Math · Aptitude · Chapter 04

⏱️ Time & Work

Workers, days, efficiency — solve any work-rate problem.

💡 Work-Rate Method

If a person can finish a job in n days, they do 1/n of the work in 1 day. This is the foundation of all time-work problems.

Single worker rate
A finishes in n days → 1 day's work = 1/n
Combined rate
A+B per day = 1/a + 1/b
Days together
Together = (a × b)/(a + b)
Inverse relation
N₁D₁ = N₂D₂ (men × days = const)

Example: A can finish a job in 12 days, B in 18 days. Together?
1 day work: A = 1/12, B = 1/18.
Together = 1/12 + 1/18 = 3/36 + 2/36 = 5/36.
Time = 36/5 = 7.2 days.
Shortcut: (12 × 18)/(12 + 18) = 216/30 = 7.2 ✓

⚡ LCM method (faster for harder problems)

Instead of fractions, assume total work = LCM of all days. Then convert.

A: 12 days, B: 18 days. LCM(12,18) = 36 units. A does 36/12 = 3 units/day. B does 36/18 = 2 units/day. Together = 5 units/day. Total work 36 ÷ 5 = 7.2 days. No fractions!

⚡ Pipes and cisterns (same logic)

Filling pipe = positive rate. Leaking/emptying pipe = negative rate. Add them all.

Pipe A fills in 6 hr, pipe B fills in 8 hr, leak C empties in 24 hr. Combined rate = 1/6 + 1/8 − 1/24 = 4/24 + 3/24 − 1/24 = 6/24 = 1/4. Tank fills in 4 hours.

⚡ Men × Days × Hours = Work

For complex problems with men, days, and hours:

(M₁ × D₁ × H₁) / W₁ = (M₂ × D₂ × H₂) / W₂

Use this whenever workers, days, hours, or amount of work changes.

🎬

Workers Filling the Bar Together

Animation
A WORKS AT 1/12 PER DAY · B WORKS AT 1/18 PER DAY 👷 Worker A — 1/12 per day 0% 👷 Worker B — 1/18 per day 0% 🤝 A + B together — 5/36 per day 0% PROGRESS Day 0.0 A finishes ≈ 12 days · B finishes ≈ 18 days · Together: 7.2 days

Two bars filling simultaneously. The "together" bar fills the fastest — that's the power of teamwork.

🧮

Time & Work Calculators

Calculator
🧑‍🤝‍🧑 Two workers together

A alone: days · B alone: days

Together: 7.2 days

Shortcut: (12 × 18)/(12 + 18) = 216/30 = 7.2 days.

🚿 Pipes & cisterns

Pipe A: hr (fill) · Pipe B: hr (fill) · Leak C: hr (empty)

Tank fills in: 4 hours

⚖️ Men × Days = Work

If men finish in days, then men finish in: 8 days

Practice (SSC): A and B together can do a piece of work in 12 days. A alone can do it in 20 days. In how many days can B alone do the work?
A+B's 1-day work = 1/12. A's 1-day work = 1/20. B's 1-day work = 1/12 − 1/20 = 5/60 − 3/60 = 2/60 = 1/30. So B alone = 30 days.
Practice (Banking): 20 men can finish a work in 30 days working 8 hr/day. How many men needed to finish the same in 20 days working 10 hr/day?
Use (M₁ × D₁ × H₁) = (M₂ × D₂ × H₂). 20 × 30 × 8 = M₂ × 20 × 10. M₂ = (20 × 30 × 8)/(20 × 10) = 4800/200 = 24 men.
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