The present ages of Amit and Sumit are in the ratio of

Solution

ATQ, The difference between the ages of Amit and Sumit after 6 years will be same as the difference between their present ages. Let the present age of Amit and Sumit be 5x years and 3x years, respectively. For I: 5x − 3x = 8 2x = 8 x = 4 So, required age of Amit 3 years ago = 5x − 3 = 5(4) − 3 = 20 − 3 = 17 years Given pair is (8, 17) → matches So, I is correct. For II: 5x − 3x = 10 2x = 10 x = 5 So, required age of Amit 3 years ago = 5x − 3 = 5(5) − 3 = 25 − 3 = 22 years But given second value is 23 → does not match So, II is incorrect. For III: 5x − 3x = 12 2x = 12 x = 6 So, required age of Amit 3 years ago = 5x − 3 = 5(6) − 3 = 30 − 3 = 27 years Given pair is (12, 27) → matches So, III is correct. For IV: 5x − 3x = 14 2x = 14 x = 7 So, required age of Amit 3 years ago = 5x − 3 = 5(7) − 3 = 35 − 3 = 32 years But given second value is 31 → does not match So, IV is incorrect. Hence, Only I and III are true.