Rates with a parameter: A can do a piece of work in x days and B can do the same work in 3x days. Working together, they finish it in 12 days. Find the value of x.

Introduction / Context:
This parameterized work problem asks you to form and solve an equation in x using combined rates. The key is to express both A’s and B’s rates in terms of x and match their sum to the given joint time of 12 days for the full job.

Given Data / Assumptions:

• A’s time = x days ⇒ rate = 1/x.
• B’s time = 3x days ⇒ rate = 1/(3x).
• Together time = 12 days ⇒ joint rate = 1/12.

Concept / Approach:
Add rates: 1/x + 1/(3x) = (4/3)*(1/x). Set equal to 1/12 and solve for x. Then select the correct numerical value from the options.

Step-by-Step Solution:
1/x + 1/(3x) = (4/3)*(1/x). Set (4/3)*(1/x) = 1/12 ⇒ 1/x = (1/12)*(3/4) = 1/16. Therefore, x = 16.

Verification / Alternative check:
Rates: A = 1/16, B = 1/48; Sum = 1/12, giving 12 days jointly, consistent with the data.

Why Other Options Are Wrong:
8, 10, 12 do not satisfy the rate equation when substituted back.

Common Pitfalls:
Averaging times (x and 3x) or averaging numbers (1 and 3) instead of adding reciprocal rates and equating to 1/12.

Final Answer:
16