A and B can complete a piece of work in 120 days. B and C can complete it in 150 days, and A and C can complete it in 200 days. In hoe many days can B alone complete two-fifths of the same work ?
80
Understanding the Work and Time Problem
This question is about work and time. We are given the time taken for pairs of people (A and B, B and C, A and C) to complete a specific piece of work. We need to find out how long it takes for one person, B, to complete a fraction (two-fifths) of the same work when working alone.
To solve this, we first need to determine the individual work rate of each person. The work rate is the amount of work done per unit of time (in this case, per day). If a person or a group can complete a work in 'd' days, their daily work rate is \( \frac{1}{d} \) of the work.
Calculating Combined Daily Work Rates
Let's denote the daily work rates of A, B, and C as \( R_A \), \( R_B \), and \( R_C \) respectively. Based on the problem statement, we have the following information about the combined daily work rates:
- A and B together can complete the work in 120 days. Their combined daily rate is \( R_A + R_B = \frac{1}{120} \).
- B and C together can complete the work in 150 days. Their combined daily rate is \( R_B + R_C = \frac{1}{150} \).
- A and C together can complete the work in 200 days. Their combined daily rate is \( R_A + R_C = \frac{1}{200} \).
Finding the Combined Daily Work Rate of A, B, and C
If we add the daily work rates of all three pairs, we get:
\( (R_A + R_B) + (R_B + R_C) + (R_A + R_C) = \frac{1}{120} + \frac{1}{150} + \frac{1}{200} \)
\( 2(R_A + R_B + R_C) = \frac{1}{120} + \frac{1}{150} + \frac{1}{200} \)
To add the fractions on the right side, we find the Least Common Multiple (LCM) of 120, 150, and 200.
- \( 120 = 2^3 \times 3 \times 5 \)
- \( 150 = 2 \times 3 \times 5^2 \)
- \( 200 = 2^3 \times 5^2 \)
- LCM is \( 2^3 \times 3 \times 5^2 = 8 \times 3 \times 25 = 600 \).
Now, we add the fractions:
\( \frac{1}{120} + \frac{1}{150} + \frac{1}{200} = \frac{1 \times 5}{120 \times 5} + \frac{1 \times 4}{150 \times 4} + \frac{1 \times 3}{200 \times 3} = \frac{5}{600} + \frac{4}{600} + \frac{3}{600} = \frac{5+4+3}{600} = \frac{12}{600} = \frac{1}{50} \)
So, \( 2(R_A + R_B + R_C) = \frac{1}{50} \). This means the combined daily work rate of A, B, and C together is:
\( R_A + R_B + R_C = \frac{1}{50} \times \frac{1}{2} = \frac{1}{100} \)
Determining B's Individual Daily Work Rate
We know the combined rate of A, B, and C, and we know the combined rate of A and C. To find B's individual rate, we subtract the rate of (A + C) from the rate of (A + B + C):
\( R_B = (R_A + R_B + R_C) - (R_A + R_C) \)
\( R_B = \frac{1}{100} - \frac{1}{200} \)
To subtract these fractions, we find the LCM of 100 and 200, which is 200.
\( R_B = \frac{1 \times 2}{100 \times 2} - \frac{1}{200} = \frac{2}{200} - \frac{1}{200} = \frac{2-1}{200} = \frac{1}{200} \)
B's daily work rate is \( \frac{1}{200} \). This means B can complete the entire work alone in 200 days.
Calculating Time for B to Complete Two-Fifths of the Work
The question asks for the time B takes to complete two-fifths (\( \frac{2}{5} \)) of the work. If B takes 200 days to complete the full work (which is 1 whole), the time to complete a fraction of the work is the total time multiplied by that fraction.
Time for B to complete \( \frac{2}{5} \) of the work = (Time for B to complete whole work) \( \times \frac{2}{5} \)
Time = \( 200 \text{ days} \times \frac{2}{5} \)
Time = \( \frac{200 \times 2}{5} \text{ days} \)
Time = \( \frac{400}{5} \text{ days} \)
Time = \( 80 \text{ days} \)
Therefore, B alone can complete two-fifths of the same work in 80 days.
Summary of Steps
Here's a quick recap of the calculation steps:
- Find the daily work rate for each pair: A+B, B+C, A+C.
- Sum these rates and divide by 2 to find the combined daily rate of A+B+C.
- Subtract the known pair rate (A+C) from the combined rate (A+B+C) to find B's individual daily rate.
- The reciprocal of B's daily rate gives the number of days B takes to complete the whole work.
- Multiply the time taken for the whole work by the fraction of the work needed (2/5) to find the final answer.
| Pair | Time Taken (days) | Daily Work Rate |
|---|---|---|
| A + B | 120 | \( \frac{1}{120} \) |
| B + C | 150 | \( \frac{1}{150} \) |
| A + C | 200 | \( \frac{1}{200} \) |
| A + B + C (combined) | \( \frac{1}{100} \) | |
| B (alone) | 200 | \( \frac{1}{200} \) |
Revision Table: Work and Time Concepts
| Concept | Description | Formula/Relation |
|---|---|---|
| Work Rate | Amount of work done per unit time. | Work Rate = \( \frac{\text{Work Done}}{\text{Time Taken}} \) |
| Time Taken | Total time to complete the work. | Time Taken = \( \frac{\text{Work Done}}{\text{Work Rate}} \) |
| If Rate is R | Time to complete whole work (Work = 1) | Time = \( \frac{1}{R} \) |
| Combined Rate | Sum of individual rates when working together. | \( R_{total} = R_1 + R_2 + ... \) |
Additional Information on Work and Time Problems
Work and Time problems often involve calculating how efficiently individuals or groups can perform a task and how long it takes them to complete it, either alone or together. Key ideas include:
- Assuming the total work is a single unit (1).
- Calculating the fraction of work done per unit of time (usually per day).
- Adding rates when people work together.
- Subtracting rates to find individual contributions from combined rates.
- Understanding that Time Taken is the reciprocal of the Work Rate.
- Calculating time for a fraction of work by multiplying the total time by the fraction.
These problems can be solved using fractions, percentages, or sometimes LCM methods to simplify calculations, especially when dealing with different completion times.
Important Questions from Work and Wages
A book has 250 pages. Person A reads 6 pages in an hour. Person B reads 8 pages in an hour. There are two chapters of 72 pages that are difficult for person B to read in the book, so person B takes double the time to read those pages. Who among them will finish the book first and how much sooner than the other?
Efficiencies of P, Q, R and S in doing a job are in the ratio 2 : 3 : 5 : 4. The wages paid for the job are Rs.4200. Who is paid the largest amount and how much is paid?
P and Q can complete a job in 6 and 8 days individually. They both finish the job with the help of R in 3 days. How much wages are to be paid to R. if the total wages are Rs. 3200?
Anjali can do a certain piece of work in 16 days. Anjali and Ayushi can together do the same work in 10 days, and Anjali, Ayushi and Ankita can do the same work together in 8 days. In how many days can Anjali and Ankita do the same work?
Pravin can do a piece of work in 6 hours. Rishi can do it in 28 hours. With the assistance of Shan, they completed the work in 4 hours. In how many hours can Shan alone do it?
