14. Efficiency-based calculations

 14. Efficiency-based calculations

Home page of Arithmetic Problems 

Below are 10 efficiency-based calculation problems, each with a step-by-step solution procedure and detailed explanation. Efficiency in arithmetic problems typically refers to work rate problems, where individuals, machines, or groups complete tasks at specific rates. The key concept is work rate, defined as the fraction of work completed per unit of time (e.g., tasks per hour or job per day). The standard formula is:
  • Work done = Rate × Time
  • Rate = Work / Time
  • Time = Work / Rate
For multiple workers, rates are added when working together, assuming one unit of work (e.g., one job) unless specified. Problems may involve pipes filling/draining tanks, workers completing tasks, or machines producing items. Units are typically days, hours, or minutes for time, and work is often normalized to 1 unit (e.g., one job, one tank). Each problem explores different scenarios, ensuring a comprehensive understanding.
Problem 1: Two Workers Completing a Job
A can complete a job in 6 days, and B can complete it in 8 days. How long will they take to complete it together?
Step-by-Step Solution:
  1. Identify given values:
    • A’s time = 6 days, so A’s rate = 1/6 job/day
    • B’s time = 8 days, so B’s rate = 1/8 job/day
  2. Calculate combined rate: 1/6 + 1/8 = 4/24 + 3/24 = 7/24 job/day
  3. Calculate time together: Time = Work / Combined rate = 1 / (7/24) = 24/7 ≈ 3.43 days
  4. State result: They take approximately 3.43 days together.
Detailed Explanation: Each worker’s rate is the reciprocal of their time to complete one job. Adding their rates gives the combined rate when working together. The time is the total work (1 job) divided by this rate. The result is less than either individual time, as collaboration increases efficiency.
Answer: ≈ 3.43 days
Problem 2: Three Workers with Different Rates
A completes a job in 10 hours, B in 15 hours, and C in 20 hours. How long will they take to complete it 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 ≈ 4.62 hours
  4. State result: They take approximately 4.62 hours.
Detailed Explanation: The combined rate is the sum of individual rates, reflecting the total work done per hour. Dividing 1 job by this rate gives the time, which is significantly less than any individual time due to the three workers’ combined effort. The LCM (60) simplifies fraction addition.
Answer: ≈ 4.62 hours
Problem 3: Partial Work by One Worker
A can complete a job in 12 days, and B can complete it in 18 days. A works alone for 4 days, then B joins. How long to finish the job?
Step-by-Step Solution:
  1. Identify given values:
    • A’s rate = 1/12 job/day
    • B’s rate = 1/18 job/day
  2. Calculate work done by A in 4 days: Work = 1/12 × 4 = 4/12 = 1/3 job
  3. Calculate remaining work: 1 - 1/3 = 2/3 job
  4. Calculate combined rate: 1/12 + 1/18 = 3/36 + 2/36 = 5/36 job/day
  5. Calculate time for remaining work: Time = (2/3) / (5/36) = 2/3 × 36/5 = 24/5 = 4.8 days
  6. State result: They take 4.8 days after B joins, total time = 4 + 4.8 = 8.8 days.
Detailed Explanation: A’s initial work reduces the job, and the remaining work is done at the combined rate. The time for the remaining work is calculated separately, and the total time includes A’s solo period. This shows how partial work affects overall efficiency.
Answer: 8.8 days total (4.8 days after B joins)
Problem 4: Pipe Filling a Tank
Pipe A fills a tank in 8 hours, and Pipe B fills it in 12 hours. How long will they take to fill the tank together?
Step-by-Step Solution:
  1. Identify given values:
    • Pipe A’s rate = 1/8 tank/hour
    • Pipe B’s rate = 1/12 tank/hour
  2. Calculate combined rate: 1/8 + 1/12 = 3/24 + 2/24 = 5/24 tank/hour
  3. Calculate time: Time = 1 / (5/24) = 24/5 = 4.8 hours
  4. State result: They take 4.8 hours.
Detailed Explanation: Pipes filling a tank are analogous to 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 less than either pipe’s individual time, showing increased efficiency.
Answer: 4.8 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 as it removes water)
  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 reduces the filling rate. The net rate is the difference, as the leak works against the pipe. The time is longer than Pipe A alone, reflecting the leak’s inefficiency. This introduces the concept of opposing rates.
Answer: 30 hours
Problem 6: Work Left After Collaboration
