Hari Ram's present age is three times his son's present age and two fifth of his father's present age. The average of the present ages of all three of them (Hari Ram, his son and his father) is 46 years. What is the difference between the present ages of Hari Ram's son and Hari Ram's father?

Introduction / Context: This three generation age problem links Hari Ram with his son and his father using two proportions and an overall average. From these relationships, we can determine each individual's present age and then compute the difference between the youngest and the oldest. It combines proportion and average concepts in a neat way.

Given Data / Assumptions:

• Hari Ram's present age is three times his son's present age.
• Hari Ram's present age is two fifth of his father's present age.
• The average of the present ages of Hari Ram, his son and his father is 46 years.
• We must find the difference between the present ages of his son and his father.

Concept / Approach: Let Hari Ram's present age be H years. Then his son's age is H / 3 and his father's age is (5 / 2) * H. The average age 46 means (H + H / 3 + (5 / 2) * H) / 3 = 46. Solving this equation yields H. We then compute the son's and father's ages and find their difference, which is the required answer.

Step-by-Step Solution: Let Hari Ram's present age be H years. Then his son's present age is H / 3 years. His father's present age is (5 / 2) * H years because H is two fifth of the father's age. The average of their present ages is 46 years, so (H + H / 3 + (5 / 2) * H) / 3 = 46. First find the total age: H + H / 3 + (5 / 2) * H. Compute the coefficients using a common denominator 6: H = 6H / 6, H / 3 = 2H / 6, (5 / 2) * H = 15H / 6. So total age = (6H + 2H + 15H) / 6 = 23H / 6. The average equation becomes (23H / 6) / 3 = 46, that is 23H / 18 = 46. Multiply both sides by 18: 23H = 46 * 18 = 828. So H = 828 / 23 = 36 years. Then son's present age = H / 3 = 36 / 3 = 12 years. Father's present age = (5 / 2) * 36 = 5 * 18 = 90 years. Difference between father's and son's present ages = 90 - 12 = 78 years.

Verification / Alternative check: Check the relationships. Hari Ram is 36, which is three times his son at 12. He is also two fifth of his father, since 2/5 of 90 is 36. The total of their ages is 36 + 12 + 90 = 138, and dividing by 3 gives an average of 46 as stated. All conditions are satisfied, so the calculations are correct and the age difference 78 is confirmed.

Why Other Options Are Wrong: Differences such as 44, 56, 67 or 52 years would require different underlying ages that would not satisfy both the three times and two fifth relations at the same time while also giving an average of 46. Only the combination 12, 36 and 90 gives a consistent solution, which produces a 78 year gap.

Common Pitfalls: Some students incorrectly take H as the son's age instead of Hari Ram's age, which leads to wrong proportions. Others misinterpret "two fifth of his father's age" and invert the fraction. Calculation mistakes in combining fractions and solving for H are also common. Using a common denominator carefully and double checking the proportion directions helps avoid these issues.

Final Answer: Therefore, the difference between the present ages of Hari Ram's son and his father is 78 years.