Solving Work Efficiency and Time Problems

This problem involves the concept of work and time, specifically dealing with the efficiency of individuals and how it affects the time taken to complete a task. The key principle here is that efficiency is inversely proportional to the time taken to complete a fixed amount of work. This means if someone is more efficient, they will take less time to finish the same work.

Understanding Efficiency and Time

Efficiency can be defined as the amount of work done per unit of time. The relationship between Work, Efficiency, and Time is given by:

\text{Work} = \text{Efficiency} \times \text{Time}

For a fixed amount of work, we can see that Efficiency is inversely proportional to Time:

\text{Efficiency} \propto \frac{1}{\text{Time}}

Step-by-Step Solution

Let's break down the problem step-by-step to find out how many days Anil will take to finish the work alone.

Step 1: Determine Sachin's Efficiency

Sachin alone can finish the work in 10 days. We can assume the total work to be done is a certain number of units. For simplicity, let's assume the total work is 1 unit (or 1 'whole' work).

Using the formula Work = Efficiency × Time, we can find Sachin's efficiency:

1 \text{ unit of work} = \text{Sachin's Efficiency} \times 10 \text{ days}

So, Sachin's Efficiency is:

\text{Sachin's Efficiency} = \frac{1 \text{ unit}}{10 \text{ days}} = \frac{1}{10} \text{ units/day}

Step 2: Calculate Anil's Efficiency

We are given that Anil is 12.5 percent more efficient than Sachin.

First, let's convert 12.5 percent to a fraction or decimal:

12.5\% = \frac{12.5}{100} = \frac{125}{1000} = \frac{1}{8}

Anil's efficiency is Sachin's efficiency plus 12.5% of Sachin's efficiency.

\text{Anil's Efficiency} = \text{Sachin's Efficiency} + 12.5\% \text{ of Sachin's Efficiency}

\text{Anil's Efficiency} = \frac{1}{10} + \frac{1}{8} \times \frac{1}{10}

\text{Anil's Efficiency} = \frac{1}{10} \left(1 + \frac{1}{8}\right)

\text{Anil's Efficiency} = \frac{1}{10} \left(\frac{8+1}{8}\right)

\text{Anil's Efficiency} = \frac{1}{10} \times \frac{9}{8}

\text{Anil's Efficiency} = \frac{9}{80} \text{ units/day}

Step 3: Calculate Time Taken by Anil

Now we can find the time Anil takes to complete the same work (1 unit). Using the formula Time = Work / Efficiency:

\text{Anil's Time} = \frac{\text{Total Work}}{\text{Anil's Efficiency}}

\text{Anil's Time} = \frac{1 \text{ unit}}{\frac{9}{80} \text{ units/day}}

\text{Anil's Time} = 1 \times \frac{80}{9} \text{ days}

\text{Anil's Time} = \frac{80}{9} \text{ days}

Alternative Method: Using Efficiency Ratio

Since efficiency is inversely proportional to time, the ratio of times taken is the inverse of the ratio of efficiencies.

Let E_A be Anil's efficiency and E_S be Sachin's efficiency. Let T_A be Anil's time and T_S be Sachin's time.

Anil is 12.5% more efficient than Sachin. If Sachin's efficiency is E_S, Anil's efficiency is:

E_A = E_S + 12.5\% \text{ of } E_S = E_S + \frac{1}{8} E_S = \left(1 + \frac{1}{8}\right) E_S = \frac{9}{8} E_S

The ratio of efficiencies is:

\frac{E_A}{E_S} = \frac{\frac{9}{8} E_S}{E_S} = \frac{9}{8}

The ratio of times taken is the inverse:

\frac{T_A}{T_S} = \frac{E_S}{E_A} = \frac{8}{9}

We know that Sachin's time (T_S) is 10 days. We want to find Anil's time (T_A).

\frac{T_A}{10 \text{ days}} = \frac{8}{9}

T_A = \frac{8}{9} \times 10 \text{ days}

T_A = \frac{80}{9} \text{ days}

Both methods give the same result.

Thus, Anil alone will finish the same work in \( \frac{80}{9} \) days.

Revision Table: Work and Time Concepts

Additional Information: Working Together

Sometimes, work and time problems involve two or more people working together. If Person A takes \( T_A \) days and Person B takes \( T_B \) days to complete a work alone, their individual efficiencies are \( \frac{1}{T_A} \) and \( \frac{1}{T_B} \) per day (assuming total work is 1 unit).

When they work together, their combined efficiency is the sum of their individual efficiencies:

\text{Combined Efficiency} = \frac{1}{T_A} + \frac{1}{T_B}

The time taken to complete the work together (T_Together) is:

T_{\text{Together}} = \frac{\text{Total Work}}{\text{Combined Efficiency}} = \frac{1}{\frac{1}{T_A} + \frac{1}{T_B}} = \frac{T_A \times T_B}{T_A + T_B}

Understanding individual efficiencies is the foundation for solving combined work problems too.