P is two times as efficient as Q. P is able to complete a piece of work in 40 days less than Q. Working together, the whole number of days taken by them to complete the work is: (Round off to the nearest integer)
27
Solving Work and Efficiency Problems: P and Q Together
This question involves a classic work and time problem, focusing on the concept of efficiency. Efficiency is inversely proportional to the time taken to complete a piece of work. If a person is more efficient, they take less time to complete the same amount of work.
Understanding Efficiency and Time
We are given that P is two times as efficient as Q. This means that for the same amount of work, P will take half the time that Q takes.
Let \( T_P \) be the time taken by P to complete the work alone, and \( T_Q \) be the time taken by Q to complete the work alone.
Since P is twice as efficient as Q, the time taken by P is half the time taken by Q:
\( T_P = \frac{1}{2} T_Q \)
Setting up Equations Based on the Problem
We are also told that P is able to complete the work in 40 days less than Q. This gives us another relationship between \( T_P \) and \( T_Q \):
\( T_P = T_Q - 40 \)
Finding Individual Times
Now we have two expressions for \( T_P \). We can set them equal to each other to solve for \( T_Q \):
\( \frac{1}{2} T_Q = T_Q - 40 \)
To solve for \( T_Q \), we can rearrange the equation:
\( 40 = T_Q - \frac{1}{2} T_Q \)
\( 40 = \left(1 - \frac{1}{2}\right) T_Q \)
\( 40 = \frac{1}{2} T_Q \)
Multiply both sides by 2:
\( T_Q = 40 \times 2 \)
\( T_Q = 80 \) days
Now that we have \( T_Q \), we can find \( T_P \) using either equation. Let's use \( T_P = T_Q - 40 \):
\( T_P = 80 - 40 \)
\( T_P = 40 \) days
Calculating Work Rate
The work rate of a person is the amount of work they complete in one day. If a person completes the entire work in \( T \) days, their work rate per day is \( \frac{1}{T} \) of the work.
- P's work rate per day = \( \frac{1}{T_P} = \frac{1}{40} \)
- Q's work rate per day = \( \frac{1}{T_Q} = \frac{1}{80} \)
Working Together
When P and Q work together, their work rates add up. The combined work rate per day is:
\( \text{Combined work rate} = \text{P's work rate} + \text{Q's work rate} \)
\( \text{Combined work rate} = \frac{1}{40} + \frac{1}{80} \)
To add these fractions, find a common denominator, which is 80:
\( \text{Combined work rate} = \frac{2}{80} + \frac{1}{80} \)
\( \text{Combined work rate} = \frac{3}{80} \)
This means that working together, P and Q complete \( \frac{3}{80} \) of the work in one day.
Finding Time Taken Together
The total time taken to complete the work when working together is the reciprocal of the combined work rate:
\( \text{Time together} = \frac{1}{\text{Combined work rate}} \)
\( \text{Time together} = \frac{1}{\frac{3}{80}} \)
\( \text{Time together} = \frac{80}{3} \) days
Rounding to the Nearest Integer
The question asks for the whole number of days taken, rounded off to the nearest integer. Let's calculate the value of \( \frac{80}{3} \):
\( \frac{80}{3} \approx 26.666... \)
Rounding 26.666... to the nearest integer gives 27.
So, working together, P and Q take approximately 27 days to complete the work.
Summary of Steps
- Used the efficiency ratio to relate the individual times of P and Q.
- Used the given time difference to create another equation relating their times.
- Solved the equations simultaneously to find the individual times \( T_P \) and \( T_Q \).
- Calculated the individual work rates per day.
- Added the individual work rates to find the combined work rate per day.
- Took the reciprocal of the combined work rate to find the total time taken when working together.
- Rounded the final answer to the nearest integer.
| Person | Individual Time (Days) | Work Rate (Work per Day) |
|---|---|---|
| P | 40 | \( \frac{1}{40} \) |
| Q | 80 | \( \frac{1}{80} \) |
| P & Q Together | \( \frac{80}{3} \approx 26.67 \) | \( \frac{3}{80} \) |
Revision Table: Work and Efficiency Concepts
| Concept | Description | Relationship |
|---|---|---|
| Work | The task to be completed (often considered as 1 unit). | - |
| Time | Duration taken to complete the work. | Inversely proportional to Efficiency |
| Efficiency / Work Rate | Amount of work done per unit of time. | Efficiency \( \propto \frac{1}{\text{Time}} \) |
| Total Work | Work Rate \( \times \) Time | Often 1 unit for the whole task |
| Combined Work Rate | Sum of individual work rates when people work together. | \( R_{total} = R_1 + R_2 + ... \) |
Additional Information: Alternative Method (LCM)
For work and time problems, you can sometimes assume a total amount of work based on the LCM (Least Common Multiple) of the individual times. This can help avoid fractions initially.
In this case, we found \( T_P = 40 \) days and \( T_Q = 80 \) days.
Let the total work be the LCM of 40 and 80, which is 80 units.
- P completes 80 units in 40 days. P's work rate = \( \frac{80 \text{ units}}{40 \text{ days}} = 2 \) units/day.
- Q completes 80 units in 80 days. Q's work rate = \( \frac{80 \text{ units}}{80 \text{ days}} = 1 \) unit/day.
Notice that P's rate (2 units/day) is indeed twice Q's rate (1 unit/day), matching the initial efficiency statement.
When P and Q work together, their combined rate is \( 2 + 1 = 3 \) units/day.
To complete 80 units of work at a combined rate of 3 units/day, the time taken is:
\( \text{Time together} = \frac{\text{Total Work}}{\text{Combined Rate}} = \frac{80 \text{ units}}{3 \text{ units/day}} = \frac{80}{3} \) days.
This gives the same result, \( \frac{80}{3} \approx 26.666... \) days, which rounds to 27 days. The LCM method can be useful for simplifying calculations with integers.
Similar Questions
X and Y can complete a work in 9 days and 36 days, respectively. X begins to do the work and they work alternately one at a time for one day each. The whole work will be complete in:
4 women or 6 boys can finish a work in the same number of days. A boy can finish it in 60 days. In how many days can 5 women finish the work, working together every day?
To do a certain work, Ajay and Bharat work on alternate days, with Bharat starting the work on the first day. Ajay can finish the work alone in 32 days. If the work gets completed in exactly 8 days, then Bharat alone can finish 7 times the same work in ____________ days.
A, B and C can separately complete a work in 12, 15 and 20 days, respectively. They worked together 4 days. What will be the remaining work?
A, B and C, working alone, can complete a job in 16, 24 and 36 days, respectively. In how many days can they complete the job if they work together?
Rakshit, Ajay, and Satish are sanitation workers in a Municipal Corporation. Rakshit alone takes 20 hours to clean a drain while Ajay takes 12 hours when working alone to do the same. All three together take only 5 hours to clean the drain. In how many hours, can Satish complete the work alone?
15 men and 25 women can complete a piece of work in 9.6 days. If 16 women can complete the same work in 27 days, find the number of days in which 16 men can complete the same work.
A,B and C can do a piece of work in 30 days, 40 days and 50 days, respectively. Beginning with A, if A, B and C do the work alternatively then in how many days will the work be finished?
Ravi can do a piece of work in 40 days and Sudha can do the same piece of work in 60 days. If they work on alternative days starting with Sudha on the first day, then in how many days will the work be completed?
Working 5 hours a day, A can complete a task in 8 days and working 6 hours a day, B can finish the same task in 10 days, working 8 hours a day, they can jointly complete the task in __________.
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?
