Video: AQA GCSE Mathematics Higher Tier Pack 2 • Paper 2 • Question 21

Yvette wants to put some books on a bookshelf. The bookshelf has a length of 43 cm, a width of 16 cm, and a height of 27 cm. These dimensions are to the nearest centimetre. The books all have a length of 2.6 cm, a width of 15.3 cm, and a height of 25.9 cm. These dimensions are all to the nearest millimetre. By continuing the pattern shown, prove that Yvette can fit at least 16 books on the bookshelf.

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Video Transcript

Yvette wants to put some books on a bookshelf. The bookshelf has a length of 43 centimetres, a width of 16 centimetres, and a height of 27 centimetres. These dimensions are to the nearest centimetre. The books all have a length of 2.6 centimetres, a width of 15.3 centimetres, and a height of 25.9 centimetres. These dimensions are all to the nearest millimetre. By continuing the pattern that’s shown, prove that Yvette can fit at least 16 books on the bookshelf.

So here we also have a diagram showing the bookshelf and some books and also showing how the books are put in the bookshelf. Now, at first, this looks like a very simple problem. The height of the bookshelf, 27 centimetres, is greater than the height of all the books, 25.9 centimetres, and so it will fit in height-wise. Also, this depth or, as it is called in the question, width of 16 centimetres is greater than the 15.3 centimetres’ width of the book. And so they’ll fit in that way.

The only question is how many of these books can be fit in side by side in the shelf. And we can get that surely just by dividing 43 centimetres, the length of the bookshelf, by 2.6 centimetres, the length of the book. But all the dimensions quoted aren’t exact. The dimensions of the bookshelf are given only to the nearest centimetre, whereas the dimensions of the books slightly more accurately are given to the nearest millimetre.

As a result, the length of the bookcase isn’t 43 centimetres exactly. It’s 43 centimetres to the nearest centimetre. And the same is true for the length or thickness of the book. This is 2.6 centimetres to the nearest millimetre. And so if the bookshelf is slightly shorter than the 43 centimetres and the books are slightly thicker than 2.6 centimetres, we might not be able to fit 43 divided by 2.6 books in the bookshelf.

Now we can’t say what the values of these dimensions are exactly, but we can use inequalities to write a range of values. The length of the bookshelf is 43 centimetres to the nearest centimetre. So when we round this length to the nearest centimetre, we get 43 centimetres. This length must therefore be less than 43.5 centimetres and greater than or equal to 42.5 centimetres. All lengths in this range are 43 centimetres to the nearest centimetre.

How about the thickness of the book? Well, we’re told that it’s 2.6 centimetres to the nearest millimetre. We have to be careful here. We’re given the length in centimetres, but the accuracy in millimetres. There are 10 millimetres in a centimetre, and so 2.6 centimetres is 2.6 times 10, which is 26 millimetres. So the thickness of the book is 26 millimetres to the nearest millimetre. So it is greater than or equal to 25.5 millimetres and less than 26.5 millimetres. We can convert these values back to centimetres by dividing by 10. 25.5 millimetres is 2.55 centimetres, and 26.5 millimetres is 2.65 centimetres.

Okay, now remember that we wanted to prove that Yvette can fit at least 16 books on the bookshelf. In the worst-case scenario, the thickness of one book is 2.65 centimetres. That’s the maximum thickness of a book. So the maximum thickness of 16 books or the maximum length if the books are placed side by side as in the diagram is 16 times this. And using a calculator, we find this to be 42.4 centimetres. And this is less than 42.5 centimetres, which is the minimum possible length of the bookshelf. So even if the books are as thick as they’re allowed to be and the bookshelf is as short as it’s allowed to be, you can still fit 16 books inside the bookshelf in the way shown in the diagram.

We should probably also check that the books really do fit in the other dimensions too. It’s a very similar process, slightly easy in fact, to show that the maximum height of a book is 25.95 centimetres, which is less than the 26.5-centimetre minimum height of the shelf, and so the book fits in height-wise, and also that the minimum width of a book is 15.35 centimetres, which is less than the 15.5-centimetre minimum width of the shelf.

We probably want to just state that we’ve proved what we set out to prove that Yvette can fit at least 16 books on the bookshelf. The key to this question was to use inequalities to find the maximum dimensions of the 16 books placed in a row as shown in the diagram and to show that each of these dimensions was less than the corresponding minimum dimension for the bookshelf.

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