Understanding the Work and Time Problem

This problem involves two individuals, Numan and Gagan, performing a certain amount of work. We are given information about their relative work efficiencies and the time they take to complete the work when working together. The goal is to find the time Gagan would take to complete the entire piece of work by himself.

In work and time problems, the key concept is the work rate or efficiency of a person. Work rate is the amount of work done per unit of time. If a person completes a total amount of work 'W' in 'T' days, their work rate 'R' is given by:

\text{Work Rate (R)} = \frac{\text{Total Work (W)}}{\text{Time Taken (T)}}

Conversely, the time taken to complete the work is $\text{T} = \frac{\text{W}}{\text{R}}$.

When two people work together, their individual work rates add up to form a combined work rate. If Numan's rate is $R_N$ and Gagan's rate is $R_G$, their combined rate is $R_{combined} = R_N + R_G$.

Analyzing the Given Information

We are given two main pieces of information:

1. Numan does half the work as Gagan in 4/5 of the time.
2. Together, they take 16 days to complete a piece of work.

Step 1: Relating Numan's and Gagan's Work Rates

Let's assume Gagan completes a certain amount of work, say $W'$, in time $T'$.

• Gagan's rate: $R_G = \frac{W'}{T'}$

According to the first statement, Numan does half of this work ($\frac{W'}{2}$) in 4/5 of the time ($\frac{4}{5}T'$).

• Numan's rate: $R_N = \frac{\text{Work done by Numan}}{\text{Time taken by Numan}} = \frac{W'/2}{(4/5)T'}$

Let's simplify Numan's rate:

R_N = \frac{W'}{2} \times \frac{5}{4T'} = \frac{5W'}{8T'}

We can see that $\frac{W'}{T'}$ is Gagan's rate, $R_G$. So, we can write Numan's rate in terms of Gagan's rate:

R_N = \frac{5}{8} R_G

This equation tells us that Numan's work rate is 5/8 times Gagan's work rate. This means Gagan is more efficient than Numan.

Step 2: Using the Information about Combined Work Time

Let the total piece of work be $W$. We know that together, Numan and Gagan take 16 days to complete this work.

Their combined rate is $R_{combined} = R_N + R_G$.

The time taken together is $\frac{\text{Total Work}}{\text{Combined Rate}} = 16$ days.

\frac{W}{R_N + R_G} = 16

Step 3: Solving for Gagan's Time to Complete the Work

We want to find the time Gagan takes to complete the work alone. This is given by $\text{Time}_G = \frac{\text{Total Work}}{R_G} = \frac{W}{R_G}$.

We have the equation $\frac{W}{R_N + R_G} = 16$. Substitute $R_N = \frac{5}{8} R_G$ into this equation:

\frac{W}{\frac{5}{8} R_G + R_G} = 16

Combine the terms in the denominator:

\frac{W}{(\frac{5}{8} + 1) R_G} = 16

\frac{W}{(\frac{5}{8} + \frac{8}{8}) R_G} = 16

\frac{W}{\frac{13}{8} R_G} = 16

Now, we need to isolate $\frac{W}{R_G}$, which represents the time Gagan takes alone. We can rewrite the left side:

\frac{W}{R_G} \times \frac{1}{\frac{13}{8}} = 16

\frac{W}{R_G} \times \frac{8}{13} = 16

Multiply both sides by $\frac{13}{8}$ to find $\frac{W}{R_G}$:

\frac{W}{R_G} = 16 \times \frac{13}{8}

\frac{W}{R_G} = \cancel{16}^2 \times \frac{13}{\cancel{8}^1}

\frac{W}{R_G} = 2 \times 13

\frac{W}{R_G} = 26

So, the time it will take Gagan to complete the work alone is 26 days.

Conclusion

Based on the analysis of their relative work rates and the time they take to complete the work together, Gagan will take 26 days to complete the piece of work by himself.

Revision Table: Work and Time Concepts

Additional Information on Work and Time Problems

Work and time problems often test your understanding of how work rates combine and how they relate to the time taken. Here are a few important points:

• If a person completes a work in 'd' days, their daily work rate is $1/d$ of the total work.
• If multiple people work together, their daily work contributions add up.
• Problems can involve varying efficiencies, different numbers of workers, or portions of work completed over certain periods.
• Always ensure that the work units and time units are consistent throughout the calculation.

Understanding the relationship between work, rate, and time ($W = R \times T$) is fundamental to solving these types of problems. By setting up equations based on the given information, you can solve for unknown rates or times, just like we did to find Gagan's time to complete the work alone.