Money good for separate wages: The same sum pays A’s wages for 21 days or B’s wages for 28 days. For how many days will it pay both A and B together if both are paid daily and the entire sum is used?

Introduction / Context:
If a fixed sum can pay one worker for a certain duration and another for a different duration, the daily wages are inversely related to those durations. To find how long the same sum can pay both together, compute the combined daily outlay and divide the total sum by that outlay.

Given Data / Assumptions:

• The sum equals 21 days of A’s wage or 28 days of B’s wage.
• Daily wages are constant for each worker.
• All payments are made daily; both work together the entire time.

Concept / Approach:
Let A’s daily wage be a and B’s be b. Then 21a = 28b = total sum M. Use the relation a/b = 28/21 = 4/3. Compute M/(a + b) to get the number of days the same sum sustains both together.

Step-by-Step Solution:

Verification / Alternative check:
Alternatively use harmonic manipulation: 1/Days = 1/21 + 1/28 = (4 + 3)/84 = 7/84 ⇒ Days = 12.

Why Other Options Are Wrong:
24 1/2 and 14 ignore combined outlay; 12 1/4 is close but incorrect; 10 1/2 does not fit the precise ratio.

Common Pitfalls:
Averaging 21 and 28 instead of using reciprocal relationships or not converting to daily outlay correctly.

Final Answer:
12 days