A is 50% more efficient then B. B worked to finish the same work in 20 days. If A and B worked together, then how much time will they take to finish the same work?
8 days
Understanding Work and Efficiency Problems
This problem involves the concepts of work, efficiency, and time. Efficiency is inversely proportional to the time taken to complete a fixed amount of work. If someone is more efficient, they take less time to finish the same work.
Analyzing the Given Information
- A is 50% more efficient than B.
- B takes 20 days to finish the entire work.
- We need to find the time A and B take together to finish the same work.
Calculating Individual Work Rates (Efficiency)
Let's think about the work done per day. If B finishes the work in 20 days, B completes $\frac{1}{20}$ of the work each day. This is B's daily work rate or efficiency relative to the total work.
- B's daily work rate = $\frac{1}{20}$ of the total work.
A is 50% more efficient than B. This means A's daily work rate is B's daily work rate plus 50% of B's daily work rate.
- 50% of B's daily work rate = $50\%$ of $\frac{1}{20} = \frac{50}{100} \times \frac{1}{20} = \frac{1}{2} \times \frac{1}{20} = \frac{1}{40}$ of the total work.
- A's daily work rate = B's daily work rate + 50% of B's daily work rate
- A's daily work rate = $\frac{1}{20} + \frac{1}{40} = \frac{2}{40} + \frac{1}{40} = \frac{3}{40}$ of the total work.
Calculating Combined Work Rate
When A and B work together, their daily work rates add up.
- Combined daily work rate of A and B = A's daily work rate + B's daily work rate
- Combined daily work rate = $\frac{3}{40} + \frac{1}{20} = \frac{3}{40} + \frac{2}{40} = \frac{5}{40} = \frac{1}{8}$ of the total work.
This means that together, A and B complete $\frac{1}{8}$ of the total work each day.
Calculating Time Taken Together
If they complete $\frac{1}{8}$ of the work each day, the total time taken to complete the entire work (which is 1 whole) is the reciprocal of their combined daily work rate.
- Time taken together = $\frac{1}{\text{Combined daily work rate}}$
- Time taken together = $\frac{1}{1/8} = 8$ days.
Alternative Approach Using Efficiency Ratios
Let the efficiency of B be $E_B$.
The efficiency of A is 50% more than B, so $E_A = E_B + 0.5 E_B = 1.5 E_B$.
The ratio of efficiencies $E_A : E_B = 1.5 : 1 = 3 : 2$.
Efficiency is inversely proportional to time. So, the ratio of time taken $T_A : T_B$ is the inverse ratio of their efficiencies.
$T_A : T_B = \frac{1}{E_A} : \frac{1}{E_B} = \frac{1}{1.5 E_B} : \frac{1}{E_B} = \frac{1}{1.5} : 1 = 1 : 1.5 = 2 : 3$.
We are given that $T_B = 20$ days. Since $T_A : T_B = 2 : 3$, we have $\frac{T_A}{T_B} = \frac{2}{3}$.
$\frac{T_A}{20} = \frac{2}{3} \implies T_A = \frac{2}{3} \times 20 = \frac{40}{3}$ days.
Let the total work be $W$.
B's daily work rate $= \frac{W}{T_B} = \frac{W}{20}$.
A's daily work rate $= \frac{W}{T_A} = \frac{W}{40/3} = \frac{3W}{40}$.
Combined daily work rate $= \frac{W}{20} + \frac{3W}{40} = \frac{2W}{40} + \frac{3W}{40} = \frac{5W}{40} = \frac{W}{8}$.
The time taken by A and B together is $\frac{\text{Total Work}}{\text{Combined daily work rate}} = \frac{W}{W/8} = 8$ days.
Summary of Steps
| Step | Description | Calculation |
|---|---|---|
| 1 | Find B's daily work rate | $\frac{1}{20}$ |
| 2 | Find A's daily work rate (50% more than B) | $\frac{1}{20} + 0.5 \times \frac{1}{20} = \frac{3}{40}$ |
| 3 | Find Combined daily work rate | $\frac{1}{20} + \frac{3}{40} = \frac{5}{40} = \frac{1}{8}$ |
| 4 | Find Time taken together | $\frac{1}{\text{Combined rate}} = \frac{1}{1/8} = 8$ days |
Therefore, if A and B worked together, they would take 8 days to finish the same work.
Revision Table: Work and Efficiency Concepts
| Concept | Relationship | Formula |
|---|---|---|
| Work Rate (Efficiency) | Amount of work done per unit of time. | Work Rate = $\frac{\text{Total Work}}{\text{Time}}$ |
| Time | Duration to complete the work. | Time = $\frac{\text{Total Work}}{\text{Work Rate}}$ |
| Work | Total task to be completed. Often assumed as 1 unit or calculated based on given rates/times. | Total Work = Work Rate $\times$ Time |
| Efficiency & Time | Inversely proportional for a fixed amount of work. | $E \propto \frac{1}{T}$ |
| Combined Work Rate | Sum of individual work rates when working together. | $R_{\text{combined}} = R_1 + R_2 + \dots$ |
Additional Information: Percentage Efficiency and Work
When efficiency is given as a percentage relative to another person, it directly affects their work rate. If someone is X% more efficient, their work rate is (100+X)% of the other person's work rate. If they are X% less efficient, their work rate is (100-X)% of the other person's work rate.
In this problem, A is 50% more efficient than B.
- Efficiency of A = 100% of Efficiency of B + 50% of Efficiency of B
- Efficiency of A = 150% of Efficiency of B
- Ratio of Efficiency of A to Efficiency of B = 150 : 100 = 3 : 2.
Since time is inversely proportional to efficiency, the ratio of time taken by A to time taken by B is the inverse of the efficiency ratio.
- Ratio of Time taken by A to Time taken by B = 2 : 3.
If B takes 20 days (which corresponds to the '3' part of the ratio), then the time taken by A (corresponding to the '2' part) can be found:
- $\frac{\text{Time}_A}{\text{Time}_B} = \frac{2}{3}$
- $\frac{\text{Time}_A}{20} = \frac{2}{3}$
- Time$_A = \frac{2 \times 20}{3} = \frac{40}{3}$ days.
A takes $\frac{40}{3}$ days to complete the work alone.
Using the formula for time taken by A and B together: $T_{\text{together}} = \frac{T_A \times T_B}{T_A + T_B}$ (This formula works when we have individual times for completing the *same* work).
- $T_{\text{together}} = \frac{\frac{40}{3} \times 20}{\frac{40}{3} + 20} = \frac{\frac{800}{3}}{\frac{40}{3} + \frac{60}{3}} = \frac{\frac{800}{3}}{\frac{100}{3}} = \frac{800}{3} \times \frac{3}{100} = \frac{800}{100} = 8$ days.
Both methods confirm that A and B working together take 8 days to finish the work.
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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?
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:
30 persons can do a piece of work in 24 days. How many more persons are required to complete the work in 20 days?
