A is three times as efficient as B, and C is twice as efficient as B. What is the ratio of the number of days taken by A, B, and C individually to complete the same piece of work?

Introduction / Context:
This question checks your understanding of how efficiency (work rate) is related to time taken to finish a job. You are given relative efficiencies of three workers A, B, and C and you need to find the ratio of time each one would take to complete the same work individually. The key idea is that higher efficiency implies less time required for the same amount of work.

Given Data / Assumptions:
We are told that A is three times as efficient as B, and C is twice as efficient as B. All three are assumed to be working on the same job. Efficiency is defined as work done per unit time, and we assume the total work to be a fixed quantity, such as one complete job.

Concept / Approach:
Time taken to finish a job is inversely proportional to the efficiency. If the efficiency of one worker is k times another, then the time taken will be 1 / k times that of the other worker. We first assign a base efficiency to B, express A and C in terms of this base, and then use the inverse proportionality to compute the ratio of times for A, B, and C. Finally, we simplify this ratio to its lowest integer form.

Step-by-Step Solution:
Step 1: Let the efficiency (work rate) of B be 1 unit of work per day. Step 2: Since A is three times as efficient as B, A’s efficiency is 3 units of work per day. Step 3: Since C is twice as efficient as B, C’s efficiency is 2 units of work per day. Step 4: Time taken to complete the same work is inversely proportional to efficiency. Thus, time for A is proportional to 1 / 3, for B to 1, and for C to 1 / 2. Step 5: Let us write the raw ratio of times: time(A) : time(B) : time(C) = 1 / 3 : 1 : 1 / 2. Step 6: To remove fractions, multiply each term by 6 (the least common multiple of 3 and 2): (1 / 3)*6 = 2, 1*6 = 6, (1 / 2)*6 = 3. Step 7: So the simplified ratio of days taken by A, B, and C is 2 : 6 : 3.

Verification / Alternative check:
As a quick check, imagine the job requires 6 units of work. At their assumed rates, A would take 6 / 3 = 2 days, B would take 6 / 1 = 6 days, and C would take 6 / 2 = 3 days. The days taken clearly form the ratio 2 : 6 : 3 which matches our derived result. This sanity check confirms that our proportional reasoning is consistent.

Why Other Options Are Wrong:
The ratio 2 : 3 : 6 incorrectly suggests that C is taking the longest time, which contradicts C being more efficient than B. The ratio 1 : 2 : 3 suggests that A takes the least time correctly but underestimates the relative difference between the workers. The ratio 3 : 1 : 2 and 3 : 2 : 1 both contradict the fact that A is the most efficient and should therefore take the least time. Only 2 : 6 : 3 preserves the correct inverse relationship between efficiency and time.

Common Pitfalls:
A common error is to directly compare times in the same ratio as efficiencies instead of taking the reciprocal. Another mistake is to mix up which worker is fastest and which is slowest when converting back from relative efficiencies to time taken. Always remember that if a worker is more efficient, they must take fewer days, and the ratio for time must be the reciprocal of the efficiency ratio.

Final Answer:
Thus, the ratio of the number of days taken by A, B, and C individually is 2 : 6 : 3.