Question

Chandni and Divakar can do a piece of work in 9 days and 12 days respectively. If they work for a day alternatively, Chandni beginning, in how many days, the work will be completed? (b) \(9 \frac{1}{5}\) (a) \(10 \frac{1}{4}\) (c) \(11.11\) (d) 10

Short answer

Answer: (a) \(10 \frac{1}{4}\) days.

Step-by-step solution

Find the individual efficiencies of Chandni and Divakar

Since Chandni can complete the work in 9 days, her efficiency (Work done in 1 day) is \(\frac{1}{9}\). Similarly, Divakar's efficiency (Work done in 1 day) is \(\frac{1}{12}\).

Calculate the work done by both in 2 days and the efficiency when they work alternatively

When they work alternatively, Chandni works on the first day, and Divakar works on the second day. So, in 2 days, the work done by them together is \(\frac{1}{9}+\frac{1}{12}\). To find the value of this expression, we need to find the LCM of 9 and 12, which is 36. So, the work done by them together in 2 days = \(\frac{4}{36}+\frac{3}{36}=\frac{7}{36}\).

Calculate the total days required to complete the work

Now, we need to find the number of days they will take to complete the work together when working alternatively. Let 'n' be the multiple of 2 (number of days) such that the work is completed. Then, \(\frac{7}{36}n=1\). Now, solve for 'n': \(n = \frac{36}{7}\) Since the value of 'n' should be a multiple of 2, therefore the smallest possible value for 'n' is 10.

Calculate the work done in 10 days

We know that both of them will do \(\frac{7}{36}\) of the work in 2 days. Therefore, in 10 days, they will do \(\frac{7}{36}\times 10 = \frac{35}{18}\) of the work.

Find the remaining work to be completed

Remaining work = \(1 - \frac{35}{18}=\frac{1}{18}\)

Calculate the extra days needed by Chandni to complete the remaining work

Since the work is done alternatively, now it's Chandni's turn to work. So, let's calculate the number of extra days needed by Chandni to complete the remaining work: Extra days needed by Chandni=\(\frac{\text{Remaining work}}{\text{Efficiency of Chandni}}=\frac{\frac{1}{18}}{\frac{1}{9}}=2 \div 1=\frac{1}{4}\)

Calculate the final total number of days to complete the work

Total number of days = 10 days (work done by both in 10 days) + \(\frac{1}{4}\) days(extra days needed by Chandni) = \(10 \frac{1}{4}\) days. So, the correct answer is (a) \(10 \frac{1}{4}\) days.

Key concepts

Efficiency in Work

Understanding efficiency is crucial when solving work and time problems. In the context of productivity, efficiency refers to the amount of work an individual can accomplish in a unit of time. For example, if Chandni can finish a job in 9 days, her efficiency is determined by dividing the total work (considered as one unit of work) by the number of days it takes her to complete it, yielding an efficiency of \( \frac{1}{9} \).

This translates into the idea that every day, Chandni completes \( \frac{1}{9} \) part of the total work. Divakar's efficiency is similarly calculated, giving \( \frac{1}{12} \) as his daily contribution to the work. When calculating the joint effort of two or more people working together, their individual efficiencies are added, demonstrating a combined effect on the work done per unit of time. Understanding efficiency allows students to approach problems systematically, accurately mapping out how long a project will take given various working scenarios.

Alternate Work Scheduling

Alternate work scheduling scenarios are common in time and work problems. This occurs when two or more individuals take turns completing a task. Such arrangements can impact the time it takes to finish the work, depending on the efficiency and scheduling of each worker. In our exercise, Chandni and Divakar work on alternate days, which requires a strategy to combine their individual efforts to complete the work together. As in the example, Chandni begins and works on the first day, followed by Divakar on the second day, and this pattern continues until the work is finished.

Figuring out how long they will take to complete the project involves analyzing the cumulative effect of their individual work days. Today's workforce often encounters alternate work scheduling, and understanding these concepts is not just academically beneficial but practically essential.

LCM Method in Time and Work

The Least Common Multiple (LCM) is an effective method for simplifying complex time and work problems, especially when workers have differing rates of efficiency. LCM helps to compare and combine different work rates on a common scale. For instance, to determine the work done by Chandni and Divakar over a period of two days, we must first find a common denominator. This is where the LCM of Chandni and Divakar's work days (9 and 12 days, respectively) comes into play.

The LCM of 9 and 12 is 36, giving us a common denominator to combine their efforts. Breaking the task into two-day increments when they work alternatively, they accomplish \( \frac{4}{36} + \frac{3}{36} = \frac{7}{36} \) parts of the work every two days. Leveraging the LCM method simplifies the calculation, allowing students to efficiently solve problems, avoid confusion, and prevent computational errors.

Quantitative Aptitude

Quantitative aptitude encompasses the ability to handle numerical and mathematical concepts effectively, a crucial skill in solving time and work problems. It allows individuals to logically deduce steps and perform the necessary calculations accurately. In our textbook example, several quantitative skills were employed, including simplifying fractions, finding the LCM, and solving equations.

Developing quantitative aptitude requires practice and understanding of basic mathematical concepts. It is vital for various competitive exams and in numerous professions. A strong quantitative aptitude helps students not only in academic environments but in making informed decisions in real-life scenarios that involve logic and numbers.