X and Y can do a piece of work in 45 days and 40 days respectively. They begin to work together, but X leaves after n days and then Y completes the remaining work in 23 days. What is n equal to ?
9
Understanding Work and Time Problems
This problem involves concepts of work, time, and individual and combined work rates. We are given the time it takes for two individuals, X and Y, to complete a piece of work independently. They start working together, but one leaves, and the other finishes the remaining work. We need to find the duration they worked together.
Calculating Individual Work Rates
The work rate of a person is the amount of work they can do in one day. If a person can complete a work in 'd' days, their daily work rate is \( \frac{1}{d} \) of the total work.
- X can do the work in 45 days.
- X's work rate per day = \( \frac{1}{45} \) of the work.
- Y can do the work in 40 days.
- Y's work rate per day = \( \frac{1}{40} \) of the work.
Work Done Working Together
X and Y work together for 'n' days. To find the work done in these 'n' days, we add their daily work rates and multiply by the number of days they worked together.
- Combined work rate per day = (X's work rate) + (Y's work rate)
- Combined work rate per day = \( \frac{1}{45} + \frac{1}{40} \)
Let's find a common denominator for the combined work rate. The least common multiple (LCM) of 45 and 40 is 360.
- Combined work rate per day = \( \frac{1 \times 8}{45 \times 8} + \frac{1 \times 9}{40 \times 9} = \frac{8}{360} + \frac{9}{360} = \frac{8+9}{360} = \frac{17}{360} \)
Work done by X and Y working together for 'n' days = (Combined work rate per day) \( \times \) n
- Work done in n days = \( n \times \frac{17}{360} = \frac{17n}{360} \)
Work Done by Remaining Worker (Y)
After 'n' days, X leaves. Y completes the remaining work in 23 days.
- Y's work rate per day = \( \frac{1}{40} \)
Work done by Y in the remaining 23 days = (Y's work rate per day) \( \times \) 23
- Work done by Y in 23 days = \( 23 \times \frac{1}{40} = \frac{23}{40} \)
Setting up the Equation for Total Work
The total work done is the sum of the work done by X and Y together and the work done by Y alone. The total work is always equal to 1 (representing the whole task).
Total work = (Work done in n days by X and Y) + (Work done in 23 days by Y)
\( 1 = \frac{17n}{360} + \frac{23}{40} \)
Solving for n
Now we need to solve this equation for 'n'.
First, isolate the term with 'n':
\( \frac{17n}{360} = 1 - \frac{23}{40} \)
To subtract the fractions on the right side, find a common denominator, which is 40.
\( 1 - \frac{23}{40} = \frac{40}{40} - \frac{23}{40} = \frac{40 - 23}{40} = \frac{17}{40} \)
So the equation becomes:
\( \frac{17n}{360} = \frac{17}{40} \)
Now, solve for 'n'. We can multiply both sides by 360:
\( 17n = \frac{17}{40} \times 360 \)
Simplify the right side:
\( 17n = 17 \times \frac{360}{40} \)
\( 17n = 17 \times 9 \)
Now, divide both sides by 17:
\( n = \frac{17 \times 9}{17} \)
\( n = 9 \)
So, X and Y worked together for 9 days before X left.
Summary of the Calculation
| Individual | Time to Complete Work | Work Rate (per day) |
|---|---|---|
| X | 45 days | \( \frac{1}{45} \) |
| Y | 40 days | \( \frac{1}{40} \) |
| Phase | Duration | Workers | Work Rate | Work Done |
|---|---|---|---|---|
| Phase 1 | n days | X and Y | \( \frac{1}{45} + \frac{1}{40} = \frac{17}{360} \) | \( n \times \frac{17}{360} = \frac{17n}{360} \) |
| Phase 2 | 23 days | Y | \( \frac{1}{40} \) | \( 23 \times \frac{1}{40} = \frac{23}{40} \) |
Total work = Work done in Phase 1 + Work done in Phase 2
\( 1 = \frac{17n}{360} + \frac{23}{40} \)
\( \frac{17n}{360} = 1 - \frac{23}{40} = \frac{40-23}{40} = \frac{17}{40} \)
\( \frac{17n}{360} = \frac{17}{40} \)
\( n = \frac{17}{40} \times \frac{360}{17} \)
\( n = \frac{360}{40} = 9 \)
Final Answer on Work Problem
The value of n, the number of days X and Y worked together, is 9.
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula |
|---|---|---|
| Work Rate | Amount of work done by a person/entity per unit of time (e.g., per day). | Work Rate \( = \frac{1}{\text{Time to complete work}} \) |
| Work Done | Total amount of work completed. | Work Done = Work Rate \( \times \) Time Worked |
| Combined Work Rate | Sum of individual work rates when people work together. | Combined Rate \( = \) Rate 1 + Rate 2 + ... |
| Total Work | Represents the entire task, usually considered as 1 unit. | Sum of work done by all parties \( = 1 \) |
Additional Information on Solving Work and Time Problems
Work and time problems often involve calculating rates, combined efforts, and portions of work completed. Here are some key takeaways:
- Always convert the time taken to complete the work into a daily (or hourly) work rate. This makes calculations easier.
- When people work together, their work rates add up.
- The total work done for completing the entire task is always 1 (or 100%).
- If someone leaves, subtract their work rate from the combined rate for the remaining period, or calculate the work done by the remaining person separately.
- Set up an equation where the sum of work done in different phases equals 1.
- Be careful with fractions and finding common denominators when adding or subtracting work rates or work done.
Similar Questions
If x men working x hours per day can do x units of work in x days, then y men working y hours per day in y days would be able to do k units of work. What is the value of k?
A, B and C can complete a work in x, 1.5x and 2x days respectively. If they complete the work together, in what ratio should they be paid ?
If A and B can finish a work in 10 days, B and C can finish the same work in 12 days, C and A can finish the same work in 15 days; then in how many days can A, B and C together finish half of the work?
Important Questions from Time and Work
Mohan can do a piece of work in 10 days and Sohan in 15 days. They started working together, but after 3 days Mohan left the work . What time will Sohan take to finish the work?
A tank is filled in 8 hours by three taps A, B and C. The tap C is thrice as fast as B and B is twice as fast as A. How much time will pipe B alone take to fill the tank?
Had been one menless, then the number of days required to do a piece of work would have been one more. If the number of Man. Days required to complete the work is 56, how many workers were there?
A can do a piece of work in 16 hours, B and C can do it in 8 hours while A and C can do it 12 hours. How long will B alone take to do it?
- lf 12 men can do a work in 20 days, in how many days will the work be done by 15 men-
