For completing a certain work, A is 50% less efficient than B and B is 50% more efficient than C. Working together A, B and C can complete the work in 48 days. A alone can complete the same work in:
208 days
Understanding Work and Time Problems with Efficiency
This problem involves calculating the time taken by an individual (A) to complete a work alone, given the efficiency relationships between three individuals (A, B, C) and the time they take to complete the work together. The key concept here is efficiency, which is inversely proportional to the time taken to complete a task. Higher efficiency means less time is needed.
Step-by-Step Solution for A, B, and C's Work
1. Establishing Efficiency Relationships
We are given the following information about the efficiency of A, B, and C:
- A is 50% less efficient than B.
- B is 50% more efficient than C.
Let's assume C's efficiency is represented by a value, say \(E_C\). Based on the problem statements:
- B's efficiency (\(E_B\)) is 50% more than C's efficiency. \[E_B = E_C + 50\% \text{ of } E_C = E_C + 0.50 \times E_C = 1.5 E_C\]
- A's efficiency (\(E_A\)) is 50% less than B's efficiency. \[E_A = E_B - 50\% \text{ of } E_B = E_B - 0.50 \times E_B = 0.50 \times E_B\] Substituting \(E_B = 1.5 E_C\), we get: \[E_A = 0.50 \times (1.5 E_C) = 0.75 E_C\]
2. Determining the Ratio of Efficiencies
The efficiencies are in the ratio \(E_A : E_B : E_C = 0.75 E_C : 1.5 E_C : E_C\). We can simplify this ratio by dividing by \(E_C\) and then multiplying by 4 to get whole numbers:
- Ratio \(E_A : E_B : E_C = 0.75 : 1.5 : 1\)
- Multiply by 4: \((0.75 \times 4) : (1.5 \times 4) : (1 \times 4) = 3 : 6 : 4\)
So, the efficiency ratio of A, B, and C is 3 : 6 : 4. This means if A does 3 units of work per day, B does 6 units per day, and C does 4 units per day.
| Person | Relative Efficiency | Ratio Unit |
|---|---|---|
| A | 0.75 \(E_C\) | 3 |
| B | 1.5 \(E_C\) | 6 |
| C | \(E_C\) | 4 |
3. Calculating Combined Efficiency
When working together, their efficiencies add up. Using the ratio units, their combined efficiency per day is:
Combined Efficiency = Efficiency of A + Efficiency of B + Efficiency of C
Combined Efficiency = \(3 + 6 + 4 = 13\) units per day.
4. Calculating Total Work
The problem states that A, B, and C working together can complete the work in 48 days. The total work can be calculated using the formula: Total Work = Combined Efficiency × Time Taken.
Total Work = \(13 \text{ units/day} \times 48 \text{ days}\)
Calculating the total work:
\[13 \times 48 = 13 \times (50 - 2) = 13 \times 50 - 13 \times 2 = 650 - 26 = 624\]Total Work = 624 units.
5. Calculating Time Taken by A Alone
To find the time taken by A alone to complete the work, we use the formula: Time Taken = Total Work / Efficiency of A.
We know the Total Work is 624 units and the efficiency of A (from the ratio) is 3 units per day.
Time taken by A alone = \(\frac{624 \text{ units}}{3 \text{ units/day}}\)
Calculating the time:
\[\frac{624}{3}\] \[624 \div 3 = (600 + 24) \div 3 = 600 \div 3 + 24 \div 3 = 200 + 8 = 208\]Time taken by A alone = 208 days.
Therefore, A alone can complete the same work in 208 days.
| Metric | Value |
|---|---|
| Efficiency Ratio (A:B:C) | 3:6:4 |
| Combined Efficiency (ratio units) | 13 units/day |
| Time together | 48 days |
| Total Work | 624 units |
| A's Efficiency (ratio units) | 3 units/day |
| Time for A alone | 208 days |
Revision Table: Work and Time Concepts
| Concept | Description | Relationship to Time |
|---|---|---|
| Efficiency | The amount of work done per unit of time. | Inversely proportional: More efficient means less time. |
| Total Work | The total units of task to be completed. | Calculated as Efficiency × Time. |
| Time Taken | The duration required to complete the total work. | Calculated as Total Work / Efficiency. |
Additional Information on Efficiency Calculations
Understanding percentage increase and decrease in efficiency is crucial for solving these types of work and time problems. If someone is 'X% more efficient', their efficiency is \((1 + X/100)\) times the base efficiency. If someone is 'X% less efficient', their efficiency is \((1 - X/100)\) times the base efficiency.
- 50% more efficient than C: Efficiency is \(1 + 50/100 = 1.5\) times C's efficiency.
- 50% less efficient than B: Efficiency is \(1 - 50/100 = 0.5\) times B's efficiency.
Using these multipliers helps in quickly setting up the relationships and ratios between the efficiencies of different individuals working on a task.
Important Questions from Work Efficiency
A and B working together can complete a job in 30 days. The ratio of their efficiencies is 3 : 2. In how many days can the faster person complete the job?
A takes 15 days to complete \(\frac{5}{7} \) of a work. With the help of B, they finish the whole work in 12 days. In how many days, B alone will complete the same work
A alone can complete a work in 14 days and B alone can complete the same work in 21 days. A and B start the work together but A leaves the work after 4 days of the starting of work. In how many days B will complete the remaining work?
30 persons can do a piece of work in 24 days. How many more persons are required to complete the work in 20 days?
X is twice as good as workman as Y. Together, they finish the work in 18 days. In how many days can it be done by each separately?
