The present ages of 'A' and 'B' are in the ratio of 5:8. Five years from now, A's age will be 25% less than C's age at that time. If the present age of 'C' is an integer and the combined present ages of 'A' and 'B' are less than 110 years, determine the maximum possible present age of 'C'.

Let the present ages of 'A' and 'B' be '5x' years and '8x' years, respectively. So, age of 'C' five years hence from now = (5x + 5) ÷ 0.75 = 5(x + 1)/(3/4) = 20(x + 1)/3 So,present age of 'C' = 20(x + 1)/3 - 5 Since, present age of 'C' is an integer, (x + 1) must be divisible by 3. (As 20 is not) So, possible values of 'x' = 2, 5, 8 and so on. At 'x' = 2, present age of 'C' = {20 X 3 ÷ 3} - 5 = 15 years And sum of ages of 'A' and 'B' = 5x + 8x = 13x = 13 X 2 = 26 years At 'x' = 5, present age of 'C' = {20 X 6 ÷ 3} - 5 = 35 years And sum of ages of 'A' and 'B' = 5x + 8x = 13x = 13 X 5 = 65 years At 'x' = 8, present age of 'C' = {20 X 9 ÷ 3} - 5 = 55 years And sum of ages of 'A' and 'B' = 5x + 8x = 13x = 13 X 8 = 104 years At 'x' = 11, present age of 'C' = {20 X 12 ÷ 3} - 5 = 75 years And sum of ages of 'A' and 'B' = 5x + 8x = 13x = 13 X 11 = 143 years Since, sum of ages of 'A' and 'B' must be less than 110, the maximum possible age of 'C' is attained at 'x' = 8. So, maximum present age of 'C' = 55 years