12. Work rate problems

12. Work rate problems

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Below are 10 work rate problems, each with a step-by-step solution procedure and detailed explanation. Work rate problems involve calculating the time, work, or rate at which individuals, machines, or systems (e.g., pipes) complete tasks. The key concept is work rate, defined as the fraction of work completed per unit of time (e.g., jobs per day). The standard formulas are:
  • Work = Rate × Time
  • Rate = Work / Time
  • Time = Work / Rate
When multiple entities work together, their rates are added. Work is typically normalized to 1 unit (e.g., one job, one tank) unless specified. Problems may involve workers, pipes filling/draining tanks, or machines, with units like days, hours, or minutes for time. Each problem covers different scenarios, ensuring a comprehensive understanding.
Problem 1: Two Workers Completing a Job
A can complete a job in 8 days, and B can complete it in 12 days. How long will they take to complete it together?
Step-by-Step Solution:
  1. Identify given values:
    • A’s time = 8 days, so A’s rate = 1/8 job/day
    • B’s time = 12 days, so B’s rate = 1/12 job/day
  2. Calculate combined rate: 1/8 + 1/12 = 3/24 + 2/24 = 5/24 job/day
  3. Calculate time: Time = Work / Combined rate = 1 / (5/24) = 24/5 = 4.8 days
  4. State result: They take 4.8 days together.
Detailed Explanation: Each worker’s rate is the reciprocal of their time to complete one job. The combined rate is their sum, reflecting their joint effort. The time is the total work (1 job) divided by this rate, resulting in less time than either alone due to increased efficiency.
Answer: 4.8 days
Problem 2: Three Workers
A completes a job in 10 hours, B in 15 hours, and C in 20 hours. How long will they take together?
Step-by-Step Solution:
  1. Identify given values:
    • A’s rate = 1/10 job/hour
    • B’s rate = 1/15 job/hour
    • C’s rate = 1/20 job/hour
  2. Calculate combined rate: 1/10 + 1/15 + 1/20 = 6/60 + 4/60 + 3/60 = 13/60 job/hour
  3. Calculate time: Time = 1 / (13/60) = 60/13 ≈ 4.615 hours
  4. State result: They take approximately 4.62 hours.
Detailed Explanation: The combined rate is the sum of individual rates. Using the LCM (60) simplifies adding fractions. The time is the reciprocal of the combined rate, showing that three workers are much faster than any one alone.
Answer: ≈ 4.62 hours
Problem 3: Partial Work
A can complete a job in 6 days, and B in 9 days. A works alone for 2 days, then B joins. How long to finish the job?
Step-by-Step Solution:
  1. Identify given values:
    • A’s rate = 1/6 job/day
    • B’s rate = 1/9 job/day
  2. Calculate A’s work in 2 days: 1/6 × 2 = 2/6 = 1/3 job
  3. Calculate remaining work: 1 - 1/3 = 2/3 job
  4. Calculate combined rate: 1/6 + 1/9 = 3/18 + 2/18 = 5/18 job/day
  5. Calculate time for remaining work: Time = (2/3) / (5/18) = 2/3 × 18/5 = 12/5 = 2.4 days
  6. Calculate total time: 2 + 2.4 = 4.4 days
  7. State result: The job takes 4.4 days total.
Detailed Explanation: A’s initial work reduces the job, and the remaining work is done at the combined rate. Total time includes A’s solo period and the joint period, showing how staggered starts impact completion time.
Answer: 4.4 days
Problem 4: Pipe Filling a Tank
Pipe A fills a tank in 12 hours, and Pipe B in 16 hours. How long to fill the tank together?
Step-by-Step Solution:
  1. Identify given values:
    • Pipe A’s rate = 1/12 tank/hour
    • Pipe B’s rate = 1/16 tank/hour
  2. Calculate combined rate: 1/12 + 1/16 = 4/48 + 3/48 = 7/48 tank/hour
  3. Calculate time: Time = 1 / (7/48) = 48/7 ≈ 6.857 hours
  4. State result: They take approximately 6.86 hours.
Detailed Explanation: Pipes are like workers, with rates as fractions of the tank filled per hour. The combined rate is their sum, and the time is the reciprocal. The result is faster than either pipe alone, reflecting joint efficiency.
Answer: ≈ 6.86 hours
Problem 5: Pipe with Leak
Pipe A fills a tank in 10 hours, and a leak empties it in 15 hours. How long to fill the tank if both are open?
Step-by-Step Solution:
  1. Identify given values:
    • Pipe A’s rate = 1/10 tank/hour
    • Leak’s rate = -1/15 tank/hour (negative for draining)
  2. Calculate net rate: 1/10 - 1/15 = 3/30 - 2/30 = 1/30 tank/hour
  3. Calculate time: Time = 1 / (1/30) = 30 hours
  4. State result: The tank takes 30 hours to fill.
Detailed Explanation: The leak’s negative rate opposes the filling, reducing the net rate. The time is longer than Pipe A alone, showing the leak’s inefficiency. This introduces opposing work rates, a common variation.
