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Work and Wages Problems

Introduction

Work and wages problems link the amount of work done by individuals or teams to the money they earn. These problems are important because they test your ability to divide payment fairly according to work contribution or time spent.

Typical applications include sharing a total payment among workers based on their efficiencies, times worked, or agreed ratios.

Pattern: Work and Wages Problems

Pattern

Key concept: Payment is distributed in the same ratio as the work done by each person (not their time), and work done = rate × time.

Steps to apply:

  1. Find each person's one-day work (efficiency) or use given ratio of efficiencies.
  2. Compute actual work done by each: (efficiency × time worked) or use proportional parts.
  3. Total payment × (individual work ÷ total work) = individual share.

Step-by-Step Example

Question

Three workers A, B and C can complete a job together. A alone takes 12 days, B alone takes 15 days and C alone takes 20 days. They finish the job together and receive ₹1080. How much does each worker get?

Solution

  1. Step 1: Compute one-day work (efficiency)

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

  2. Step 2: Convert to a common denominator

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

  3. Step 3: Write efficiency ratio

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

  4. Step 4: Find total parts

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

  5. Step 5: Compute payment per part

    Payment per part = ₹1080 ÷ 12 = ₹90.

  6. Step 6: Compute individual shares

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

  7. Final Answer:

    A gets ₹450, B gets ₹360, C gets ₹270

  8. Quick Check:

    Sum = 450 + 360 + 270 = ₹1080 ✅; ratio 5 : 4 : 3 consistent with efficiencies.

Quick Variations

1. Workers are paid based on time (if they are paid hourly rather than by work) - then distribute proportionally to time worked.

2. Some workers join later or leave early - compute individual work done (rate × time) and share payment accordingly.

3. One worker is paid a fixed amount first, remaining shared among others by work done - subtract fixed part then distribute remainder by work ratio.

4. Problems where wages include bonuses or deductions - apply wage adjustments after computing base shares.

Trick to Always Use

  • Step 1 → Convert times to efficiencies (one-day work) or use given efficiency ratio directly.
  • Step 2 → Compute actual work done by each (efficiency × time worked); if all work whole job together, use efficiencies as parts.
  • Step 3 → Payment share = Total pay × (individual work ÷ total work).

Summary

Summary

For Work and Wages problems:

  • Always base payment on work done unless the problem explicitly says "paid by time".
  • Work done = rate × time; efficiencies are reciprocals of individual times for the whole job.
  • Reduce ratios to smallest integer parts using LCM to make division simple.
  • Quick sanity check: individual shares must sum to total payment and preserve the computed ratio.

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.

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