fëanor said...

Glad you liked it, Abha. Welcome to the blog.

Hari Batti said...

Wonderful!

Fëanor said...

Thanks, Hari.

Banno said...

Lovely maths story. Lots of fun reading it. And I'm sure it would make lots of sense to a kid with their convoluted maths problems.

Fëanor said...

Thanks, Banno. How about that? The Russians introduce simultaneous linear equations in 3rd grade!