Question

If \(A\) takes \(4 / 5\) days as \(B\) takes and working together thej require \(\frac{20}{3}\) days to complete the whole work What is the efficiency of \(B\) ? (a) \(6 \frac{2}{3} \%\) (b) \(16 \%\) (c) 5.55\% (d) \(8 \frac{1}{3} \%\)

Short answer

Answer: (a) 6 2/3 %

Step-by-step solution

Find work done by A and B together in 1 day

To find the work done by A and B together in 1 day, we will take the reciprocal of the number of days they take to complete the whole work when working together. $$\text{Work done by A and B together in 1 day} = \frac{1}{\frac{20}{3}} = \frac{3}{20}$$

Find work done by A in 1 day

Since A takes 4/5 days as B takes to complete the task, we can write this relationship as a ratio: $$\frac{\text{Work done by A in 1 day}}{\text{Work done by B in 1 day}} = \frac{4}{5}$$ Let the work done by A in 1 day be x. Then the work done by B in 1 day is 5x/4.

Calculate work done by B in 1 day

We know that the combined work of A and B in 1 day is 3/20. We can write an equation representing this, using the work done by A and B in 1 day. $$x + \frac{5x}{4} = \frac{3}{20}$$ Now, we can solve this equation for x (work done by A in 1 day). $$\frac{9x}{4} = \frac{3}{20}$$ $$x = \frac{1}{20}$$

Calculate work done by B in 1 day using x

We previously found that the work done by B in 1 day is 5x/4. Now that we know the value of x, we can find the work done by B in 1 day. $$\text{Work done by B in 1 day} = \frac{5x}{4} = \frac{5}{4} \times \frac{1}{20} = \frac{1}{16}$$

Calculate the efficiency of B

To find the efficiency of B, we need to express the work done by B in 1 day as a percentage of their combined work in 1 day. We already know that the combined work of A and B in 1 day is 3/20. $$\text{Efficiency of B} = \frac{\text{Work done by B in 1 day}}{\text{Combined work of A and B in 1 day}} \times 100 \%$$ $$\text{Efficiency of B} = \frac{\frac{1}{16}}{\frac{3}{20}} \times 100 \% = \frac{5}{3} \times \frac{1}{5} \times 100 \% = \frac{20}{3} \% = 6\frac{2}{3} \%$$ The efficiency of worker B is \(6 \frac{2}{3} \%\). Therefore, the correct answer is (a) \(6 \frac{2}{3} \%\).

Key concepts

Time and Work

Understanding the concept of time and work is crucial when addressing efficiency and productivity problems. This concept helps us analyze how long it takes for individuals or machines to complete a specific task by themselves or in a group. In the context of the exercise, we compare the completion times of two workers, A and B, and determine their relative efficiencies.

Practically, knowing how time and work relate allows us to allocate resources effectively. It's important to note that not all workers have the same efficiency, which is why in tasks involving multiple workers, calculating individual and collective work rates is essential before making any conclusions about their output.

Reciprocal Method in Work

The reciprocal method in work problems is a powerful technique that simplifies calculations involving rates, work, and time. It is based on the principle that the reciprocal of the time taken to complete a task represents the part of the task completed in a unit time. When two or more subjects work together, as demonstrated in the given exercise, their combined work per unit time is the sum of their individual work rates.

We determine the efficiency of workers by calculating how much work each accomplishes individually within a set period. By focusing on what fraction of the work is done in one day, which is the reciprocal of the total time, we streamline the computation process.

Ratio and Proportion

The concept of ratio and proportion is crucial for understanding relationships between different quantities. In work and time problems, we often compare workers’ efficiency, which translates to comparing how much work each can complete in the same amount of time. Using ratios, we express these comparisons mathematically, while proportions help us determine the relative quantities or rates at which workers complete tasks.

In the provided exercise, we compared the amount of work done by worker A to that of worker B using ratios. These ratios help us determine each worker’s contribution to the task and are key in solving for their individual efficiencies.

Percentage Efficiency

When it comes to assessing performance, percentage efficiency provides a comprehensible measure, indicating how much work can be completed by an individual or a tool in terms of the whole. In essence, it quantifies the ratio of a particular worker’s output against an ideal or combined output as a percentage.

Applying this to labor or machinery allows for a straight comparison in terms of work done, making it easier to understand and compare efficiencies across different entities. In our exercise, determining worker B’s efficiency involved calculating their work rate as a percentage of the total (combined) work rate, which provides not just a clear indication of B’s contribution, but also a simple means to evaluate and compare work done by different workers.