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Work and Wages Problems

Introduction

Work and wages problems में काम की मात्रा को workers या teams द्वारा कमाए गए पैसों से जोड़ा जाता है। ये problems इसलिए important हैं क्योंकि ये यह जांचते हैं कि आप payment को work contribution या time spent के अनुसार fair तरीके से बांट सकते हैं या नहीं।

Common cases में total payment को workers के efficiencies, worked time या दिए गए ratios के आधार पर share करना शामिल होता है।

Pattern: Work and Wages Problems

Pattern

Key concept: Payment हमेशा उसी ratio में बांटा जाता है जिसमें प्रत्येक व्यक्ति का work done हो (time के आधार पर नहीं), और work done = rate × time.

लागू करने के steps:

  1. हर व्यक्ति का one-day work (efficiency) निकालें या दिया हुआ efficiency ratio इस्तेमाल करें।
  2. Actual work done निकालें: (efficiency × time worked) या proportional parts का उपयोग करें।
  3. Total payment × (individual work ÷ total work) = individual share।

Step-by-Step Example

Question

तीन workers A, B और C मिलकर एक काम पूरा करते हैं। A अकेले 12 days, B 15 days और C 20 days में काम करते हैं। वे तीनों मिलकर पूरा काम करते हैं और ₹1080 कमाते हैं। हर worker को कितना भुगतान मिलेगा?

Solution

  1. Step 1: One-day work (efficiency) निकालें

    A = 1/12, B = 1/15, C = 1/20.

  2. Step 2: Common denominator में बदलें

    LCM(12, 15, 20) = 60: A = 5/60, B = 4/60, C = 3/60.

  3. Step 3: Efficiency ratio लिखें

    Efficiency ratio A : B : C = 5 : 4 : 3.

  4. Step 4: Total parts निकालें

    Total parts = 5 + 4 + 3 = 12 parts.

  5. Step 5: Payment per part

    ₹1080 ÷ 12 = ₹90.

  6. Step 6: Individual shares निकालें

    A = 5 × 90 = ₹450.
    B = 4 × 90 = ₹360.
    C = 3 × 90 = ₹270.

  7. Final Answer:

    A = ₹450, B = ₹360, C = ₹270

  8. Quick Check:

    Total = 450 + 360 + 270 = ₹1080 ✅; ratio 5 : 4 : 3 efficiency से match करता है।

Quick Variations

1. Workers को time के अनुसार भुगतान मिले (hourly wage) - तब payment time-proportional होगा।

2. कुछ workers बीच में join या leave करें - individual work (rate × time) से payment share तय करें।

3. एक worker को fixed amount पहले दिया जाए और बाकी को work ratio से - तब fixed part subtract करके बाकी distribute करें।

4. Wages में bonuses या deductions हों - base shares निकालने के बाद adjustments करें।

Trick to Always Use

  • Step 1 → Times को efficiencies (one-day work) में बदलें या दिया हुआ efficiency ratio लें।
  • Step 2 → Actual work done निकालें (efficiency × time worked); अगर सभी पूरा काम साथ करें तो efficiencies सीधे parts मान लें।
  • Step 3 → Payment share = Total pay × (individual work ÷ total work).

Summary

Summary

Work and Wages problems में:

  • Payment हमेशा work done पर आधारित होता है, जब तक कि explicitly “paid by time” न लिखा हो।
  • Work done = rate × time; efficiencies whole job के individual times के reciprocals होते हैं।
  • Ratios को simplify करने के लिए LCM का उपयोग करके smallest integer parts बनाएं।
  • Quick check: सभी shares मिलाकर total payment आना चाहिए और ratio match होना चाहिए।

Practice

(1/5)
1. A can complete a work in 10 days and B in 20 days. They together earn ₹1200 for completing the work. What is A’s share of wages?
easy
A. ₹800
B. ₹600
C. ₹900
D. ₹700

Solution

  1. Step 1: Compute individual efficiencies

    A’s one-day work = 1/10; B’s one-day work = 1/20.

  2. Step 2: Form work ratio

    Work ratio A : B = (1/10) : (1/20) = 2 : 1.

  3. Step 3: Convert parts into payment

    Total parts = 3 ⇒ value per part = 1200 ÷ 3 = ₹400.

