Maths conundrum

[Christiana]

This comes from the Past Times Puzzler Book. A young rider on a merry-go-round dreamt up this problem while enjoying his ride. One third of the number of kids riding ahead of me, added to three-quarters of those riding behind me gives the correct number of children on this merry-go-round. How many children were riding the merry-go-round. I realised you had to count in the child himself but still couldn't work it out. The answers are in the back but no explanation and I am still puzzled. Is it possible to do this sum or have they made a mistake?

[grasscarp]

There must be 13. 12 others give a third 4 plus 9 (3/4)

[gen2]

I got the same answer (13) So what was the answer in the back of the book Christiana?

[gen2]

Let N = number of other riders
Merrygoround is circular so number in front = number behind = N

(1/3)N + (3/4)N = N + 1
LCD=12 so:
(4/12)N + (9/12)N = N + 1
(13/12)N = N + 1
multiply both sides by 12
13N = 12N + 12
Subtract 12N from both sides
N = 12
So there are 12 other riders and with the person themself that makes 13 all together.

[Christiana]

The answer is 13 - you're right, but I still don't get it. One third of 12 is 4 but that is the number of kids riding ahead, leaving none to ride behind. What am I missing?

[grasscarp]

You are in a circle so the number in front and behind is the same....

[Christiana]

Okay, now I get it - it was the circular bit I'd missed. Most of the puzzles have explanations but this one didn't. I tried to use a method similar to yours gen2 but failed at the first hurdle because I hadn't recognised that basic fact.

[grasscarp]

Number of kids ahead is 12 - a third is 4. number of kids behind is 12 and 3/4 of them is 9.....

[grasscarp]

I didnt use N or any calculations, just thought what is the lowest number divisible by 3 and 4.

[Christiana]

Thanks to both of you. It was really annoying me - now, of course, I could kick myself. Still I'm about halfway through the book, which has 366 Brain Teasers in all, so plenty of time t oredeem myself or to shout for help again.