Question

\(B\) and \(C\) are equally efficient, but the efficiency of \(A\) is half of each \(B\) and \(C . A\) and \(B\) started a work and 3 days later \(c\) joined them. If \(A\) alone can do the work in 14 days, then in how many more days the work will be completed? (a) 1 (b) 2 (c) 3 (d) \(4.5\)

Short answer

Answer: 1

Step-by-step solution

Determine A's Efficiency

Since A can complete the work in 14 days, A's efficiency can be calculated as the reciprocal of 14, which is \(\frac{1}{14}\).

Determine B's and C's Efficiency

According to the problem, the efficiency of B and C is twice that of A. So, we can calculate B's and C's efficiency by doubling A's efficiency: \(\frac{1}{14}\times 2=\frac{2}{14}=\frac{1}{7}\).

Calculate Combined Efficiency of A and B

Sum the efficiencies of A and B: \(\frac{1}{14}+\frac{1}{7}=\frac{1+2}{14}=\frac{3}{14}\).

Calculate Work Done in 3 Days

Multiply the combined efficiency of A and B by 3 (days they worked together before C joined) to get the completed fraction of work: \(\frac{3}{14}\times 3 = \frac{9}{14}\).

Calculate Remaining Work

Subtract the work completed by A and B in 3 days from the total work to get the remaining fraction of work: \(1 - \frac{9}{14} = \frac{5}{14}\).

Calculate Combined Efficiency of A, B, and C

Sum the efficiencies of all three workers: \(\frac{1}{14}+\frac{1}{7}+\frac{1}{7}=\frac{1+2+2}{14}=\frac{5}{14}\).

Find the Remaining Days to Complete the Work

Divide the remaining work by the combined efficiency of A, B, and C to find the number of remaining days to complete the work: \(\frac{\frac{5}{14}}{\frac{5}{14}}=1\). So, the work will be completed in 1 more day. The correct answer is (a) 1.

Key concepts

Efficiency in Work Problems

Understanding efficiency in work problems is crucial for solving questions related to time and work. In the context of such problems, 'efficiency' refers to the amount of work a person or machine can complete in a unit of time. It's like understanding how fast a tap can fill a tank, but instead of water, we measure in tasks or units of work.

In our exercise, each person's efficiency is the key to solving the problem. If person A can finish a job in 14 days, we say A's daily work efficiency is the reciprocal of the time taken, which is \( \frac{1}{14} \). This efficiency acts as a rate at which work is done.

Efficiency can often be compared. If B and C are twice as efficient as A, their efficiencies are simply double of A's, leading us to calculate B's and C's efficiency as \( \frac{1}{7} \). This concept is pivotal as it allows us to add together the efficiencies of different workers to find out how quickly they can complete a task when working together, which leads us to our next concept, combined work efficiency.

Combined Work Efficiency

When multiple workers or factors contribute to completing a task, their combined work efficiency becomes the sum of individual efficiencies. It's a bit like a team sport where each player's skill contributes to the overall performance of the team. For instance, if two taps fill a tank faster when opened together, it's a result of their combined flow rate.

In our exercise, after calculating individual efficiencies of A and B, and knowing C is joining later, we add A's and B's efficiencies to get a combined rate. This is represented as \( \frac{1}{14} \)+\( \frac{1}{7} \)=\( \frac{3}{14} \). Here's the clever part: when C joins in, the combined efficiency of A, B, and C, now all working simultaneously, is summed up to give us a new efficiency rate of \( \frac{5}{14} \).

This form of adding efficiencies together allows us to understand the impact of collaboration on work completion rates. Remember, combined efficiency can only be calculated when the workers are contributing to the same task simultaneously.

Time to Complete Work

The time to complete work is the final piece of the puzzle. Once we know how efficient workers are individually or together, we can calculate the time needed to finish a task. It's like knowing the speed of a car and determining how long it will take to travel a certain distance.

In our problem, we first identified the completed work by A and B together before C joined, and then calculated the remaining work. With the combined efficiency of A, B, and C, the remaining work quantity is divided by this efficiency to find out how many more days are required to complete the work. The formula here is Remaining Work ÷ Combined Efficiency = Remaining Time to Complete Work, which in our case is \( \frac{5}{14} \)/\( \frac{5}{14} \) yielding 1. Hence, 1 more day is needed to finish off the task at hand.

Understanding this final step is essential as it directly answers the 'how long' question that often concludes problems in the topic of work and time. Keeping track of what portion of work is finished and what remains is vital for accurate calculation of the completion time.