Who won the backstroke?

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Quandaries & Queries

Three girls held a series of swimming races among themselves. They
decided to award a positive whole number of points for finishing first,
second and third, where the number of points for first was more than the
number for second, which was more than the number for third. The same
number of points was awarded for the place in each race. There were no
ties.

The first race was Breaststroke. Altogether Sarah accumulated 20 points,
Michelle 10 points and Catherine 9 points. If Sarah did not win the
backstroke race, who did win the backstroke?

Hi Timothy,

I can get you started.

Altogether the three of them accumulated 20 + 10 + 9 = 39 points. The prime factorization of 39 is 3 13 so the series of races was either 3 races with a total of 13 points per race or 13 races with a total of 3 points per race. The second option isn't possible (Can you see why?) and hence you know that

There were three races and the total number of points in each race was 13.

Can you see how to proceed?
Penny

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