Seven years ago, the average age of A, B and C was 51 years. If at present A is 3 years older than B and B is 3 years older than C, then what are the present ages (in years) of A, B and C respectively?

Introduction / Context:
This is a typical ages and averages problem. We are given the average age of three people at a past time and the age differences between them in the present. Using these facts, we need to determine their current ages. Such questions test the ability to translate age relationships and average information into algebraic equations.

Given Data / Assumptions:

• Seven years ago, the average age of A, B and C was 51 years.
• At present, A is 3 years older than B.
• At present, B is 3 years older than C.
• We assume ages are in whole years, but the method works for any real ages.

Concept / Approach:
From the given average age some years ago, we can find the total of their ages at that time. We then add 7 years to each person to obtain present ages. In parallel, we express the present ages of A and B in terms of C using the age differences. Equating the total present age obtained from these two methods allows us to solve for the current ages.

Step-by-Step Solution:
Step 1: Compute total age seven years ago. Average age seven years ago = 51 years. Number of persons = 3, so total age then = 51 * 3 = 153 years. Step 2: Express present ages in terms of C. Let present age of C be c. Then present age of B = c + 3. Present age of A = (c + 3) + 3 = c + 6. Step 3: Relate present total age to past total age. Seven years have passed since the ages summed to 153. Each person has gained 7 years, so total gain = 3 * 7 = 21 years. So present total age = 153 + 21 = 174 years. Step 4: Form equation for present total in terms of c. Present total age = A + B + C = (c + 6) + (c + 3) + c = 3c + 9. Thus 3c + 9 = 174. Step 5: Solve for c and find all ages. 3c = 174 - 9 = 165, so c = 55. B = c + 3 = 55 + 3 = 58. A = c + 6 = 55 + 6 = 61.

Verification / Alternative check:
Seven years ago, their ages would have been A = 61 - 7 = 54, B = 58 - 7 = 51, C = 55 - 7 = 48. The average of 54, 51 and 48 is (54 + 51 + 48) / 3 = 153 / 3 = 51, matching the given information. The age gaps A minus B = 3 and B minus C = 3 are also satisfied. Hence the result is correct.

Why Other Options Are Wrong:
Option b and option d reverse the age order or give values that do not differ by exactly 3 years between consecutive persons. Option c lists ages in increasing order 55, 58 and 61, which does not match the requirement that A is the oldest. Moreover, if we test these other triplets seven years ago, the average is not 51 in any of those cases. Only 61, 58 and 55 satisfy all the given conditions.

Common Pitfalls:
Students sometimes subtract or add the 7 years incorrectly, or they mistakenly assume the average remains 51 at present. Another error is to assign the age differences backwards, for example making A younger than C. A disciplined approach is to define one person age as a variable, express others in terms of it, and then use the total from the average to solve the equations.

Final Answer:
The present ages of A, B and C are 61 years, 58 years and 55 years respectively.