Maths Week Scotland 2023 - Problem 1 - Books and magazines

CHRIS: This problem is about working out how many books there are in different boxes.

Eilidh is a second hand bookseller and she needs to clear some space in her shop. She has six boxes of books and a box of magazine she wants to move.

Each box shows how many items there are in it.

Eilidh makes the space she needs by selling off some books to one customer, twice that number to another customer and keeping the magazines for herself.

Can you work out how many magazines Eilidh held on to?

One book buyer is gonna get twice as many books as the other person. Yeah? So think about what that might tell you, anybody got any ideas what that tells us?

PUPIL: What can you work out about the total number of books?

CHRIS: So you'd need a thirty one. Can you make 31 from the numbers that are available?

PUPIL: Can you find out how many books each customer buys?

CHRIS: Every one book he's got, you've got two books, yeah?

PUPIL: Can you find different totals from adding different boxes together?

CHRIS: So that's a useful thing to know, though.

The amount of books in one pile has got to be double the amount of books in the other pile.

PUPIL: Good luck everyone and try your hardest.

Did you manage to divide up Eilidh's boxes and find out how many magazines she kept?

Here's one way you could have solved it.

There are seven boxes in total, one of them contains magazines so the other six all contain books.

Since the books can be divided so that one customer gets twice as many as the other, the total number of books must be a multiple of three.

One person's getting double the amount of the other person.

So altogether that's gonna add up to, like, a multiple of three books.

Love it. Love it!

To solve this problem, let's find the total number of all the books plus magazines.

So add all the numbers together: 16 plus 20 plus 19 plus 24 plus 17 plus 25 plus 16 which equals 137. If you divide 137 by 3, you're left with a remainder of two. The total number of books is a multiple of three, so that means that the number of magazines in the box you remove must also have a remainder of two when you divide by three.

Divide each number by the three.

Most of the numbers don't work.

The only ones that do are twenty, because twenty divided by three is six remainder two, and seventeen, because seventeen divided by three is five remainder two.

Now try removing each number from the total, 137.

If we remove twenty we're left with one hundred and seventeen. One hundred and seventeen divided by three is thirty nine, but no combination of the remaining books gives thirty nine so it can't be that one.

If we remove seventeen, we're left with one hundred and twenty. One hundred and twenty divided by three is forty.

It looks like Eilidh kept the box with seventeen magazines in it but let's check.

We can get to forty books by adding together the box with sixteen items and the box with twenty four items and the other customer gets twice that number: eighty books. And sure enough the remaining boxes, twenty plus nineteen plus twenty five plus sixteen, add up to eighty, so that confirms the answer that Eilidh keeps the box with seventeen magazines in it.

Well done if you worked that out.