Question

Kavita, Babita and Samita started a work. 5 days later Samita left the work and Babita left the work after working 8 days. In how many more days Kavita would have completed the rest work if they take 20,60 and 30 days individually to finish a work? (a) 4 (b) 5 (d) 8 (c) 6

Short answer

Answer: Kavita does not need any more days to complete the work, as the work has already been finished during the time Kavita worked together with Samita and Babita.

Step-by-step solution

Find the work done by each person per day

Since each person takes a different amount of time to finish the work individually, we can calculate the work done by each person in one day by finding the reciprocal of the amount of days it takes for them to finish the work individually. Kavita takes 20 days, so she completes 1/20 of the work in a day. Babita takes 60 days, so she completes 1/60 of the work in a day. Samita takes 30 days, so she completes 1/30 of the work in a day.

Calculate the work done while all three work together for 5 days

For the first 5 days, all three of them work together. To find out their combined work for these 5 days, we need to sum up the daily work done by each person and then multiply it by the number of working days. Daily work done while all three work together: (1/20 + 1/60 + 1/30) = (3/60 + 1/60 + 2/60) = 6/60 = 1/10 Work done in 5 days: 5 * 1/10 = 1/2 So, half of the work is completed in these 5 days.

Calculate the work done by Kavita and Babita after Samita leaves for 8 days

Now, Samita leaves after 5 days, and Babita works for 8 more days. We need to find the combined work of Kavita and Babita for these 8 days. Daily work done while Kavita and Babita work together: (1/20 + 1/60) = 4/60 = 1/15 Work done in 8 days: 8 * 1/15 = 8/15 So, 8/15 of the work is completed in these 8 days.

Calculate the remaining work to be done by Kavita alone

To find the remaining work, we will subtract the work done in steps 2 and 3 from the total work, which is assumed to be 1. Remaining work = 1 - (1/2 + 8/15) = 1 - (15/30 + 16/30) = 1 - 31/30 = -1/30 Since the remaining work is negative, it indicates that the work must have already been completed during the time Kavita worked together with Samita and Babita. Therefore, Kavita does not need any more days to complete the work. However, there seems to be an error in the problem statement, as none of the given options match this conclusion. It's possible that a typo or omission occurred in the original exercise.

Key concepts

Problem Solving

Tackling word problems in mathematics often involves understanding the scenario presented, identifying the required solution, and determining the steps necessary to arrive at it.
This exercise is a classic example where we have to understand how different people working together impact the completion of a task.

• Step-by-Step Approach: When you encounter such problems, break them down into smaller parts. Consider each person's work contribution individually and then collectively.
• Analysis: Always cross-verify the problem with the available options to check for any discrepancies or errors in the question.

Problem solving in work and time exercises is like piecing together a puzzle. By logically following the described scenario, you will reach the solution systematically.

Work Calculation

Work calculation is a crucial aspect of solving problems involving time and work. It involves determining how much of a task, represented as a single unit of work, is completed by individuals when working alone or together.
In this exercise, we looked at how to calculate the amount of work completed per day by each person:

• Individual Contribution: Calculate individual work rates as reciprocals of their total time taken to complete the work. For example, if Kavita takes 20 days alone, her daily work rate is \(\frac{1}{20}\).
• Combined Work Rates: Summate these rates to find out how much work can be completed when they work as a team. For the initial days, when all participants worked together, we summed up their rates, \(\frac{1}{20} + \frac{1}{60} + \frac{1}{30}\), to get a combined rate.

This method allows for a precise understanding of how collaboration affects overall task completion.

Efficiency

Efficiency refers to the effectiveness of working together versus individually, measured by the speed at which work is completed. In this problem, we explore how efficiency is maximized by cooperation.

• Working Together: When Kavita, Babita, and Samita work together, their collective effort results in a quicker completion of the work. The combined work rate is much higher, resulting in a substantial portion of work being completed in just 5 days.
• Impact of Leaving Members: After Samita left, we saw a decrease in overall efficiency, which is evident from the recalculated work rate for Kavita and Babita working together. Yet, they still effectively finish a good chunk of the work.
• Real World Implication: In practical scenarios, it's often beneficial to have multiple people working together on time-sensitive projects to increase total efficiency, assuming coordinated effort.

Understanding efficiency helps in planning how people can be organized best to achieve goals in less time.