Time and Work

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Time and Work — Key Facts for NABE (Pakistan)

• Work = Time × Rate | If A can do work in T days, A’s rate = 1/T work/day
• Combined Work: If A takes a days and B takes b days, together they take (ab)/(a+b) days
• Man-Days Concept: Work required = Number of workers × Number of days
• Pipes and Cisterns: Inlet pipes fill (positive work), Outlet pipes empty (negative work)
• ⚡ Exam tip: If A is twice as efficient as B, A’s time is half of B’s time

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Time and Work — NABE (Pakistan) Study Guide

Basic Concept

Work is measured in units (usually “1 job” or “whole work”).

Rate of Work: The fraction of the job completed per unit time.

Formulas:

Work = Rate × Time
Rate = Work / Time = 1/Time (for complete job)
Time = Work / Rate

Individual Work

If A can complete a job in n days:

• A’s 1 day’s work = 1/n of the job
• A’s work in x days = x/n of the job

Example: If Asad can complete a task in 5 days:

• Asad’s rate = 1/5 work per day
• In 3 days: 3 × (1/5) = 3/5 of the work

Two-Person Work

If A can do job in ‘a’ days and B can do it in ‘b’ days:

Together (A + B) time:

(A + B)'s 1 day work = 1/a + 1/b = (a+b)/ab
Time = (ab)/(a+b) days

Example: Asad can do job in 6 days, Babar in 8 days. How long together?

• Together: (6 × 8)/(6 + 8) = 48/14 = 24/7 ≈ 3.43 days

Three-Person Work

Formula for A, B, C with times a, b, c respectively:

(A+B+C)'s 1 day work = 1/a + 1/b + 1/c
Time = 1 / (1/a + 1/b + 1/c)

Work Equivalence

If P₁ persons can do W₁ work in D₁ days working t₁ hours per day And P₂ persons can do W₂ work in D₂ days working t₂ hours per day

Then: P₁ × D₁ × t₁ × W₂ = P₂ × D₂ × t₂ × W₁

NABE Exam Pattern

Common question types:

1. Individual work time calculations
2. Combined work time (two or more persons)
3. Pipes and cisterns problems
4. Men, women, and children work comparisons
5. Work equivalence problems

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Time and Work — Comprehensive NABE (Pakistan) Notes

Detailed Theory

1. Fundamental Principles

The Unit Work Concept:

• Assume total work = 1 unit (or LCM of time periods)
• All rates are expressed as fractions of this unit
• This approach simplifies calculations

Efficiency and Time Relationship:

• Efficiency is inversely proportional to time
• If A is twice as efficient as B:
  • A takes half the time of B
  • A’s rate is twice B’s rate

Proportionality:

• Work ∝ Number of workers (when time is constant)
• Work ∝ Time (when workers are constant)
• Workers × Time × Efficiency = Constant Work

2. Individual Work Problems

Basic Approach:

1. Find the rate of work per day
2. Multiply by number of days to find fraction completed
3. Set up equation for the required condition

Example Problem:

• Asad can complete work in 20 days
• He works for 5 days, then Babar completes remaining in 12 days
• Find total time

Solution:

• Asad’s 5 days work = 5/20 = 1/4
• Remaining work = 3/4
• Babar’s rate = (3/4)/12 = 1/16
• Babar needs 16 days to complete alone (which matches)

3. Combined Work — Complete Derivation

For two workers:

• A’s 1 day work = 1/a
• B’s 1 day work = 1/b
• Together = 1/a + 1/b = (a+b)/ab
• Time = ab/(a+b) days

Proof: Let total work = 1 unit Time taken together = T days A’s work in T days = T/a B’s work in T days = T/b T/a + T/b = 1 T(1/a + 1/b) = 1 T = 1/(1/a + 1/b) = ab/(a+b)

Example: A and B together can complete in 8 days what A alone does in 24 days and B alone in 16 days. Verify.

• 1/24 + 1/16 = (2+3)/48 = 5/48
• Time = 48/5 = 9.6 days (not 8!)

4. Three Workers — Extended Formula

For three workers:

Time = 1 / (1/a + 1/b + 1/c) = abc/(ab + bc + ca)

Proof:

• Let total work = 1
• (A+B+C)‘s 1 day work = 1/a + 1/b + 1/c
• Time = 1/(1/a + 1/b + 1/c)
• = abc/(bc + ac + ab)

5. Pipes and Cisterns

Types of Pipes:

• Inlet Pipe: Fills the tank (positive contribution)
• Outlet Pipe: Empties the tank (negative contribution)
• Leak: Acts like outlet pipe (negative contribution)

Net Rate:

Net 1 hour work = Sum of inlet rates - Sum of outlet rates

Example: Tank capacity = 1. Inlet fills in 6 hrs, outlet empties in 8 hrs. Both open?

• Net = 1/6 - 1/8 = (4-3)/24 = 1/24
• Time to fill = 24 hours (tank actually fills despite outlet!)

With Leak: Example: A fill pipe fills in 10 hrs. There’s a leak that empties full tank in 20 hrs. How long to fill empty tank?

• Fill rate = 1/10
• Leak rate = 1/20 (emptying)
• Net = 1/10 - 1/20 = 1/20
• Time = 20 hours

6. Men, Women, and Children Work

Concept: Different workers have different efficiencies.

Standard Approach:

• Express all work in “man-days” or “child-days”
• Find ratio of efficiencies

Example:

• 4 men = 6 women = 9 children (work equivalence)
• Therefore: 1 man : 1 woman : 1 child = 9 : 6 : 4

Problem: If 2 men, 4 women, 6 children work together for 5 days and complete 1/4 of work. Find time for 1 man, 1 woman, 1 child to complete remaining.

• Combined efficiency = 2×9 + 4×6 + 6×4 = 18+24+24 = 66 child-days per day
• Work done = 66×5 = 330 child-days
• Total work = 330 × 4 = 1320 child-days
• Remaining = 1320 × 3/4 = 990 child-days
• New group: 1 man+1 woman+1 child = 9+6+4 = 19 child-days/day
• Time = 990/19 ≈ 52 days

7. Work and Wages

Principle: Work done ∝ wages earned (when rates are equal)

Example: A and B work for 3:5 days and complete a job. If total wages = Rs. 640, divide.

• Work ratio = Time ratio (if same efficiency)
• A:B = 3:5
• A’s share = 640 × 3/8 = Rs. 240
• B’s share = 640 × 5/8 = Rs. 400

8. Chain Rule in Work

Principle: More workers → Less time (inverse proportion)

Problem Types:

1. Workers reduced → Time increases
2. Workers increased → Time decreases
3. Work increased → Time increases

Formula:

P₁ × D₁ × W₂ = P₂ × D₂ × W₁

9. Common Mistakes to Avoid

1. Efficiency Confusion: More efficient worker takes less time
2. Sign Errors in Pipes: Inlet positive, outlet/leak negative
3. Combining Wrong Rates: Use reciprocals correctly
4. Ignoring Partial Work: Track fraction completed

Practice Questions for NABE

1. Asad can complete a work in 15 days and Babar in 20 days. If they work together, how many days will they take?
2. A pipe fills a tank in 10 hours but there’s a leak that empties it in 20 hours. How long to fill the tank with leak?
3. 12 men can complete a work in 18 days. After 6 days, 6 more men join. How many additional days are needed?
4. If 4 women can do a work in 16 days and 6 children can do the same in 24 days, how long will 2 women and 3 children take together?
5. A can do a work in 10 days. He works for 3 days, then B finishes remaining in 14 days. How long would B take alone?

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