13.171. In a family, there is a father, a mother, and three daughters; together they are 90 years old. The age difference between the girls is 2 years. The mother's age is 10 years more than the sum of the daughters' ages. The difference in age between the father and the mother is equal to the age of the middle daughter. How old is each member of the family?
Solution.

Let $x$ years be the age of the youngest daughter. Then the age of the middle daughter is $-x+2$ years, the oldest is $-x+4$ years, the mother is $x+x+2+x+4+10=3x+16$ years old, and the father is $3x+16+x+2=4x+18$ years old. According to the condition, $x+x+2+x+4+3x+16+4x+18=90$, from which $x=5$ years is the age of the youngest daughter, $5+2=7$ years for the middle daughter, $5+4=9$ years for the oldest, $3 \cdot 5+16=31$ years for the mother, and $4 \cdot 5+18=38$ years for the father.

Answer: $38, 31, 5, 7$ and 9.