  4. Step 4: Compute A’s share

    A’s share = 2 × 400 = ₹800.

  5. Final Answer:

    ₹800 → Option A.

  6. Quick Check:

    B gets 1 × 400 = ₹400; 800 + 400 = 1200 ✅
Hint: Wages are proportional to work done; use efficiencies (1/time) to get ratio.
Common Mistakes: Splitting money equally instead of proportional to efficiency.
2. A and B can complete a job in 15 and 10 days respectively. They earn ₹600 for the whole work. Find B’s share of wages.
easy
A. ₹240
B. ₹360
C. ₹400
D. ₹300

Solution

  1. Step 1: Compute efficiencies

    A = 1/15; B = 1/10.

  2. Step 2: Find work ratio

    Work ratio = (1/15):(1/10) = 2:3.

  3. Step 3: Convert to payment

    Total parts = 5 ⇒ per part = 600 ÷ 5 = ₹120.

  4. Step 4: Compute B’s share

    B gets 3 × 120 = ₹360.

  5. Final Answer:

    ₹360 → Option B.

  6. Quick Check:

    A=240; B=360; sum=600 ✅
Hint: Use reciprocal of time to form efficiency ratio, not the time ratio itself.
Common Mistakes: Using 15 : 10 directly instead of inverting.
3. A, B, and C can finish a work in 12, 18, and 24 days respectively. They earn ₹1170 for completing it. What is C’s share?
easy
A. ₹180
B. ₹360
C. ₹270
D. ₹300

Solution

  1. Step 1: Compute efficiencies

    A = 1/12, B = 1/18, C = 1/24.

  2. Step 2: Convert to integer parts

    LCM(12,18,24) = 72 ⇒ A=6, B=4, C=3 ⇒ ratio = 6:4:3.

  3. Step 3: Compute per-part value

    Total parts = 13 ⇒ per part = 1170 ÷ 13 = ₹90.

  4. Step 4: Calculate C’s share

    3 × 90 = ₹270.

  5. Final Answer:

    ₹270 → Option C.

  6. Quick Check:

    540 + 360 + 270 = 1170 ✅
Hint: Use LCM to simplify fractional efficiencies into whole number parts.
Common Mistakes: Directly using times instead of reciprocals.
4. A can complete a work in 15 days while B takes 10 days. A worked alone for 5 days and then B joined. Total payment for the job is ₹600. How much does A get?
medium
A. ₹200
B. ₹240
C. ₹300
D. ₹360

Solution

  1. Step 1: Compute efficiencies

    A=1/15; B=1/10.

  2. Step 2: Compute solo work

    A’s 5-day work = 1/3; remaining = 2/3.

  3. Step 3: Joint work

    Combined rate = 1/6 ⇒ time for remaining = 4 days.

  4. Step 4: Compute A’s total work

    A works 9 days ⇒ 9×(1/15)=3/5 of work.

  5. Step 5: Compute share

    A gets (3/5)×600 = ₹360.

  6. Final Answer:

    ₹360 → Option D.

  7. Quick Check:

    B’s work=2/5 ⇒ share=240; 360+240=600 ✅
Hint: Always convert contributions to fractions of work before multiplying by payment.
Common Mistakes: Splitting wages by number of days instead of actual work.
5. A and B can do a work in 12 and 18 days respectively. They get ₹540 for completing the work. If A works alone for the first 6 days and then both complete the rest together, what is A’s share?
medium
A. ₹432
B. ₹300
C. ₹320
D. ₹250

Solution

  1. Step 1: Compute efficiencies

    A=1/12; B=1/18.

  2. Step 2: Solo work

    A’s 6-day work = 1/2; remaining = 1/2.

  3. Step 3: Combined work

    Combined rate = 5/36 ⇒ time for remaining = 18/5 days.

  4. Step 4: Compute shares

    A’s work = 0.5 + 0.3 = 0.8 = 4/5 of job.

  5. Step 5: Multiply by payment

    A’s share = (4/5)×540 = ₹432.

  6. Final Answer:

    ₹432 → Option A.

  7. Quick Check:

    B’s work = 0.2 ⇒ share=108; 432+108=540 ✅
Hint: Work = rate × time; sum all contributions before distributing wages.
Common Mistakes: Ignoring B’s contribution duration while calculating shares.

Mock Test

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