Question

Aisha, Ben and Cheng entered a weekly school competition. In each week, the three of them came first second and third in some order. The scores for the top 3 places are positive integers and the same each week. The score for first place was more than the score for second place which was more than the score for third place. Aisha won in the first week. At one stage the total score for Aisha was 12, for Ben was 34 and for Cheng 9. How many week(s) did they take to achieve these scores?

Answer 1

Whatever the scores are for first, 2nd and 3rd, we know they add up to some number x handed out each week. So after N weeks, the total number of points handed out is
Nx= 12+34+9= 55
and N= 55/x
We know the points are positive integers and the number of weeks is a positive integer, so the only choices for x are: 1, 5, 11, 55
We know it is not 1 week, because A won 1st place at least once, but is not first now.
It is not 55, because the minimum score would be 55 for winning 3rd place each time.
(3rd place must be worth at least 1 point, 1 being the smallest positive integer)
By the same logic, 11 is also too large.
So 5 weeks must be the answer.

Answer 2

ok thanks