Answer: 30 hours
Problem 6: Finding Individual Time
A and B together complete a job in 5 days. A alone takes 8 days. How long does B take alone?
** Step-by-Step Solution:**
  1. Identify given values:
    • Combined rate = 1/5 job/day
    • A’s rate = 1/8 job/day
  2. Calculate B’s rate: 1/5 = 1/8 + B’s rate
    • B’s rate = 1/5 - 1/8 = 8/40 - 5/40 = 3/40 job/day
  3. Calculate B’s time: Time = 1 / (3/40) = 40/3 ≈ 13.33 days
  4. State result: B takes approximately 13.33 days.
Detailed Explanation: Subtracting A’s rate from the combined rate isolates B’s rate. The reciprocal gives B’s time. This reverses the typical problem, solving for an individual’s efficiency based on joint work.
Answer: ≈ 13.33 days
Problem 7: Machine Production
Machine A produces 500 units in 4 hours, and Machine B produces 300 units in 3 hours. How long to produce 2000 units together?
Step-by-Step Solution:
  1. Identify given values:
    • A’s rate = 500 / 4 = 125 units/hour
    • B’s rate = 300 / 3 = 100 units/hour
  2. Calculate combined rate: 125 + 100 = 225 units/hour
  3. Calculate time: Time = 2000 / 225 = 80/9 ≈ 8.89 hours
  4. State result: They take approximately 8.89 hours.
Detailed Explanation: Each machine’s rate is units per hour, and the combined rate is their sum. The time is the total units divided by the combined rate, adapting work rate to production contexts where work is measured in units.
Answer: ≈ 8.89 hours
Problem 8: Worker Leaving
A and B complete a job in 6 days. After 3 days, B leaves, and A takes 5 more days to finish. How long would B take alone?
Step-by-Step Solution:
  1. Identify given values:
    • Combined rate = 1/6 job/day
    • Together for 3 days, A alone for 5 days
  2. Calculate work in 3 days: 3 × 1/6 = 3/6 = 1/2 job
  3. Calculate remaining work: 1 - 1/2 = 1/2 job
  4. Calculate A’s rate: A’s time for 1/2 job = 5 days
    • A’s rate = (1/2) / 5 = 1/10 job/day
  5. Calculate B’s rate: 1/6 = 1/10 + B’s rate
    • B’s rate = 1/6 - 1/10 = 5/30 - 3/30 = 2/30 = 1/15 job/day
  6. Calculate B’s time: Time = 1 / (1/15) = 15 days
  7. State result: B takes 15 days alone.
Detailed Explanation: The work done together and by A alone allows calculating A’s rate. Subtracting A’s rate from the combined rate gives B’s rate, and the reciprocal is B’s time. This handles scenarios where a worker leaves mid-task.
Answer: 15 days
Problem 9: Staggered Start
Pipe A fills a tank in 9 hours, and Pipe B in 12 hours. Pipe B starts alone for 3 hours, then A joins. How long to fill the tank?
Step-by-Step Solution:
  1. Identify given values:
    • Pipe A’s rate = 1/9 tank/hour
    • Pipe B’s rate = 1/12 tank/hour
  2. Calculate B’s work in 3 hours: 1/12 × 3 = 3/12 = 1/4 tank
  3. Calculate remaining work: 1 - 1/4 = 3/4 tank
  4. Calculate combined rate: 1/9 + 1/12 = 4/36 + 3/36 = 7/36 tank/hour
  5. Calculate time for remaining work: Time = (3/4) / (7/36) = 3/4 × 36/7 = 27/7 ≈ 3.857 hours
  6. Calculate total time: 3 + 3.857 ≈ 6.857 hours
  7. State result: The tank takes approximately 6.86 hours.
Detailed Explanation: B’s initial work reduces the tank to be filled. The remaining work is done at the combined rate. Total time includes B’s solo period and the joint period, showing how staggered starts affect completion.
Answer: ≈ 6.86 hours
Problem 10: Efficiency Change
A worker completes a job in 15 days. If their efficiency increases by 20%, how long will it take?
Step-by-Step Solution:
  1. Identify given values:
    • Original time = 15 days
    • Efficiency increase = 20%
  2. Calculate original rate: Rate = 1/15 job/day
  3. Calculate new rate: New rate = 1.2 × 1/15 = 1.2 / 15 = 6/75 = 2/25 job/day
  4. Calculate new time: Time = 1 / (2/25) = 25/2 = 12.5 days
  5. State result: The worker takes 12.5 days.
Detailed Explanation: A 20% efficiency increase means the rate increases by 20% (1.2 times original). The new time is the reciprocal of the new rate, showing that higher efficiency reduces time proportionally.
Answer: 12.5 days
Summary
These 10 work rate problems cover two and three workers, partial work, pipes with leaks, finding individual times, production, workers leaving, staggered starts, and efficiency changes. Each uses work rate formulas, with rates added for collaboration or subtracted for opposition. Step-by-step solutions ensure clarity, and detailed explanations highlight how rates combine, how partial work affects totals, and how efficiency impacts time. These problems build a strong understanding of work rate calculations across various scenarios.

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