I am stuck with writing ratios as a fraction

Question

My teacher gave me this question and said you need to understand these types of questions and I have no clue about this question.

Dan gets pic 'n mix when at the cinema and makes his bag up with 3 types of sweet. He picks 3 times as many cola bottles as smarties. He also picks twice as many marshmallows as smarties. What proportion of the bag of sweets are smarties?

Give your answer in its simplest form.

PLEASE help me ASAP I need help because I have a test next week and I need to understand this

Thanks

Answer 1

Let c = cola bottles, s = smarties, and m = marshmallows.

He picks 3 times as many cola bottles as smarties, so we have: s = 3c

He picks twice as many marshmallows as smarties, so we have: s = 2m

An easy way to understand this is using sample numbers.

We know 2m = 3c, so let's consider m = 3, and c = 2.

This gives us s = 6. The total number of sweets is s + c + m = 6 + 3 + 2 = 11.

Therefore, the proportion of smarties is 6/11.

A more abstract way to think about it is like this: we write c and m in terms of s, so we have c = s/3, and m = s/2. The total number of sweets is s + c + m = s + s/3 + s/2 = (11s/6). So we know the total number of sweets in the bag is 11/6 times the number of smarties. Therefore, taking the reciprocal, there are 6/11 smarties in the bag.