Question

\(A\) and \(B\) can complete a task in 30 days when working together after \(A\) and \(B\) have been working together for 11 days, \(B\) is called away and \(A\), all by himself completes the task in the next 28 days. Had \(A\) been working alone, the number of days taken by him to complete the task would have been : (a) \(33 \frac{3}{19}\) (b) \(19 \frac{6}{25}\) (c) \(44 \frac{4}{19}\) (d) none of these

Short answer

Answer: A would take 44 and 4/19 days to complete the task alone.

Step-by-step solution

Find Task Completed by A and B in 11 Days

As we know, A and B together can complete the task in 30 days. So, in one day, they complete \(\frac{1}{30}\) of the task. Therefore, in 11 days, they will complete: $$ \text{Task Completed in 11 Days} = \frac{1}{30} \times 11 = \frac{11}{30} $$

Calculate Remaining Task and Task Completed by A Alone

Since A and B together completed \(\frac{11}{30}\) of the task, the remaining task to be completed is: $$ \text{Remaining Task} = 1 - \frac{11}{30} = \frac{19}{30} $$ We are given that A completes this remaining task alone in 28 days. Therefore, A completes \(\frac{19}{30}\) of the task in 28 days.

Find the Number of Days A Needs to Complete the Task Alone

A completes \(\frac{19}{30}\) of the task in 28 days. To find the number of days A needs to complete the whole task, we can set up a proportion: $$ \frac{\frac{19}{30}}{28} = \frac{1}{\text{Days A Needs Alone}} $$ Cross-multiplying, we get: $$ \text{Days A Needs Alone} \times \frac{19}{30} = 28 $$ $$ \text{Days A Needs Alone} = 28 \times \frac{30}{19} = 44 \frac{4}{19} $$ So, A would take \(44\frac{4}{19}\) days to complete the task alone. The correct option is (c).

Key concepts

Time and Work Efficiency

Understanding time and work efficiency is crucial when dealing with problems that involve multiple workers or resources completing a task. Efficiency, in this context, refers to the rate at which a person or machine can complete a portion of a given task within a certain time frame.

For instance, if two people, A and B, are working together to complete a task, their combined efficiency is the sum of their individual efficiencies. This concept allows us to calculate how much work can be done collectively in a certain period. In the provided exercise, A and B’s joint efficiency allows them to finish a task in 30 days. When examining a situation where only one person is working, like A working alone for the remainder of the task, we investigate that person's individual efficiency to understand how their performance impacts the completion of the task.

Proportion and Rate of Work

The proportion and rate of work relate directly to a worker's efficiency. The proportion refers to how much work is done relative to the whole task, while the rate assesses how quickly this work is accomplished. These concepts are vital when we need to compare the work done by different workers over various time periods.

To utilize these concepts, we often calculate the part of work done per day, which acts as the worker's rate. This was demonstrated in the exercise when A's rate was determined over the remaining 28 days to complete \(\frac{19}{30}\) of the task. We can then employ this rate to establish proportions and solve for unknown quantities, such as the total number of days A would need to complete the entire task alone.

Concept of Man-Days

The concept of man-days (or person-days) is a way to quantify the amount of work done by one person in one day. It's a unit of measurement that converts work efficiency into tangible outputs. This helps in planning and estimating the time it takes to complete a project with a certain number of workers.

In the case of A and B working together for 11 days, we can say they have contributed 11 man-days towards completion of the task, since they work as a unit. However, once B leaves, we start calculating the man-days for A alone, which helps us to determine A's individual input to finish the task. This concept is a convenient method to illustrate the collaboration and solo contributions in the context of time and work problems.

Algebraic Method in Time and Work

Applying the algebraic method in time and work problems allows us to use equations to represent the relationships between time taken, work done, and efficiency. This method is especially efficient in solving problems with variables representing unknown quantities.

In our exercise, we use algebra to form an equation expressing A's work rate. We find that if \(\frac{19}{30}\) of a task is completed in 28 days, we can utilize proportions to calculate the total duration required for A to finish the task alone. This algebraic equation is cross-multiplied to reveal the unknown variable, which is the total number of days A needs to complete the task solo. Employing algebra in such contexts streamlines the process of solving complex work and time scenarios, transforming word problems into solvable mathematical equations.