A lot of you had a really good attempt at this question. In addition to Ruth, Tom and Ollie (whose solutions are given below), well done to those of you who got the answer correct:

Talia and Maelona from Jebel Ali school, Dubai; Tom from C.P.S.; Katie, Oliver and Daniella from Portsmouth Grammar School; Amy from Stoke by Nayland Middle School; Daniel from Camberwell South Primary School, Melbourne, Australia; Zak from Clavering Primary School; Andrew Mott from Aldermaston C.E. primary school; Alice and Miriam from Aldermaston Primary; Alara from IK Primary School and Jason (school unknown).

Ruth from Oxford High sent in this clearly explained solution:

In the first problem Rachel has double the number of peanuts to cashew nuts. She has 8 cashew nuts in her bag so you double the number of cashew nuts and you get 16 peanuts.

In the second problem Marianne also has a bag of nuts but this time she has 3 peanuts for every 2 cashew nuts. Marianne has 12 peanuts which is 4 times bigger than 3. So she will have 8 cashew nuts because this is 4 times bigger than 2.

When you add the two bags together they make the following:

 Bag Owner Cashews Peanuts Rachel 8 16 Marianne 8 12 Total 16 28

16 : 28 = 4 : 7
Because both 16 and 28 can be divided by 4, the ratio in its simplest form is 4 : 7

Tom and Ollie of Ardingly College had a slightly different way of explaining it:

Q1: Rachel's bag of nuts should contain 8 cashews and 16 peanuts. We found this out by doubling the cashews and converting them to peanuts.

Q2: Marianne's bag of nuts should contain 12 peanuts and 8 cashews we found out how many cashews there were by working through the amounts in stages. These are our workings out:
2-3
4-6
6-9
8-12

Q3: Rachel and Marianne's bag of mixed nuts should contain a cancelled down ratio of 4:7 (left = cashews right = peanuts). We worked this out by adding the different nuts up in groups; the uncancelled ratio was 16:28. We cancelled the ratio down twice.

How to do the last section:
To do this last part you must know all the amounts of peanuts and cashew nuts in both girls' bags. The nuts from Rachel's bag were 8 cashew nuts and 16 peanuts and from Marianne's bag were 8 cashew nuts and 12 peanuts. Next you must add the amount of peanuts together which makes 28 peanuts and the amount of cashew nuts which gives the answer of 16 peanuts. This should be written as 28:16 or 16:28 which can be reduced by the number which goes into them both by dividing them. You will know your answer when no number goes into both of them.