Question

\(A\) can do a piece of work in 8 days, \(B\) can do it in 16 days, while \(C\) can do it in 80 days. In how many days they can complete the whole work, working together? (a) 5 (b) 6 (c) \(8 \frac{2}{3}\) (d) \(20 \frac{2}{5}\)

Short answer

Answer: (a) 5 days

Step-by-step solution

Determine the fraction of work completed in one day by each individual

In one day, \(A\) completes \(\frac{1}{8}\) of the work, \(B\) completes \(\frac{1}{16}\) of the work, and \(C\) completes \(\frac{1}{80}\) of the work.

Calculate the total fraction of work completed by all three in one day

To find the total fraction of work completed by \(A\), \(B\), and \(C\) in one day, add the fractions from Step 1: \(\frac{1}{8} + \frac{1}{16} + \frac{1}{80} = \frac{10+5+1}{80} = \frac{16}{80}\).

Find the number of days needed to complete the work

To find the number of days needed to complete the work, take the reciprocal of the total fraction of work completed in one day: \(\dfrac{1}{\frac{16}{80}} = \dfrac{80}{16} = 5\). So, working together, \(A\), \(B\), and \(C\) can complete the work in 5 days. The correct answer is (a) 5.

Key concepts

Understanding Work Efficiency

When we talk about work efficiency in the context of time and work problems, we are referring to the ability of a person to complete a task within a given time frame. It's a measure of productivity, meaning how much work a person or a machine can accomplish in a unit of time. The greater the work efficiency, the more work gets done in less time.

In the provided textbook exercise, we see 'A' can do a piece of work in 8 days, which implies that 'A's efficiency allows them to complete the work within this period. 'B' and 'C', having higher number of days to finish the same task, have lower efficiency compared to 'A'. When combined, their collaborative work efficiency increases, allowing them to complete the task much quicker than any of them would individually.

Fractional Work Rate Explained

A fractional work rate refers to the part of the whole task completed in a unit of time, usually one day. This concept is crucial as it allows us to break down the entire work into manageable parts that can be attributed to workers or machines.

Fractional Work Rate in Practice

For instance, if 'A' can complete the work in 8 days, 'A's fractional work rate is \(\frac{1}{8}\) per day, meaning each day 'A' finishes one-eighth of the total task. 'B's rate being \(\frac{1}{16}\) and 'C's rate at \(\frac{1}{80}\) indicates they contribute less to the task per day. It's the division of the whole into equal fractional parts based on the time taken by each worker.

Collaborative Work Rate

The collaborative work rate is the sum of individual work rates when multiple entities are working together on a task. It provides insight into the combined productivity and is essential for planning and time management.

Combining Individual Efforts

Let's consider the combined work rate from our exercise. By adding \(\frac{1}{8}\), \(\frac{1}{16}\), and \(\frac{1}{80}\), which are the individual rates of 'A', 'B', and 'C', we obtain a collaborative work rate of \(\frac{16}{80}\) or \(\frac{1}{5}\) of the work done per day. This indicates collectively, every day, the trio completes a fifth of the entire job.

Reciprocal Work Time

Reciprocal work time is the inverse of the total work rate and is used to calculate the total time taken by all workers combined to complete a task. The reciprocal essentially tells us how many units of time are needed for the group to finish the work.

Reciprocal in Action

In this scenario, after finding the collaborative work rate to be \(\frac{1}{5}\), we calculate the reciprocal of this fraction to determine the total time needed. The reciprocal of \(\frac{1}{5}\) is 5, meaning 'A', 'B', and 'C' working together will finish the task in 5 days. Remember, the reciprocal work time gives us the total duration when workers pool their efforts.