Question

\(C\) is twice efficient as \(A . B\) takes thrice as many days as \(C . A\) takes 12 days to finish the work alone. If they work in pairs (i.e.. \(A B, B C . C A\) ) starting with \(A B\) on the first day then \(B C\) on the second day and \(A C\) on the third day and so on, then how many days are required to finish the work? (a) \(6 \frac{1}{5}\) days (b) \(4.5\) days (c) \(5 \frac{1}{9}\) days (d) 8 days

Short answer

Answer: 6 days and 1/5 of a day.

Step-by-step solution

Calculate Efficiencies of A, B, and C

According to the given information, A takes 12 days to complete the work. Since C is twice as efficient as A, let C's efficiency be 2A. B takes thrice as many days as C, hence B's efficiency will be (1/3)C. Now let's calculate their efficiencies: Efficiency of A = (1/12) Efficiency of C = 2*(1/12) = (1/6) Efficiency of B = (1/3)*(1/6) = (1/18)

Calculate Work Done When Working in Pairs

When A and B work together, their combined efficiency will be: Efficiency of A + Efficiency of B = (1/12) + (1/18) = (5/36) When B and C work together, their combined efficiency will be: Efficiency of B + Efficiency of C = (1/18) + (1/6) = (4/18) = (2/9) When A and C work together, their combined efficiency will be: Efficiency of A + Efficiency of C = (1/12) + (1/6) = (1/4)

Calculate Days Required to Finish the Work

Now, the work is done in pairs and the pairs are rotated every day. Let x be the number of days required to finish the work. Since three pairs complete one rotation in three days, we can write an equation representing the total work done: x * (5/36) + x * (2/9) + x * (1/4) = 1 (x/3)[(5/12) + (2/9) + (1/4)] = 1 (x/3)(15/36) = 1 x = 3 * (36/15) x = 6*(6/5) x = 6 + 6*(1/5) So, the given number of days required to finish the work is 6 days and 1/5 of a day, which is option (a).

Key concepts

Work and Time

Understanding the relationship between work and time is fundamental in solving efficiency problems. In the context of these problems, 'work' usually refers to a measurable amount of tasks that can be completed by a person or a group. The 'time' is the duration required to complete that work. In mathematical terms, we often consider the 'rate' at which work is done, which is simply the amount of work done per unit of time.

For example, if person A can complete a job in 12 days, A's work rate is the reciprocal of the time taken, which translates to \( \frac{1}{12} \) of the work done per day. To find out how much work A does in a day, we multiply A's daily work rate by 1 day. Hence, A's one-day work output is \( \frac{1}{12} \) of the total work. When we talk about two or more individuals working together, we sum their individual work rates to know their combined efficiency. This forms the basis of our equation when calculating the time it will take for a group to finish a task.

Efficiency Calculation

Efficiency calculation involves determining how capable a person or machine is in terms of output produced over a given time. It's quantified by the amount of work each can do per unit of time. This becomes particularly useful when comparing the productivity of several workers and planning collaboration.

To quantify efficiency, we establish rates as we did in our previous example where A's efficiency rate was \( \frac{1}{12} \) of work per day. In our exercise, we're provided additional information: C is twice as efficient as A, and B takes thrice as many days as C to complete the same work. The equations derived from these conditions allow us to determine the rate of work for B and C.

• Efficiency of A: \( \frac{1}{12} \) per day
• Efficiency of C: \( \frac{1}{6} \) per day (since C is twice as efficient as A)
• Efficiency of B: \( \frac{1}{18} \) per day (since B takes thrice as long as C)

Knowing these, we can plan and predict the output when these individuals collaborate, which directly impacts the completion time for any work.

Completing Work in Pairs

When it comes to completing work in pairs, the combined efficiency is greater than that of each individual working alone. This synergy is because the pair can potentially cover for each other's weaknesses and capitalize on each other's strengths.

In our given problem, individuals A, B, and C work in pairs in a rotating system. To determine how quickly they can complete the assigned work, we first calculate the efficiency of each pair by summing the individual efficiencies.

• A and B together: \( \frac{1}{12} + \frac{1}{18} = \frac{5}{36} \) per day
• B and C together: \( \frac{1}{18} + \frac{1}{6} = \frac{2}{9} \) per day
• A and C together: \( \frac{1}{12} + \frac{1}{6} = \frac{1}{4} \) per day

With these efficiencies, we find that the total work completed by the rotating pairs over three days is equivalent to the sum of these rates multiplied by a day's work. By setting up an equation that equals 1 (the total work), we can solve for x, the number of days required to finish the task. This teaches us how the efficiency of collaborative work patterns can affect the overall completion time for any given work.