Simplifying more than three ratios

Question

I was asked by s student to solve this question based on ration proportion and my answer was 5:1 although not sure. The question was, The ratio of goats to sheep is 4:3.sheep to cows 5:2.cows to donkeys 3:2. Find the ratio of g:d? Kindly,is it correct?

Answer 1

If you want an answer that rehearses the logic behind why you can just multiply the ratios that stays in the integers, here is such an answer:

For every 2 donkeys, you have 3 cows. For every 2 cows, you have 5 sheep. So for every 6 cows you have 15 sheep, or 4 donkeys. For every 3 sheep you have 4 goats, so for 15 sheep (or 4 donkeys, or 6 cows) you have 20 goats.

So, we've shown that for every 20 goats you have 4 donkeys, so for every 5 goats you have 1 donkey, so goats, donkeys are in ratio 5:1.

You can generalise this argument to persuade yourself that you can always just multiply the ratios A:B, B:C to get to ratio A:C, or indeed that $$A_1:A_n=(A_1:A_2 )\times (A_2:A_3) \times \cdots \times (A_{n-1}:A_n)$$

Answer 2

Because $$g:d=\frac{g}{s}\cdot\frac{s}{c}\cdot\frac{c}{d}=\frac{4}{3}\cdot\frac{5}{2}\cdot\frac{3}{2}=5=5:1.$$