Calculating Additional Days to Complete Work

This problem is about understanding the relationship between the amount of work done and the time taken, assuming a constant work rate. We are given that Rajan completes a fraction of the total work in a certain number of days and need to find out how many more days he requires to finish the remaining part.

Understanding the Work Done and Remaining Work

Rajan has completed $\frac{7}{11}$ of the total work. The total work can be considered as 1 whole or $\frac{11}{11}$.

The fraction of work remaining is calculated by subtracting the completed work from the total work:

Remaining Work = Total Work - Work Done

Remaining Work = $1 - \frac{7}{11}$

To subtract these fractions, we find a common denominator, which is 11.

Remaining Work = $\frac{11}{11} - \frac{7}{11}$

Remaining Work = $\frac{11 - 7}{11}$

Remaining Work = $\frac{4}{11}$

So, $\frac{4}{11}$ of the work is still left to be done by Rajan.

Relating Work Done and Time Taken

We are told that Rajan does $\frac{7}{11}$ of the work in 21 days. Assuming Rajan works at a constant pace, the time taken is directly proportional to the amount of work done. This means if he does more work, he takes more time, and vice versa, in direct proportion.

We can set up a proportion to find the total time required to complete the entire work, or directly find the time needed for the remaining work.

Method 1: Find total time first

Let $T$ be the total number of days required to complete the entire work ($\frac{11}{11}$ or 1).

The proportion is: $\frac{\text{Fraction of Work Done}}{\text{Time Taken}} = \text{Constant}$

Using the given information:

Simplifying the left side:

Cross-multiplying to solve for $T$:

The total time required to complete the entire work is 33 days.

Rajan has already worked for 21 days. The number of more days he will take to complete the remaining work is:

More Days = Total Time - Time Already Spent

More Days = $33 \text{ days} - 21 \text{ days}$

More Days = $12 \text{ days}$

Method 2: Find time for remaining work directly

We know Rajan does $\frac{7}{11}$ of the work in 21 days.

Let $D$ be the number of days required to complete the remaining $\frac{4}{11}$ of the work.

Using the proportion: $\frac{\text{Fraction of Work 1}}{\text{Time 1}} = \frac{\text{Fraction of Work 2}}{\text{Time 2}}$

We can cancel the common denominator 11 from both numerators:

Now, solve for $D$ by cross-multiplying:

Divide by 7:

Both methods show that Rajan will take 12 more days to complete the remaining work.

Conclusion on Additional Days Needed

Based on the calculations, Rajan needs 12 more days to complete the $\frac{4}{11}$ fraction of the work that is remaining.

Revision Table: Work and Time Concepts

Additional Information: Solving Work and Time Problems

Work and time problems often involve calculating the time taken by individuals or groups to complete a task, or the rate at which they work. A key assumption is usually that the work rate is constant.

• If a person completes a work in $N$ days, their daily work rate is $\frac{1}{N}$ of the total work per day.
• If a person completes a fraction of work, say $\frac{a}{b}$, in $D$ days, the time taken for the entire work can be found using proportionality or by finding the rate. Rate = $\frac{a/b}{D} = \frac{a}{bD}$ per day. Total time = $\frac{1}{\text{Rate}} = \frac{bD}{a}$ days.
• When multiple people work together, their individual work rates are often added to find the combined work rate.
• Efficiency can sometimes be a factor, which affects the work rate.

In problems like Rajan's, where a fraction of work is done, it's helpful to determine the rate of work. Rajan's rate is $\frac{7/11 \text{ work}}{21 \text{ days}} = \frac{7}{11 \times 21} = \frac{7}{231} = \frac{1}{33}$ work per day. This means he takes 33 days for the entire work. Since he finished 7/11 in 21 days, the remaining time for 4/11 work would be $\frac{4/11}{1/33} = \frac{4}{11} \times 33 = 4 \times 3 = 12$ days. This confirms the result.