A can finish a certain piece of work in 7 days, while B can finish the same work in 9 days. If they work together from the beginning, in how many days will they complete the entire work?

Introduction / Context:
This problem asks for the time taken when two people, A and B, work together on the same task. Each person has a known individual time to finish the work alone. Time and work questions like this rely on converting these times into daily work rates and then combining the rates to find the total time when working jointly.

Given Data / Assumptions:
- A alone can finish the work in 7 days.
- B alone can finish the work in 9 days.
- Both A and B start together and work continuously until the job is done.
- Total work is taken as 1 unit.

Concept / Approach:
When total work is 1 unit, the work rate of a person is the reciprocal of the time taken to complete the work alone. The combined work rate of two people is the sum of their individual rates. Once the combined rate is known, the total time taken to complete one unit of work is simply the reciprocal of this combined rate.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: Rate of A = 1 / 7 work per day. Step 3: Rate of B = 1 / 9 work per day. Step 4: Combined rate of A and B = 1 / 7 + 1 / 9. Step 5: Take LCM of 7 and 9 which is 63, so 1 / 7 = 9 / 63 and 1 / 9 = 7 / 63. Step 6: Combined rate = 9 / 63 + 7 / 63 = 16 / 63 work per day. Step 7: Time taken together = 1 / (16 / 63) = 63 / 16 days.

Verification / Alternative check:
As a quick check, note that A alone takes 7 days and B alone takes 9 days, so the combined time must be less than 7 days. The value 63 / 16 is approximately 3.94 days, which is less than both 7 and 9 days, so it is reasonable. Also, if they work for 63 / 16 days at a rate of 16 / 63 per day, total work equals 63 / 16 * 16 / 63 = 1 unit, confirming correctness.

Why Other Options Are Wrong:
- 31/16 and 47/16: These correspond to significantly shorter times and would imply an unrealistically high combined rate, greater than the sum of the individual rates.
- 79/16: This is about 4.94 days, which is greater than A’s time of 7 days but still incorrect when recomputed with the combined rate.
- 4 days: This is close but not exact. Only 63 / 16 satisfies the precise arithmetic.

Common Pitfalls:
Many learners mistakenly average the times (7 and 9) instead of working with rates. Another frequent error is neglecting to use the LCM for adding fractions and making mistakes in fraction addition. Always convert times to rates, sum the rates, and then take the reciprocal to find the combined time.

Final Answer:
Working together from the start, A and B will complete the work in 63/16 days.