Unit 1: Efficiency and Inlet-Outlet Pipes - Subjective Questions
PEA308 — Advanced Analytical Skills-Ii • Practice Questions with Detailed Answers
20 questions
Define efficiency in time-and-work problems. A can complete a task in 24 days and B can complete the same task in 36 days. Find their individual efficiencies and the efficiency when they work together.
Solution:
- Total work can be taken as the least common multiple of 24 and 36:
- A's efficiency:
- B's efficiency:
- Combined efficiency:
- Time taken together:
Therefore, A's efficiency is units per day, B's efficiency is units per day, and their combined efficiency is units per day.
A, B, and C can complete a piece of work in 20, 30, and 60 days respectively. They work together for 5 days, after which C leaves. Find the total time required to complete the work.
Solution:
- Work done by A, B, and C in one day:
- Work completed in 5 days:
- Remaining work:
- Work done by A and B in one day:
- Time taken by A and B to complete the remaining work:
Hence, the total time required is:
Explain the relationship between efficiency and time. If the efficiency of a worker increases by , by what percentage does the time required to complete the same work decrease?
Solution:
For a fixed amount of work:
Therefore, efficiency and time are inversely proportional:
Let the original efficiency be units and the new efficiency be units. Then:
The decrease in time is:
Thus, when efficiency increases by , the time required decreases by .
A contractor agrees to pay workers a total wage of $18,000 for completing a project in $15$ days. After $5$ days, only one-fourth of the work is complete. How many additional workers of the same efficiency are required to finish the remaining work in the next $6$ days?
Solution:
- Work completed in 5 days by 12 workers:
- Therefore, the work done by one worker in one day is:
- Remaining work:
- Let the required number of workers for the next 6 days be .
- Solving:
- Additional workers required:
Therefore, 18 additional workers are required. The wage information is not needed for the worker calculation, but it can be used to determine the average wage per worker per day:
Six men or eight women can complete a task in 24 days. Find the ratio of the efficiency of one man to that of one woman. Also determine how many days 3 men and 4 women will take to complete the task.
Solution:
Since 6 men and 8 women complete the same work in the same time:
Thus:
Therefore, the efficiency ratio of one man to one woman is:
Take the efficiency of one man as units and one woman as units.
- Efficiency of 3 men and 4 women:
- Efficiency of 6 men:
Hence, 3 men and 4 women have the same efficiency as 6 men. They will therefore complete the task in 24 days.
Explain the chain rule for work problems. If 8 workers working 6 hours per day complete a project in 15 days, how many workers working 10 hours per day will complete twice the project in 12 days, assuming all workers have equal efficiency?
Solution:
The amount of work is proportional to:
Let the required number of workers be . Using the chain rule:
Therefore:
Thus, 12 workers working 10 hours per day for 12 days are required.
A group of 20 workers can complete a construction project in 18 days by working 8 hours per day. After 6 days, the workers are reduced to 15, and the remaining work must be completed in 10 days. How many hours per day should the remaining workers work?
Solution:
- Total work in worker-hours:
- Work completed in the first 6 days:
- Remaining work:
- Let the required working hours per day be .
- Therefore:
Hence, the remaining workers must work 12.8 hours per day, or 12 hours 48 minutes per day.
A can complete a work in 12 days and B can complete it in 18 days. They work on alternate days, beginning with A. In how many days will the work be completed?
Solution:
- A's one-day work:
- B's one-day work:
- Work completed in a two-day cycle:
- In 7 two-day cycles, or 14 days, work completed is:
- Remaining work:
- A works next and completes work in:
Therefore, the work will be completed in:
A, B, and C can complete a job alone in 10, 15, and 30 days respectively. A works every day, while B and C work on alternate days, with B working on the first day. Find the time required to complete the job.
Solution:
- A's daily work:
- B's daily work:
- C's daily work:
In two consecutive days, the work done is:
Thus, in 6 days, three such cycles complete:
Therefore, the job is completed in 6 days.
A and B together can complete a work in 8 days, B and C together in 12 days, and C and A together in 24 days. Find the time taken by each person working alone.
Solution:
Let the daily efficiencies of A, B, and C be , , and respectively.
Given:
Adding all three equations:
Therefore:
Now:
- A's efficiency:
- B's efficiency:
- C's efficiency:
Hence:
- A alone takes 24 days.
- B alone takes 12 days.
- C alone takes 24 days.
What is meant by an efficiency-based wage problem? A, B, and C work for 6, 8, and 10 days respectively. Their daily efficiencies are in the ratio . If the total wage is $7,000, distribute the wage among them in proportion to the work done.
