Anil is 12.5 percent more efficient than Sachin. If Sachin alone can finish a work in 10 days, then in how many days will Anil alone finish the same work?
(80/9) days
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 EA be Anil's efficiency and ES be Sachin's efficiency. Let TA be Anil's time and TS be Sachin's time.
Anil is 12.5% more efficient than Sachin. If Sachin's efficiency is ES, 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 (TS) is 10 days. We want to find Anil's time (TA).
\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.
| Person | Efficiency (units/day) | Time Taken (days) |
|---|---|---|
| Sachin | \( \frac{1}{10} \) | 10 |
| Anil | \( \frac{9}{80} \) | \( \frac{80}{9} \) |
Thus, Anil alone will finish the same work in \( \frac{80}{9} \) days.
Revision Table: Work and Time Concepts
| Concept | Formula/Relation | Notes |
|---|---|---|
| Work | Efficiency × Time | Total amount of task |
| Efficiency | \( \frac{\text{Work}}{\text{Time}} \) | Work done per unit time |
| Time | \( \frac{\text{Work}}{\text{Efficiency}} \) | Time taken to complete work |
| Efficiency & Time | Inversely Proportional (for constant work) | If Efficiency increases, Time decreases proportionally |
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 (TTogether) 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.
Important Questions from Time and Work
A and B can do a piece of work in 18 days. B and C together can do it in 30 days. If A is twice as good a workman as C, find in how many days B alone can do the work?
Two men and 7 women can complete a work in 28 days, whereas 6 men and 16 women can do the same work in 11 days. In how many days will 5 men and 4 women, working together, complete the same work?
A man and a woman, working together can do a work in 66 days. The ratio of their working efficiencies is 3 ∶ 2. In how many days 6 men and 2 women together can do the same work?
Each one of five men independently can complete a work in 20 days. The work is started by one person. Next day one more person joins and every next day one more person joins. From the fifth day, five persons continued working as a team. In how many days, will the work be completed?
Numan does half the work as Gagan in 4/5 of the time. If together they take 16 days to complete a piece of work, then how long will it take Gagan to complete the work?
