Worker A can complete a certain piece of work in 24 days. Worker B is 20% more efficient than A, so B can finish the same work faster. Worker C can complete the work in 10 more days than B takes. How many days will A and C, working together from the beginning, take to complete the entire work?

Introduction / Context:
This time and work problem involves three workers A, B, and C with varying efficiencies. The efficiency of B is related to A by a percentage, and C is slower than B by a fixed number of days. The question asks for the time taken by A and C together to complete the task, which requires converting efficiency relationships into exact work rates and then combining those rates appropriately.

Given Data / Assumptions:

• A alone can complete the work in 24 days.
• B is 20% more efficient than A.
• C takes 10 more days than B to complete the same work alone.
• A and C work together from the beginning.
• Total work is considered as 1 unit, and all workers have constant rates.

Concept / Approach:
We first determine A's work rate, then use the fact that B is 20% more efficient to find B's work rate and corresponding time. Using B's time, we find C's time, which is 10 days more than B's. We then convert C's time into a rate. The combined rate of A and C is the sum of their individual rates, and taking the reciprocal of this combined rate gives the total time required when A and C work together.

Step-by-Step Solution:
Step 1: Assume total work is 1 unit. Step 2: A can do the work in 24 days, so A's rate = 1/24 units per day. Step 3: B is 20% more efficient than A, so B's rate = 1.2 * (1/24) = 1/20 units per day. Hence B's time = 20 days. Step 4: C takes 10 more days than B, so C's time = 20 + 10 = 30 days. Therefore C's rate = 1/30 units per day. Step 5: A and C work together, so combined rate = 1/24 + 1/30. Step 6: Use the least common multiple of 24 and 30, which is 120. Then 1/24 = 5/120 and 1/30 = 4/120. Step 7: Combined rate = 5/120 + 4/120 = 9/120 = 3/40 units per day. Step 8: Time taken by A and C together = total work / combined rate = 1 / (3/40) = 40/3 days.

Verification / Alternative check:
We can estimate numerically. A alone would take 24 days, and C alone would take 30 days, so their average is about 27 days. When they work together, the time must be less than either 24 or 30 days. The answer 40/3 is about 13.33 days, which is comfortably smaller and reasonable since two workers are sharing the workload. Rechecking the rates confirms that the arithmetic is consistent.

Why Other Options Are Wrong:

• 43/3 days: This is more than 14 days and does not match the correctly calculated combined rate.
• 50/3 days: This is around 16.67 days, corresponding to a slower combined rate than 3/40 and does not align with the data.
• 41/2 days: This represents 20.5 days, which is even closer to the time of a single worker and clearly inconsistent with two workers combining their efforts.

Common Pitfalls:
A common error is to treat "20% more efficient" as adding 20% of the time instead of 20% of the rate. Another mistake is to average days instead of combining rates. Students may also mis-handle fractions when adding 1/24 and 1/30, so using the correct least common multiple and carefully adding the converted fractions is very important.

Final Answer:
Workers A and C together will finish the work in 40/3 days.