Nine years later, the age of B will be equal to the present age of A. The sum of A age 3 years later and B age 4 years ago is 76. If C is currently half of the present age of B, then what will be C age in years after 10 years?

Introduction / Context:
This is a multi person age problem involving three individuals A, B and C. Two separate conditions relate the ages of A and B at different times, and a third condition defines C age in terms of B present age. The question then asks for C age after 10 years. Such questions test your ability to combine multiple time shifted age relationships and then propagate the result to a third person.

Given Data / Assumptions:

• Nine years later, B age will be equal to the present age of A.
• The sum of A age 3 years later and B age 4 years ago is 76.
• C is currently half of B present age.
• We are required to find C age after 10 years.
• All ages change linearly and are measured in full years.

Concept / Approach:
We set up variables for the present ages of A and B. The first condition gives a direct equation linking A and B by comparing B future age to A present age. The second condition provides a second equation that involves both A and B again but now in terms of future and past ages. Solving this system gives the present values of A and B. Then, using the relation that C is half of B present age, we can find C now and then add 10 years to get C future age. This method follows a simple algebraic framework but requires careful attention to time shifts.

Step-by-Step Solution:
Step 1: Let the present ages of A and B be a and b years respectively. Step 2: Nine years later, B age will be b + 9, and this is equal to the present age of A, so a = b + 9. Step 3: A age 3 years later is a + 3 and B age 4 years ago is b − 4. Their sum is given as 76, so (a + 3) + (b − 4) = 76. Step 4: Simplify the second equation to a + b − 1 = 76, so a + b = 77. Step 5: Solve the system a = b + 9 and a + b = 77. This gives a = 43 years and b = 34 years. Step 6: C is half of B present age, so C now is 34 ÷ 2 = 17 years. After 10 years, C age will be 17 + 10 = 27 years.

Verification / Alternative check:
Verify the conditions with a = 43 and b = 34. Nine years later, B will be 34 + 9 = 43 years, equal to A present age, satisfying the first condition. Three years later, A will be 46 and four years ago, B was 34 − 4 = 30. Their sum is 46 + 30 = 76, which matches the second condition. Since C is half of 34, C is 17 now and will be 27 after 10 years. All parts of the problem are consistent, confirming the correctness of the solution.

Why Other Options Are Wrong:
Option 32, 36, 31, and 29: All of these values assume incorrect present ages for A and B when traced back into the equations. Substituting them leads to sums or equalities that do not match the stated conditions, so they cannot be right for C age after 10 years.

Common Pitfalls:
It is easy to mix up whether an age is present, past, or future. Many learners mistakenly set a = b − 9 instead of a = b + 9 or incorrectly handle the sum involving future and past ages. A good practice is to sketch a small timeline for each person, label present, future, and past points, and only then translate the statements into equations. This reduces confusion and makes the algebra much more reliable.

Final Answer:
C age after 10 years will be 27 years.