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?
X = 27 days, Y = 54 days
Understanding Work and Efficiency
This question involves a classic work and time problem where the efficiency of two workers, X and Y, is related. We are told that X is twice as good a workman as Y. This means that in the same amount of time, X can do twice the amount of work that Y can do.
We are also given that working together, X and Y can finish the entire work in 18 days. We need to find out how many days each person would take to finish the work individually.
Defining Work Rate
In work and time problems, it's helpful to think about the 'work rate' or the amount of work done per unit of time (in this case, per day). The work rate is inversely proportional to the time taken to complete the work. If someone takes fewer days, their work rate is higher.
- Let the work done by Y in one day be represented by \(W_Y\).
- Since X is twice as good as Y, the work done by X in one day is \(W_X = 2 \times W_Y\).
Calculating Combined Work Rate
When X and Y work together, their individual work rates add up to form a combined work rate. The amount of work done by X and Y together in one day is:
\(W_{\text{combined}} = W_X + W_Y\)
Substituting the relationship between \(W_X\) and \(W_Y\):
\(W_{\text{combined}} = 2 W_Y + W_Y = 3 W_Y\)
So, together, X and Y do three times the amount of work that Y does alone in one day. This aligns with X being twice as efficient as Y, meaning together they are effectively as efficient as three Ys.
Finding the Total Work
We know that together, X and Y finish the entire work in 18 days. The total work can be calculated by multiplying their combined work rate by the number of days they worked together:
\(\text{Total Work} = W_{\text{combined}} \times \text{Number of days worked together}\)
\(\text{Total Work} = (3 W_Y) \times 18\)
\(\text{Total Work} = 54 W_Y\)
The total work is equivalent to the amount of work Y can do in 54 days.
Calculating Individual Times
Now that we know the total work and the individual work rates, we can find the time each person takes to complete the work alone.
The time taken by a person is given by:
\(\text{Time} = \frac{\text{Total Work}}{\text{Individual Work Rate}}\)
Time taken by Y:
Y's work rate is \(W_Y\).
\(\text{Time}_Y = \frac{\text{Total Work}}{W_Y}\)
\(\text{Time}_Y = \frac{54 W_Y}{W_Y}\)
\(\text{Time}_Y = 54 \text{ days}\)
Time taken by X:
X's work rate is \(W_X = 2 W_Y\).
\(\text{Time}_X = \frac{\text{Total Work}}{W_X}\)
\(\text{Time}_X = \frac{54 W_Y}{2 W_Y}\)
\(\text{Time}_X = \frac{54}{2} \text{ days}\)
\(\text{Time}_X = 27 \text{ days}\)
Summary of Results
Based on our calculations:
- Time taken by X alone to finish the work = 27 days.
- Time taken by Y alone to finish the work = 54 days.
This makes sense because X is twice as good as Y, so X should take half the time Y takes to complete the same work (27 is half of 54).
Let's verify the combined work rate: In 1 day, X does \(\frac{1}{27}\) of the work, and Y does \(\frac{1}{54}\) of the work. Together, they do \(\frac{1}{27} + \frac{1}{54} = \frac{2}{54} + \frac{1}{54} = \frac{3}{54} = \frac{1}{18}\) of the work per day. If they do \(\frac{1}{18}\) of the work per day, they will finish the work in 18 days, which matches the information given in the question.
Conclusion
The time taken by X to finish the work separately is 27 days, and the time taken by Y is 54 days.
Revision Table: Work and Time Concepts
| Concept | Explanation | Relationship with Time |
|---|---|---|
| Work Rate | Amount of work done per unit of time (e.g., per day). | Inversely proportional to time taken. Higher rate means less time. |
| Efficiency | A measure of how quickly work is done. Directly proportional to work rate. | A more efficient worker has a higher work rate and takes less time. |
| Total Work | The total amount of work to be completed. Often considered '1 unit' or a fixed amount. | Total Work = Work Rate × Time |
| Combined Work Rate | The sum of individual work rates when multiple people work together. | If workers A and B have rates \(R_A\) and \(R_B\), combined rate is \(R_A + R_B\). |
Additional Information: Work and Time Formulas
Here are some key formulas used in solving work and time problems:
- If a person can complete a work in \(d\) days, their work rate is \(\frac{1}{d}\) work per day.
- If the work rate is \(R\) per day, the time taken to complete the work is \(\frac{1}{R}\) days.
- If Person 1 takes \(d_1\) days and Person 2 takes \(d_2\) days to complete a work, and they work together, the time taken is \(\frac{1}{\frac{1}{d_1} + \frac{1}{d_2}} = \frac{d_1 \times d_2}{d_1 + d_2}\) days.
- If the efficiency of person A is \(k\) times the efficiency of person B, then the time taken by A is \(\frac{1}{k}\) times the time taken by B (assuming they do the same amount of work).
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?
