A question of shapes and ratio

I am going to answer a question today about shapes and ratio.

We would be getting to exam season now, if exams had not been cancelled this year. But I am going to post some exam questions with answers in this diary to keep me – and you – on your toes.

Try the question yourself before reading my answer. All questions will be from recent GCSE exams.

 

What does ‘Similar’ mean here?

A key work in this question is similar. This means the formulas for area and volume will be the same. The ratios between the height, length and any other measurement of the shape will be constant.

This means that if we take h, the height, as one measurement, and the ratios between the heights of A, B, and C as a:b:c, then the rations between the areas will be a²:b²:c²,  and the ratios between the volumes will be a³:b³:c³

See here for more information on the volume and areas of solid shapes

 

Working out the ratios

We have been given the Surface areas of A and B, so we can see the ratio is 4:25,  so the ratio a:b is 2:5 (by taking square roots)

We are given the ratio between the volumes of B and C as 27:64, so the ration b:c is 3:4 (by taking cube roots)

To find a ratio a:b:c using whole numbers, we need a LCM of 3 and 5, so that b can be the same in both. The LCM of 3 and 5 is 15, so a:b:c is 6:15:20

So the ratio of heingths a:c is 6:20, which can be simplified to 3:10.

Remember ratios work just like fractions – we get the simplest form by ‘cancelling’