Time & Work - Theory & Concepts

⏰ Time & Work - Complete Theory

Master efficiency, man-days, and work calculations!

🎯 Basic Concept

Work = Time × Efficiency

Key Idea: If A can complete work in ’n’ days:

Work done in 1 day = 1/n
Total work = 1 (complete job)

📐 Basic Formulas

Formula 1: One Day Work

If A completes work in 'a' days:
A's 1 day work = 1/a

Formula 2: Together Work

If A takes 'a' days and B takes 'b' days:
Together in 1 day = 1/a + 1/b
Time together = 1/(1/a + 1/b) = ab/(a+b) days

Example: A in 10 days, B in 15 days

Together = 1/10 + 1/15 = 5/30 = 1/6
Time = 6 days

Formula 3: Efficiency

Efficiency ∝ 1/Time

If Time ratio = m:n
Then Efficiency ratio = n:m (inverse!)

⚡ LCM Method (Best for IBPS!)

Steps:

Find LCM of all given days
LCM = Total Work (in units)
Efficiency = Total Work / Days
Calculate based on question

Example: A in 12 days, B in 15 days, find time together.

Solution:

LCM(12,15) = 60 units (Total Work)

A's efficiency = 60/12 = 5 units/day
B's efficiency = 60/15 = 4 units/day

Together = 5 + 4 = 9 units/day
Time = 60/9 = 6.67 days

💡 Solved Examples

Example 1: Basic Together Work

Q: A can do work in 20 days, B in 30 days. How long together?

LCM Method:

LCM(20,30) = 60 units

A = 60/20 = 3 units/day
B = 60/30 = 2 units/day
Together = 5 units/day

Time = 60/5 = 12 days

Example 2: Work with Different Efficiencies

Q: A is twice as efficient as B. A completes work in 12 days. How long for B?

Solution:

Efficiency ratio A:B = 2:1
Time ratio A:B = 1:2 (inverse!)

If A takes 12 days:
B takes 12 × 2 = 24 days

Example 3: Alternate Days

Q: A alone in 10 days, B alone in 15 days. They work on alternate days starting with A. Find time.

LCM Method:

LCM = 30 units

A = 3 units/day
B = 2 units/day

In 2 days (one cycle):
Work done = 3 + 2 = 5 units

30/5 = 6 cycles = 12 days

Example 4: Work Left After Some Days

Q: A in 15 days, B in 20 days. A works for 5 days then leaves. B finishes remaining work in how many days?

Solution:

LCM = 60 units

A = 4 units/day
B = 3 units/day

A works 5 days = 4 × 5 = 20 units
Remaining = 60 - 20 = 40 units

B finishes 40 units in = 40/3 = 13.33 days

🔄 Important Variations

1. Pipes & Cisterns

Same as Time & Work!

Filling pipe = +ve work
Emptying pipe = -ve work

Net work = Filling rate - Emptying rate

Example: Pipe A fills in 6 hours, Pipe B empties in 8 hours. Both open, time to fill?

LCM = 24 units

A fills = +4 units/hour
B empties = -3 units/hour
Net = +1 unit/hour

Time = 24/1 = 24 hours

2. Men-Days Formula

M₁ × D₁ × H₁ × E₁ / W₁ = M₂ × D₂ × H₂ × E₂ / W₂

Where:
M = Number of men
D = Days
H = Hours per day
E = Efficiency
W = Work done

Shortcut: If all parameters same except two, they are inversely proportional.

3. Women & Children

If 1 man = x women = y children:
Convert all to one unit, then solve

Example: 1 man = 2 women = 3 children

Men : Women : Children = 1 : 1/2 : 1/3
                       = 6 : 3 : 2 (multiply by LCM)

⚡ Quick Shortcuts

Shortcut 1: Same Efficiency

If n people of same efficiency, each taking 't' days:
Together time = t/n days

Shortcut 2: Efficiency Ratio

Time ratio m:n
Efficiency ratio = n:m (just flip!)

Shortcut 3: Half Work

If A and B together complete work in 't' days:
Half work in t/2 days

Shortcut 4: Negative Work

If A does +ve work and B does -ve work:
Net = |A - B|

📊 Common Patterns

Pattern 1: One Leaves Midway

1. Calculate work done by both together for given time
2. Find remaining work
3. Calculate time for remaining person

Pattern 2: Wages Distribution

Wages are distributed in ratio of work done
Work done ∝ Efficiency × Days worked

Pattern 3: Additional Workers Join

1. Calculate work done before joining
2. Calculate combined efficiency after joining
3. Find time for remaining work

⚠️ Common Mistakes

❌ Mistake 1: Adding Times Directly

Wrong: A in 10 days + B in 15 days = 25 days together ✗
Right: Use 1/10 + 1/15 = 1/6, so 6 days ✓

❌ Mistake 2: Efficiency = Time

Wrong: If A takes 2× time, A is 2× efficient ✗
Right: If A takes 2× time, A is 1/2 efficient ✓

❌ Mistake 3: Negative Work

In pipes: Outlet is negative, not positive!

📝 Practice Problems

Level 1:

A in 8 days, B in 12 days. Time together?
A is thrice as efficient as B. B takes 24 days. Time for A?
6 men complete work in 10 days. How many men for 5 days?

Level 2:

A in 15 days, B in 20 days. A works 5 days, then both work together. Total time?
Pipe A fills in 4 hours, B empties in 6 hours. Both open, time to fill?
A and B together in 12 days. A alone in 20 days. Time for B alone?

Level 3:

A, B, C together in 10 days. A alone in 30 days, B in 40 days. Time for C?
12 men or 18 women complete work in 20 days. 8 men and 12 women together?
A and B start together. After 4 days, A leaves and B finishes in 10 more days. If A alone, how long?

Prerequisites: