Mr and Mrs Black took their three children from Norwich to Cromer by train.The tickets for the five people totalled £31.50.All the children travelled at half price .how much was an adult ticket?

Probably best looked at by considering it in units : so an adult ticket is 2 units and a child's 1 unit

so we have 2+2+1+1+1=7

does that help?

Mr and Mrs Black = 2 adult fares.

Child fare = 1/2 of adult fare so 3 children = 1.5 adult fares

so £31.50 for 3.5 adult fares

They should have bought a family railcard!

Good point! and sometimes they do a discount ticket for 4 and they could have bought one more ticket for one of the kids

C2C do a group discount of a third off for 3-9 people travelling together and I think you can turn a child into an adult if necessary. Greater Anglia ( presumably the operator relevant to the question) also at least used to do something similar - travelling from Chelmsford to Colchester with the DC when they were much younger but still liable to pay fares, it was quite amusing to have at least one of them made honorary adults for the day ,

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actually the bit about honorary adult rings a bell now -

we had a day trip from Blaenau Ffestiniog to Llandudno - it was cheaper to buy a single ticket for 4 adults than 2 adults / 2 children.

we can explain to kids this way :

consider
adult is X
child is Y

Now adults are 2 so 2X
children 3 so 3Y

2X + 3Y = 31.50 ( next step hint : now needs to convert Y in to X then only we can get value of Adult Fare)

child fare is half of adult fare so Y = 1/2 X

So replace Y to 1/2 X

2X +3/2 X = 31.50
3.5 X = 31.50
X = 9
Adult fare is £9

( Adult fare - £9 , child fair is half of adult fare - £4.5 , 2 adult fare - £18 , 3 child fare - £13.5 , so total = adult fare + child fare , total = 18+13.5 , total = £31.5 , so answer is correct )

Don't know this is the easy way or not

It is not an easy way to explain this problem to a 10-year-old child. They know just very beginning of algebra. Diagram of the trial-and-error methods work here.

I would get primary children to use something like the way I suggested.
Papers/school would not be expecting a KS2 child to immediately use algebra.

Diagrams help.

Get the child/yourself to actually think around the problem. What information do you already know? How will you use that to find out the answer? This makes you a better mathematician rather than just asking someone to work it out ir give you all the details.