The present ages of A, B and C add up to 54 years. Five years ago, the ratio of the ages of A to B was 5 : 3. Three years from now, the ratio of the ages of B to C will be 4 : 3. What is the present age of C?

ATQ, Let present ages be A, B, C. Given: A + B + C = 54 …(1) Five years ago: (A − 5) : (B − 5) = 5 : 3 ⇒ 3(A − 5) = 5(B − 5) 3A − 15 = 5B − 25 3A − 5B = −10 …(2) Three years hence: (B + 3) : (C + 3) = 4 : 3 ⇒ 3(B + 3) = 4(C + 3) 3B + 9 = 4C + 12 3B − 4C = 3 …(3) From (2): 3A = 5B − 10 ⇒ A = (5B − 10)/3 From (3): 4C = 3B − 3 ⇒ C = (3B − 3)/4 Substitute in (1): A + B + C = 54 (5B − 10)/3 + B + (3B − 3)/4 = 54 LCM(3,4) = 12 [4(5B − 10) + 12B + 3(3B − 3)] / 12 = 54 (20B − 40 + 12B + 9B − 9) / 12 = 54 (41B − 49)/12 = 54 41B − 49 = 648 41B = 697 B = 697 / 41 = 17 Then: A = (5×17 − 10)/3 = (85 − 10)/3 = 75/3 = 25 C = (3×17 − 3)/4 = (51 − 3)/4 = 48/4 = 12 So C’s present age = 12 years.