If I have two friends of different ages, I would ask you to guess "what their ages are." and you would say "He is 21, and I think she is 25." The use of AGES requires are. It is a plural age(s). If you use HE for both it is easier to understand. Guess what his hobby is? Guess what his hobbies are? Now use their. Guess what Their favourite hobby is? Is their favourite hobby fishing? Yes, their favourite hobby is fishing. Guess what Their favourite hobbies are? They like fishing and swimming. Their favourite hobbies are swimming and fishing. "Age is" would probably only be used in a context where someone is trying to guess a combined age. What is their combined age? Their combined age is 42. What are their ages? He is 20 and she is 22. If you are trying to differentiate two different people with different ages the first one is correct Guess what their ages are. If the two people were twins, you might get away with the second, but even then, people would say "guess how old they are?" Their is sometimes used in place of he/she (like on Facebook: see what THEY have posted on their wall, yet it is ONE person). So instead of seeing Write on his wall, you will see write on their wall. Use something other than age and it becomes clearer. Guess what their ideas are, guess what their idea is. If you are using their to refer to TWO distinct people, the first one makes more sense. The second only makes sense if they share an age (they are both 31 years old). You can still use the first sentence in that case EXAMPLE: Hey Bob, you see Steve and Jenny over there? Try to guess what their ages are. Are they 21 and 24? No. Are they the same age? ARE they both 24? Yes they ARE both 24. Age should be used with is and ages with are, however as you are using they they should be proceeded by are. They ARE this age. As opposed to this AGE is when people are allowed to vote. So guess what their ages are? or guess what his/her age is?

They are actually both correct! It depends on if you are referring to multiple people, or one. For example: If I was asking you about the age of Bob, I would ask you "Guess what their age is?" If I asked you about the age of Bob and Sarah (more than one person), I would say "Guess what their ages are?". Another more common way of asking this question though is probably: "Guess how old he is?" if I was talking about Bob, or "Guess how old she is?" if I was talking about Sarah. If I asked you about both Bob and Sarah, I would probably say "Guess how old they are?" So your answers are not wrong, but these ones above are probably more common.