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Efficiency-Based Questions

Introduction

Time and Work problems में efficiency यह दिखाती है कि कोई व्यक्ति (या machine) दूसरे की तुलना में कितना effectively काम करता है। Efficiency-based questions में work rates की comparison, relative performance निकालना और ratios की मदद से time या पूरा किया गया work find करना होता है।

Efficiency समझना ज़रूरी है क्योंकि इससे आप जल्दी पता लगा सकते हैं कि efficiency बदलने पर किसी काम को पूरा करने में लगने वाला total time कैसे बदलता है।

Pattern: Efficiency-Based Questions

Pattern

Key concept: Efficiency उसी work के लिए लिए गए time के उल्टा proportional होती है।

अगर A और B वही work TA और TB days में पूरा करते हैं, तो:
Efficiency Ratio = TB : TA
(जो कम दिन लेता है, वही ज़्यादा efficient होता है।)

दूसरे तरीके से, Work done ∝ Efficiency × Time - यानी efficiency की मदद से partial या combined work आसानी से निकाला जा सकता है।

Step-by-Step Example

Question

A कोई काम 12 days में और B वही काम 18 days में करता है। उनकी efficiencies का ratio निकालें और बताएं कि अगर B की efficiency 50% बढ़ जाए तो A कितने दिनों में काम पूरा करेगा?

Solution

  1. Step 1: Efficiency ratio निकालें।

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

  2. Step 2: B की efficiency 50% बढ़ाएं।

    B की नई efficiency = 2 × 1.5 = 3 (A के बराबर)।

  3. Step 3: नई efficiencies compare करें।

    अब A और B दोनों की efficiency बराबर है (3-3), यानी दोनों को एक जैसा time लगेगा।

  4. Step 4: A का नया time निकालें जब B की बढ़ी efficiency A के बराबर हो।

    A की efficiency बदली नहीं है, इसलिए A अभी भी काम 12 days में पूरा करेगा।

  5. Final Answer:

    Efficiency ratio = 3 : 2; A काम 12 days में ही पूरा करेगा।

  6. Quick Check:

    Inverse ratio से verify करें: time ratio = 2 : 3 → efficiency ratio = 3 : 2 ✅

Quick Variations

1. कई workers की efficiencies compare करके पता लगाना कि कौन तेज़ या धीमा है।

2. Work के बीच में efficiency बदल जाए (जैसे 25% increase) तो work done निकालना।

3. जब workers एक साथ काम करें, तो efficiencies को जोड़कर combined काम निकालना।

4. Efficiency की मदद से पता लगाना कि एक worker के कुछ work पूरा करने के बाद दूसरे को कितना time लगेगा।

Trick to Always Use

  • Step 1 → Efficiency ratio हमेशा time ratio का उल्टा लिखें।
  • Step 2 → Partial या combined work के लिए “Work = Efficiency × Time” का उपयोग करें।
  • Step 3 → Efficiency में किसी % increase/decrease के बाद नया time या work फिर से calculate करें।

Summary

Summary

  • Efficiency, उसी work के लिए लिए गए time के उल्टा proportional होती है।
  • Efficiency ratio = time ratio का reciprocal।
  • Efficiency x% बढ़े तो time x% उतना ही उल्टा कम होता है (और vice versa)।
  • Formula links: Work = Efficiency × Time और Time = Work / Efficiency.
  • Efficiency की मदद से combined, comparative और percentage-based work questions जल्दी solve होते हैं।

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.

Mock Test

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