11+ Questions: 4. Water, ice and all things nice

Last week’s 11+ question was a bit of an epic, so for this week I’ve got something much more concise. This question comes from a Sevenoaks School 11+ sample paper, and it’s the final question in the whole test. Let’s have a look:

‘When freezing, water increases its volume by 1/11. By what part of its volume will ice decrease when it melts and turns back into water?’

So they’ve put it last because it’s one of these tricky puzzle-like questions that tend to crop up in the 11+. On the face of it it’s quite an intimidating question, but the way to solve it is quite straightforward.

Let’s begin, as always, with what we know:

• Water increases its volume by 1/11^th when freezing.
• When ice melts it loses the volume it gained when (as water) it froze in the first place.
• We need to work out what proportion of the volume of ice is lost when it turns back into water.

Now a strategy that I recommend to 11+ pupils when they haven’t got a clue is to get some numbers involved. These numbers could be a complete guess, but it doesn’t matter. Just by getting something down on paper you’re getting closer to the answer.

So let’s give our water a volume. Let’s say it has a volume of 100cm³ .

Now, when the water freezes it’s going to gain 1/11^th of its volume. So what’s 1/11^th of 100?

We could divide 100 by 11, but that would be a) very difficult, b) very time consuming, and c) very difficult to integrate into the next part of the question. We can see that 100 wasn’t a very sensible choice for the volume of our water. It would be much more sensible to choose a number that’s a multiple of 11, rather than a multiple of 10 like 100.

Let’s start again, and this time say our water is a multiple of 11. Let’s say it’s 99cm³ .

So, when water freezes it gains 1/11^th of its volume. So what is 1/11^th of 99?

That’s easy: 99 / 11  = 9. So 1/11^th of 99 is 9.

So that means that when 99cm³ of water freezes, it gains 9cm³ . The new volume of this frozen water, or 'ice' as we might call it, is 108cm³.

This means that when ice melts, it goes from being 108cm³ to being 99cm³. The question is asking us ‘what proportion of the original volume of ice (108cm³) is going to be lost when it becomes water (99cm³)?’

Well, from 108, 9 has been lost. So the question then becomes, ‘what proportion of 108 is 9?’

Well, we write it out as a fraction: 9/108 and then cancel both sides by 9, and we end up with:

1/12.

So there’s our answer!

If water gains 1/11 of its volume when freezing, ice will lose 1/12 of its volume when turning back into water!

Only a short one this time round, but it demonstrates the power of simply getting some numbers down on paper and seeing what happens. Until next week!