Question

Krishna can do a work in 10 days while Mohan can do the same work in 20 days. They started work together. After 3 days Krishna left the work and Mohan completed it. For how many days Mohan worked alone more than the number of days required when both worked together? (a) \(4 \frac{1}{3}\) (b) \(3 \frac{1}{4}\) (c) \(2 \frac{3}{5}\) (d) \(3 \frac{2}{3}\)

Short answer

Answer: Mohan worked an additional \(2 \frac{2}{3}\) days when he worked alone compared to when they worked together.

Step-by-step solution

Calculate Work Per Day for Each Person

To determine the work done by each person in one day, divide 1 by the total number of days taken by each person to complete the work alone. Let the work done by Krishna in one day be \(K_w\) and the work done by Mohan in one day be \(M_w\). \(K_w = \frac{1}{10}\) and \(M_w = \frac{1}{20}\)

Calculate Combined Efficiency

Sum the daily work done by Krishna and Mohan to find their combined efficiency. \(Combined\_efficiency = K_w + M_w = \frac{1}{10} + \frac{1}{20} = \frac{3}{20}\)

Calculate Work Done by Krishna

Find the work done by Krishna in the 3 days he worked together with Mohan. \(K\_work = K_w \times 3 = \frac{1}{10} \times 3 = \frac{3}{10}\)

Calculate Remaining Work

Subtract the work done by Krishna from the total work (1) to find the remaining work to be completed by Mohan. \(Remaining\_work = 1 - K\_work = 1 - \frac{3}{10} = \frac{7}{10}\)

Calculate Mohan's Working Days

Divide the remaining work by Mohan's daily work to find how many days it takes for him to complete the remaining task. \(Mohan\_days = \frac{Remaining\_work}{M_w} = \frac{\frac{7}{10}}{\frac{1}{20}} = 14\)

Calculate Additional Days

Subtract the original number of days Mohan and Krishna would have taken together from the total days Mohan took to complete the task alone. \(Additional\_days = Mohan\_days - \frac{1}{Combined\_efficiency} = 14 - \frac{1}{\frac{3}{20}} = 14 - \frac{20}{3} = 2 \frac{2}{3}\) So Mohan worked alone for another \(2 \frac{2}{3}\) days than when they worked together. The correct answer would be close to option (c) \(2 \frac{3}{5}\), but none of the given options is exactly the right value.

Key concepts

Work Efficiency

Work efficiency is all about determining how much work a person can complete in a given time. Here, we measure this efficiency by finding out the fraction of the whole work each person can finish in one day. For example:

• If Krishna can finish the entire work in 10 days, she completes \( \frac{1}{10} \) of the work in one day, which gives us Krishna's work efficiency \( (K_w) \).
• Similarly, Mohan, who can complete the work in 20 days, finishes \( \frac{1}{20} \) of the work each day, which is Mohan's work efficiency \( (M_w) \).

This shows how quickly each person completes their portion of the work daily, offering a glimpse into their individual efficiencies. The higher this daily fraction, the more efficient that person is in completing the task quickly.

Combined Work Rate

The combined work rate is the total amount of work two or more individuals can accomplish together in a specific period.To find this rate, we simply add each person's one day work efficiency. For Krishna and Mohan working together:\[ Combined\_efficiency = K_w + M_w = \frac{1}{10} + \frac{1}{20} = \frac{3}{20} \]This means that together, Krishna and Mohan can complete \( \frac{3}{20} \) of the work in one day. This combined rate is useful for understanding how quickly the task can be completed when both contribute simultaneously.Understanding the combined work rate helps in drawing a clear picture of the collective effort and time saved when working together, rather than individually.

Fraction of Work Completed

The fraction of work completed refers to how much of the total task has been done at a particular point in time. This is essential to determine progress and estimate the remaining work.When Krishna worked together with Mohan for 3 days, finding the exact fraction completed aids in visualizing how much has been accomplished before Krishna left:\[ K\_work = K_w \times 3 = \frac{1}{10} \times 3 = \frac{3}{10} \]So, Krishna completed \( \frac{3}{10} \) of the work during the initial days. Now, the remaining work for Mohan is:\[ Remaining\_work = 1 - K\_work = 1 - \frac{3}{10} = \frac{7}{10} \]By calculating these fractions, we can easily track work progress and plan the completion efficiently.

Individual Work Contribution

Knowing the individual work contribution means understanding what portion of the entire job each person has done or must do. This insight ensures clarity about each person's responsibility in completing a task.After Krishna left, Mohan was left with part of the task:- The remaining work Mohan needed to complete was \( \frac{7}{10} \).- Given Mohan's daily efficiency \( M_w = \frac{1}{20} \), it took him: \[ Mohan\_days = \frac{Remaining\_work}{M_w} = \frac{\frac{7}{10}}{\frac{1}{20}} = 14 \text{ days} \]Here, we see that Mohan took a significant chunk of the work, illustrating his contribution after Krishna's departure. Understanding this helps in recognizing the individual responsibility and contribution towards completing a task.