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Efficiency-Based Questions

Introduction

In Time and Work problems, efficiency represents how effectively a person (or machine) performs a task compared to another. Efficiency-based questions focus on comparing work rates, finding relative performance, and using ratios to determine time or work completed.

Understanding efficiency is essential because it allows you to quickly calculate how changes in efficiency affect the total time required to complete a task.

Pattern: Efficiency-Based Questions

Pattern

Key concept: Efficiency is inversely proportional to time taken for the same work.

If A and B complete the same work in TA and TB days respectively, then:
Efficiency Ratio = TB : TA
(The one who takes fewer days is more efficient.)

Alternatively, Work done ∝ Efficiency × Time - meaning efficiency can be used to find partial or combined work completion easily.

Step-by-Step Example

Question

A can complete a work in 12 days and B can complete it in 18 days. Find the ratio of their efficiencies and determine in how many days A will complete the work if B’s efficiency increases by 50%.

Solution

  1. Step 1: Find efficiency ratio.

    Efficiency ∝ 1 / Time.
    Hence, A : B = (1/12) : (1/18) = 18 : 12 = 3 : 2.

  2. Step 2: Increase B’s efficiency by 50%.

    B’s new efficiency = 2 × 1.5 = 3 (same as A).

  3. Step 3: Compare new efficiencies.

    Now A and B have equal efficiency (3 each), meaning both will take equal time.

  4. Step 4: Compute new time for A if B’s improved efficiency matches A’s.

    Since efficiency is unchanged for A, A still completes work in 12 days.

  5. Final Answer:

    Efficiency ratio = 3 : 2; A completes the work in 12 days.

  6. Quick Check:

    Verify using inverse ratio: time ratio = 2 : 3 → efficiency ratio = 3 : 2 ✅

Quick Variations

1. Comparing multiple workers’ efficiencies to find who’s faster or slower.

2. Finding work done when efficiency changes mid-way (e.g., 25% increase).

3. Combining efficiencies when workers collaborate on the same job.

4. Using efficiency to determine how long one will take after another completes part of the work.

Trick to Always Use

  • Step 1 → Write efficiency ratio as inverse of time ratio.
  • Step 2 → Use “Work = Efficiency × Time” for partial or combined work.
  • Step 3 → Adjust efficiency for any % increase or decrease, then recalculate time or work done.

Summary

Summary

  • Efficiency is inversely proportional to time taken for the same work.
  • Efficiency ratio = Reciprocal of time ratio.
  • When efficiency changes by x%, time changes inversely by x% in opposite direction.
  • Formula links: Work = Efficiency × Time and Time = Work / Efficiency.
  • Efficiency helps solve combined, comparative, and percentage-based work questions quickly.

Practice

(1/5)
1. A is twice as efficient as B. If B can complete a work in 12 days, in how many days can A complete it?
easy
A. 8 days
B. 6 days
C. 10 days
D. 5 days

Solution

  1. Step 1: Identify the values.

    B's time = 12 days → B's one-day work = 1/12.

  2. Step 2: Use efficiency relation.

    A is twice as efficient ⇒ A's one-day work = 2 × (1/12) = 1/6.

  3. Step 3: Compute A's time.

    A's time = 1 ÷ (1/6) = 6 days.

  4. Final Answer:

    A completes the work in 6 days → Option B.

  5. Quick Check:

    Efficiency ratio A:B = 2:1 ⇒ time ratio A:B = 1:2 → 6 is half of 12 ✅
Hint: If A is k times efficient, A's time = B's time ÷ k.
Common Mistakes: Multiplying time by efficiency instead of dividing.
2. A is 25% more efficient than B. If B can finish a work in 20 days, how long will A take to finish the same work?
easy
A. 15 days
B. 18 days
C. 16 days
D. 12 days

Solution

  1. Step 1: Identify the values.

    B's time = 20 days.

  2. Step 2: Convert efficiency relation.

    A is 25% more efficient ⇒ A's efficiency = 125% of B's = 5/4 × B's efficiency.

  3. Step 3: Use inverse relation with time.

    Time ∝ 1/Efficiency ⇒ A's time = 20 × (4/5) = 16 days.

  4. Final Answer:

    A can finish the work in 16 days → Option C.

  5. Quick Check:

    Efficiency ratio A:B = 5:4 ⇒ Time ratio A:B = 4:5; 16:20 = 4:5 ✅
Hint: If A is p% more efficient, A's time = B's time × 100/(100+p).
Common Mistakes: Adding percentages to time instead of adjusting via reciprocals.
3. A is 50% more efficient than B. Together they can complete a task in 6 days. How many days will A alone take to complete it?
easy
A. 10 days
B. 12 days
C. 15 days
D. 9 days

Solution

  1. Step 1: Set variables for efficiencies.

    Let B's one-day work = x. Then A = 1.5x (50% more).

  2. Step 2: Use combined time to find x.

    Combined one-day work = x + 1.5x = 2.5x = 1/6 → x = 1/(6 × 2.5) = 1/15.

  3. Step 3: Compute A's time.

    A's one-day work = 1.5x = 1.5 × (1/15) = 1/10 → A's time = 1 ÷ (1/10) = 10 days.

  4. Final Answer:

    A alone takes 10 days → Option A.

  5. Quick Check:

    B's time = 15 days (1/15). Combined rate = 1/10 + 1/15 = (3+2)/30 = 5/30 = 1/6 → reciprocal 6 days ✅
Hint: Express efficiencies as multiples (x and kx), solve combined = 1/time.
Common Mistakes: Mixing up which variable represents efficiency vs time.
4. A and B’s efficiency ratio is 5:4. If B can complete the work in 30 days, how long will A take to complete it?
medium
A. 24 days
B. 18 days
C. 20 days
D. 15 days

Solution

  1. Step 1: Write the ratio relations.

    Efficiency A:B = 5:4 ⇒ Time A:B = 4:5 (inverse relation).

  2. Step 2: Use B's time to find A's time.

    If B = 30 days, then A = 30 × (4/5) = 24 days.

  3. Final Answer:

    A will take 24 days → Option A.

  4. Quick Check:

    Efficiency ratio 5:4 ⇒ times 4:5; 24:30 = 4:5 ✅
Hint: Invert the efficiency ratio to get the time ratio before scaling.
Common Mistakes: Applying the same ratio to time (instead of the inverse).
5. A is 40% less efficient than B. If together they can complete a job in 7.5 days, how many days will A alone take to finish it?
medium
A. 12.5 days
B. 15 days
C. 17.5 days
D. 20 days

Solution

  1. Step 1: Express efficiencies.

    Let B's efficiency = x. A is 40% less efficient ⇒ A = 0.6x.

  2. Step 2: Use combined time to find x.

    Combined rate = x + 0.6x = 1.6x = 1/7.5 → x = 1 / (7.5 × 1.6) = 1/12.

  3. Step 3: Compute A's time.

    A's one-day work = 0.6x = 0.6 × (1/12) = 1/20 → A's time = 20 days.

  4. Final Answer:

    A alone takes 20 days → Option D.

  5. Quick Check:

    B's time = 12 days (1/12). 1/20 + 1/12 = (3 + 5)/60 = 8/60 = 1/7.5 ✅
Hint: Convert percentage change to decimal (e.g., 40% less = 0.6×), add rates, then invert.
Common Mistakes: Reducing both efficiencies instead of expressing one relative to the other.

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