A and B together can complete a job in 4 days, and A alone takes 6 days. How long would B take alone?
Step-by-Step Solution:
  1. Identify given values:
    • A’s rate = 1/6 job/day
    • Combined rate = 1/4 job/day
  2. Calculate B’s rate: Combined rate = A’s rate + B’s rate
    • 1/4 = 1/6 + B’s rate
    • B’s rate = 1/4 - 1/6 = 3/12 - 2/12 = 1/12 job/day
  3. Calculate B’s time: Time = 1 / (1/12) = 12 days
  4. State result: B takes 12 days alone.
Detailed Explanation: The combined rate is given, and A’s rate is known. Subtracting A’s rate from the combined rate isolates B’s rate, and the reciprocal gives B’s time. This problem reverses the usual setup, solving for an individual’s efficiency.
Answer: 12 days
Problem 7: Machine Production
Machine A produces 600 units in 5 hours, and Machine B produces 400 units in 4 hours. How long to produce 2000 units together?
Step-by-Step Solution:
  1. Identify given values:
    • A’s rate = 600 / 5 = 120 units/hour
    • B’s rate = 400 / 4 = 100 units/hour
  2. Calculate combined rate: 120 + 100 = 220 units/hour
  3. Calculate time: Time = 2000 / 220 = 100/11 ≈ 9.09 hours
  4. State result: They take approximately 9.09 hours.
Detailed Explanation: Each machine’s rate is units per hour, and their combined rate is the sum. The time to produce 2000 units is the total work divided by the combined rate. This adapts the work rate concept to production, with units as the “work.”
Answer: ≈ 9.09 hours
Problem 8: Worker Leaving Mid-Task
A and B can complete a job in 5 days. After 2 days, A leaves, and B takes 9 more days to finish. How long would A take alone?
Step-by-Step Solution:
  1. Identify given values:
    • Combined rate = 1/5 job/day
    • Together for 2 days, B alone for 9 days
  2. Calculate work done in 2 days: 2 × 1/5 = 2/5 job
  3. Calculate remaining work: 1 - 2/5 = 3/5 job
  4. Calculate B’s rate: B’s time for 3/5 job = 9 days
    • B’s rate = (3/5) / 9 = 3/45 = 1/15 job/day
  5. Calculate A’s rate: 1/5 = A’s rate + 1/15
    • A’s rate = 1/5 - 1/15 = 3/15 - 1/15 = 2/15 job/day
  6. Calculate A’s time: Time = 1 / (2/15) = 15/2 = 7.5 days
  7. State result: A takes 7.5 days alone.
Detailed Explanation: The work done together and by B alone allows calculating B’s rate. Subtracting B’s rate from the combined rate gives A’s rate, and the reciprocal is A’s time. This problem shows how to handle workers leaving mid-task.
Answer: 7.5 days
Problem 9: Pipes with Different Start Times
Pipe A fills a tank in 6 hours, and Pipe B fills it in 8 hours. Pipe A starts alone for 2 hours, then B joins. How long to fill the tank?
Step-by-Step Solution:
  1. Identify given values:
    • Pipe A’s rate = 1/6 tank/hour
    • Pipe B’s rate = 1/8 tank/hour
  2. Calculate work by A in 2 hours: 1/6 × 2 = 2/6 = 1/3 tank
  3. Calculate remaining work: 1 - 1/3 = 2/3 tank
  4. Calculate combined rate: 1/6 + 1/8 = 4/24 + 3/24 = 7/24 tank/hour
  5. Calculate time for remaining work: Time = (2/3) / (7/24) = 2/3 × 24/7 = 16/7 ≈ 2.29 hours
  6. Calculate total time: 2 + 2.29 ≈ 4.29 hours
  7. State result: The tank takes approximately 4.29 hours to fill.
Detailed Explanation: A’s initial work reduces the tank to be filled. The remaining work is done at the combined rate. Total time includes A’s solo period and the joint period, showing how staggered starts affect efficiency.
Answer: ≈ 4.29 hours
Problem 10: Efficiency Increase
A machine completes a task in 12 hours. If its efficiency increases by 25%, how long will it take?
Step-by-Step Solution:
  1. Identify given values:
    • Original time = 12 hours
    • Efficiency increase = 25%
  2. Calculate original rate: Rate = 1/12 task/hour
  3. Calculate new efficiency: New efficiency = 1.25 × original = 1.25 × 1/12 = 5/48 task/hour
  4. Calculate new time: Time = 1 / (5/48) = 48/5 = 9.6 hours
  5. State result: The machine takes 9.6 hours.
Detailed Explanation: A 25% efficiency increase means the rate increases by 25% (1.25 times original). The new time is the reciprocal of the new rate, showing that higher efficiency reduces time proportionally.
Answer: 9.6 hours
Summary
These 10 efficiency-based problems cover workers, pipes, machines, and production, addressing scenarios like collaboration, opposing rates (leaks), partial work, staggered starts, and efficiency changes. Each uses work rate formulas (Work = Rate × Time), 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 changes impact time. These problems build a strong understanding of work rate calculations in various contexts. 

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