A can complete a piece of work in 30 days. He works alone on the job for 6 days and then B finishes the remaining work in 18 days. In how many days can A and B together complete the same work if both work at their usual rates from the beginning?

Introduction / Context:
This time and work problem gives the individual time of A, a partial working period by A, and then the time taken by B to finish the remaining portion of the job. Using this information, you must determine how long A and B would take to complete the same job working together from the start at their usual rates.

Given Data / Assumptions:
A can complete the entire work in 30 days when working alone. A works alone for 6 days on the job and then stops. B then completes the remaining work alone in 18 days. We assume that both A and B work at constant rates and that the total work is one complete job. The goal is to find the total time if A and B work together on the entire job from the beginning.

Concept / Approach:
We use the concept of work rates: A’s rate is the reciprocal of 30 days. By calculating how much of the job A completes in 6 days, we determine what fraction of the work remains for B. From this, we can calculate B’s rate. Once we know both A and B’s rates, we add them to get the combined rate and then take the reciprocal to obtain the time required when they work together from the start.

Step-by-Step Solution:
Step 1: Let the total work be 1 unit. Step 2: A alone completes the work in 30 days, so A’s rate is 1 / 30 of the work per day. Step 3: In 6 days, the amount of work done by A is 6 * (1 / 30) = 6 / 30 = 1 / 5 of the total work. Step 4: Therefore, the remaining work after A stops is 1 - 1 / 5 = 4 / 5 of the total work. Step 5: B completes this remaining 4 / 5 of the work in 18 days. Step 6: B’s daily rate is (4 / 5) / 18 = 4 / (5 * 18) = 4 / 90 = 2 / 45 of the work per day. Step 7: Now we know A’s rate is 1 / 30 and B’s rate is 2 / 45. The combined rate when working together is (1 / 30 + 2 / 45). Step 8: To add the rates, use a common denominator of 90. Then 1 / 30 = 3 / 90 and 2 / 45 = 4 / 90. Step 9: Combined rate = 3 / 90 + 4 / 90 = 7 / 90 of the work per day. Step 10: Time taken when working together is the reciprocal of this combined rate: time = 1 / (7 / 90) = 90 / 7 days. Step 11: 90 / 7 days can be written as 12 6/7 days.

Verification / Alternative check:
To verify, we compute how much work A and B would do in 12 6/7 days. In fractional form, 12 6/7 days is 90 / 7 days. At the combined rate of 7 / 90 per day, total work done = (7 / 90) * (90 / 7) = 1 unit of work, which matches the definition of the complete job. The internal consistency check using the derived rates confirms that our answer is correct.

Why Other Options Are Wrong:
14 1/2, 11, 13 1/4, and 15 days all correspond to different combined rates that do not match the individual rates deduced from the given partial work scenario. For instance, 11 days would imply a combined rate of 1 / 11, which is significantly larger than 7 / 90, and would not be consistent with B needing 18 days to complete 4 / 5 of the work alone. Only 12 6/7 days arises naturally from the algebraic calculations based on the given information.

Common Pitfalls:
Students sometimes forget to subtract the work already done by A before assigning the remainder to B, or they miscalculate the fraction 4 / 5. Another common mistake is to average the times 30 and 18 instead of properly calculating the work completed and the rates. Always move step by step: compute partial work, compute remaining work, determine individual rates, add them, and then compute the reciprocal for total time.

Final Answer:
A and B working together from the start will complete the work in 12 6/7 days.