Solution:
Wages are distributed according to the amount of work done, where:
The work ratios are:
- A:
- B:
- C:
Thus, the wage ratio is:
Total ratio:
Therefore:
- A's wage:
- B's wage:
- C's wage:
Thus, the wages are approximately $2,863.64, $2,545.45, and $1,590.91 respectively.
A contractor pays $24,000 for a project. A completes $\frac{1}{3}$ of the work, B completes $\frac{1}{4}$, and C completes the remainder. If their wages include a fixed allowance of $1,000 each in addition to payment proportional to work done, find the wage received by each worker.
Solution:
- Total fixed allowance:
- Amount remaining for distribution according to work:
- C's share of work:
The work-based payments are:
- A:
- B:
- C:
Adding the fixed allowance:
- A's wage:
- B's wage:
- C's wage:
The total is:
Distinguish between an inlet and an outlet in tank problems. An inlet fills a tank in 12 hours, while an outlet empties it in 18 hours. If both are opened simultaneously, find the time required to fill the empty tank.
Solution:
- An inlet adds water to the tank and has a positive rate.
- An outlet removes water from the tank and has a negative rate.
The inlet's rate is:
The outlet's rate is:
Net filling rate:
Therefore, the tank will be filled in:
An inlet can fill a tank in 20 minutes, another inlet can fill it in 30 minutes, and an outlet can empty it in 60 minutes. If all three are opened together, determine the time taken to fill the tank.
Solution:
The net rate is:
Taking the denominator as :
Thus, the tank is filled at the rate of tank per minute.
Therefore, the filling time is:
A tank is filled by pipe A in 16 hours and by pipe B in 24 hours. A leakage can empty the full tank in 48 hours. If A, B, and the leakage are open simultaneously, how long will it take to fill the tank?
Solution:
- Rate of A:
- Rate of B:
- Rate of leakage:
Net rate:
Using denominator :
Therefore, the tank will be filled in:
Pipe A fills a tank in 10 hours and pipe B fills it in 15 hours. Both pipes are opened together, but pipe B is closed after 2 hours. Find the total time required to fill the tank.
Solution:
- Combined rate of A and B:
- Work completed in the first 2 hours:
- Remaining part of the tank:
- A alone fills at a rate of tank per hour.
- Time required for the remaining part:
Total time:
Therefore, the tank will be filled in hours, or 8 hours 40 minutes.
Two pipes can fill a tank in 18 minutes and 24 minutes respectively. A third pipe can empty the tank in 36 minutes. The first two pipes are opened for 6 minutes, and then the outlet is also opened. Find the total time required to fill the tank.
Solution:
- Combined filling rate of the first two pipes:
- Work completed in 6 minutes:
- Remaining work:
- Net rate after opening the outlet:
- Time to complete the remaining work:
Hence, total time required is:
A tank is already full. An inlet fills the entire tank in 20 hours, while an outlet empties the entire tank in 30 hours. If both are opened simultaneously, how long will it take to fill the remaining portion?
Solution:
- Remaining portion of the tank:
- Net rate of filling:
- Time required:
Therefore, it will take 24 hours to fill the remaining portion.
An inlet fills of a tank in 3 hours, while an outlet empties of the tank in 2 hours. If the tank is initially half full and both pipes are opened, determine whether the tank fills or empties and find the time until it becomes empty or full.
Solution:
- Inlet rate:
- Outlet rate:
Since the rates are equal:
The amount of water remains unchanged at of the tank. Therefore, the tank neither fills nor empties, and it will remain half full indefinitely while both pipes remain open.
Pipe A fills a tank in 8 hours, pipe B in 12 hours, and pipe C empties it in 24 hours. A and B are opened together for 2 hours, then C is opened while all three remain open. Find the time taken to fill the tank from the beginning.
Solution:
- Rate of A and B together:
- Work done in the first 2 hours:
- Remaining work:
- Net rate when all three pipes are open:
- Time for the remaining work:
Total time:
Therefore, the tank is filled in 5.5 hours, or 5 hours 30 minutes.
Define efficiency in time-and-work problems. A can complete a task in 24 days and B can complete the same task in 36 days. Find their individual efficiencies and the efficiency when they work together.
Solution:
- Total work can be taken as the least common multiple of 24 and 36:
- A's efficiency:
- B's efficiency:
- Combined efficiency:
- Time taken together:
Therefore, A's efficiency is units per day, B's efficiency is units per day, and their combined efficiency is units per day.
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