At present, the sum of the ages of four people A, B, C, and D is 76 years. After 7 years, their ages will be in the ratio 7:6:5:8 respectively. What is the present age of person C in years?

Introduction / Context:
This aptitude question on problems related to ages involves four different people whose total present age is known. We are also given a future ratio of their ages after a fixed number of years. The task is to determine the present age of person C. Questions of this type test the ability to convert ratio conditions and total sum information into algebraic equations and then solve them logically.

Given Data / Assumptions:
- There are four people A, B, C, and D.
- The sum of their present ages is 76 years.
- After 7 years, their ages will be in the ratio 7:6:5:8 in the order A, B, C, D.
- All ages are measured in complete years.
- We must find the present age of person C.

Concept / Approach:
The key idea is that when all four ages are shifted by the same number of years, the ratio of their new ages can be represented using a common multiplying factor. We express each person's future age in terms of this factor and then convert back to present ages. Using the fact that the total present age is known, we set up an equation in the common factor and solve for it. Once this factor is known, the present age of each person can be calculated easily.

Step-by-Step Solution:
Step 1: Let the common multiplying factor for the ratio after 7 years be k.Step 2: After 7 years, the ages of A, B, C, and D will be 7k, 6k, 5k, and 8k years respectively.Step 3: Therefore, their present ages are A = 7k - 7, B = 6k - 7, C = 5k - 7, and D = 8k - 7.Step 4: The sum of their present ages is given as (7k - 7) + (6k - 7) + (5k - 7) + (8k - 7) = 76.Step 5: Simplify this expression: 7k + 6k + 5k + 8k = 26k and the constants sum to -28, so 26k - 28 = 76.Step 6: Add 28 to both sides to get 26k = 104. Thus k = 104 / 26 = 4.Step 7: Substitute k = 4 in C = 5k - 7 to get C = 5 * 4 - 7 = 20 - 7 = 13 years.

Verification / Alternative check:
Using k = 4, the present ages are A = 7 * 4 - 7 = 21 years, B = 6 * 4 - 7 = 17 years, C = 13 years, and D = 8 * 4 - 7 = 25 years. The sum 21 + 17 + 13 + 25 equals 76 years, which matches the given total. After 7 years, their ages will be 28, 24, 20, and 32 years. The ratio 28:24:20:32 simplifies by dividing all terms by 4 to 7:6:5:8, which matches the required ratio. This confirms that the value of C is correct.

Why Other Options Are Wrong:
Values like 11 years, 12 years, 15 years, or 17 years for C do not produce a total present age of 76 when combined with consistent ages for A, B, and D that maintain the future ratio 7:6:5:8. When tested, they either break the total sum of 76 or fail to give the correct ratio after 7 years. Therefore, they cannot be correct answers for this problem.

Common Pitfalls:
One common mistake is to try to apply the ratio directly to the present ages instead of to the ages after 7 years. Another error is to forget that the same number, 7, is subtracted from each future age to obtain the present ages. Some learners also mismanage the algebra when summing and simplifying the expressions. Working step by step with a clear definition of k helps avoid these errors.

Final Answer:
The present age of person C is 13 years.