Worker A is 50% more efficient than worker B. They start working together on a job, and after 8 days, 7/12 of the work is still left unfinished. In how many days would B alone be able to complete the entire work from the beginning?

Introduction / Context:
This question examines how to use relative efficiency information and partial work done to determine the time required by a single worker. Knowing that A is 50% more efficient than B lets us express one worker's rate in terms of the other. The question also provides how much work remains after a specific time period when both work together, which allows us to solve for B's individual time to complete the full job alone.

Given Data / Assumptions:

• Worker A is 50% more efficient than worker B.
• A and B work together for 8 days.
• After 8 days, 7/12 of the work is still left.
• That means 5/12 of the work has been completed in 8 days.
• We assume total work is 1 unit and all work rates are constant.

Concept / Approach:
First, we express A's work rate in terms of B's rate because efficiency is proportional to rate. If B's rate is b units per day, then A's rate is 1.5b units per day. Their combined rate is then 2.5b. We use the information that 5/12 of the job is completed in 8 days to calculate b. Once we know B's rate, we can invert it to find the number of days B alone would take to do the whole job.

Step-by-Step Solution:
Step 1: Let B's rate be b units per day. Step 2: A is 50% more efficient, so A's rate = 1.5b units per day. Step 3: Combined rate of A and B = b + 1.5b = 2.5b units per day. Step 4: In 8 days, the work done by them together = 8 * 2.5b = 20b units. Step 5: We know that this equals 5/12 of the total work (since 7/12 is left). So, 20b = 5/12. Step 6: Solve for b: b = (5/12) / 20 = 5 / 240 = 1/48 units per day. Step 7: B's time to complete the full job alone = 1 / b = 1 / (1/48) = 48 days.

Verification / Alternative check:
Check the combined work done in 8 days using the found rates. A's rate = 1.5b = 1.5 * (1/48) = 1/32 units per day. B's rate = 1/48 units per day. Combined rate = 1/32 + 1/48. Using a common denominator of 96, we get 3/96 + 2/96 = 5/96 units per day. In 8 days, total work = 8 * (5/96) = 40/96 = 5/12, which matches the given information that 5/12 of the work is completed. Hence the calculation is correct.

Why Other Options Are Wrong:

• 36 days: This would give B a rate of 1/36, which does not produce the correct fraction of work completed in 8 days.
• 40 days: Implies a rate of 1/40, again inconsistent with 5/12 of work being done in 8 days by both workers.
• 60 days: This is slower than the correct value and does not satisfy the relationship derived from the partial work information.

Common Pitfalls:
Some students confuse "50% more efficient" with "works in 50% of the time." Efficiency relates to rate, not directly to days, so we must be careful to scale rates, not times. Others forget that only 5/12 of the work is completed, not 7/12, which leads to incorrect equations. Being consistent about which fraction of work has been done and carefully solving the algebraic equation are crucial steps.

Final Answer:
B alone would take 48 days to complete the entire work from the beginning.