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The sister couples club

The sister couples club

A club consists of 10 girls, whose ages are all different and range from 4 to 13 years. In addition, it has the peculiarity that girls are two to two sisters.

The sums of the ages of the sister couples are 10, 13, 17, 22 and 23 years respectively.

Clara, who is 7 years old, is one of the club members. How old is your sister Berta?

Solution

We write all the possible combinations of sisters classifying them according to sum 10, 13, 17, 22, 23 years so that we have:

Couples that add 10: 6-4
Couples totaling 13: 9-4, 8-5, 7-6
Couples totaling 17: 13-4, 12-5, 11-6, 10-7, 9-8
Couples that add 22: 13-9, 12-10
Couples that add 23: 13-10, 12-11

Therefore, girls of 4 and 6 years old must be sisters, since they are the only ones whose ages add up to 10. Now we suppress those couples that contain 6 or 4. And we have left that girls of 8 and 5 years old have to be sisters because they are the only ones left, whose ages add up to 13.

We repeat the process and eliminate the couples that contain the 5 or the 8 and we have left that the girls of 7 and 10 years have to be sisters for the same reason: they are the only ones whose ages add up to 17.

So Berta, Clara's sister, is 10 years old.