Time & Work - All Formulas & Shortcuts

Quick reference guide for all Time & Work formulas with the powerful LCM method

📘 Basic Formulas

1. Fundamental Relationship

Work = Time × Efficiency OR Work Done = Number of Days × Work per Day

Key Concept:

If A can complete work in ’n’ days: Work done by A in 1 day = 1/n

If A does 1/n work per day: A completes whole work in ’n’ days

2. Work Done in Given Time

If A completes work in ‘a’ days:

Work done in 1 day = 1/a Work done in ‘x’ days = x/a Work remaining = 1 - x/a = (a-x)/a

Example: A completes work in 10 days

• In 1 day: 1/10 work
• In 3 days: 3/10 work
• Remaining: 7/10 work

3. Time Required to Complete Remaining Work

If x/y work is done:

Remaining work = 1 - x/y = (y-x)/y

Time to complete remaining = (Remaining Work) / (Work per day)

🎯 Working Together

4. Two People Working Together

If A completes in ‘a’ days, B in ‘b’ days:

Combined work per day:

1/a + 1/b = (a+b)/(ab)

Time to complete together:

T = ab/(a+b) days

Example: A = 10 days, B = 15 days

Together = (10×15)/(10+15) = 150/25 = 6 days

5. Three or More People Working Together

If A, B, C complete in a, b, c days:

Combined per day:

1/a + 1/b + 1/c

Time together:

T = 1/(1/a + 1/b + 1/c) = abc/(bc+ac+ab)

⚡ LCM Method (THE BEST METHOD!)

6. Why LCM Method?

Instead of fractions, work with whole numbers!

Steps:

1. Find LCM of all given days
2. LCM = Total work (in units)
3. Efficiency = Total work / Days
4. Apply: Work = Efficiency × Time

7. LCM Method Example

Problem: A completes in 10 days, B in 15 days. Together?

Traditional Method:

1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6 Time = 6 days

LCM Method:

LCM(10, 15) = 30 units (Total work)

Efficiency: A = 30/10 = 3 units/day B = 30/15 = 2 units/day Together = 3 + 2 = 5 units/day

Time = 30/5 = 6 days

Advantage: No fractions! Easy calculations!

💡 Advanced Formulas

8. Alternate Days Working

A works on Day 1, B on Day 2, A on Day 3, B on Day 4…

Work in 2 days = 1/a + 1/b

Number of 2-day cycles:

Cycles = ⌊Work/(1/a + 1/b)⌋

Then calculate remaining work separately

9. Work & Wages

When payment is proportional to work done:

Wage ratio = Work ratio = Efficiency ratio

If A:B efficiency = 3:2

Wage A:B = 3:2

If total wage = W:

A’s share = W × 3/5 B’s share = W × 2/5

10. Man-Days Concept

M₁ × D₁ = M₂ × D₂ (Men₁ × Days₁ = Men₂ × Days₂)

Example: 10 men complete in 6 days. How many days for 15 men?

10 × 6 = 15 × D D = 60/15 = 4 days

General Formula:

M₁ × D₁ × H₁ × E₁ = M₂ × D₂ × H₂ × E₂

Where:

• M = Men, D = Days, H = Hours/day, E = Efficiency

🔥 Pipes & Cisterns

11. Inlet & Outlet Pipes

Inlet pipe (fills tank):

If fills in ‘a’ hours: Part filled in 1 hour = 1/a

Outlet pipe (empties tank):

If empties in ‘b’ hours: Part emptied in 1 hour = -1/b (negative!)

12. Combined Inlet & Outlet

Inlet fills in ‘a’ hours, Outlet empties in ‘b’ hours:

Both open:

Net fill per hour = 1/a - 1/b = (b-a)/(ab)

Time to fill = ab/(b-a) hours

Note: This assumes b > a (outlet slower than inlet)

If a > b: Tank will never fill (outlet faster!)

13. Multiple Pipes

Two inlets (a, b hours) and one outlet (c hours):

All open:

Net = 1/a + 1/b - 1/c

Time = 1/(1/a + 1/b - 1/c)

14. Pipe Opens/Closes at Intervals

Use LCM method with time intervals!

Example: A fills in 4 hrs, B empties in 6 hrs. Both open alternately for 1 hr each.

Total = LCM(4,6) = 12 units

A’s efficiency = +3 units/hr B’s efficiency = -2 units/hr

In 2 hours: 3 - 2 = 1 unit For 12 units: 12 cycles × 2 hrs = 24 hrs

🎓 Complex Scenarios

15. Work Done by Group

M men or W women can do work in D days:

1 man’s 1 day work:

1/(M × D)

1 woman’s 1 day work:

1/(W × D)

Efficiency ratio (Man:Woman):

W:M

16. Work Done in Parts

A and B together for ‘x’ days, then A alone for ‘y’ days:

Total work done:

x(1/a + 1/b) + y(1/a)

Set equal to 1 (whole work) and solve

17. Efficiency Ratio Method

If A is x% more efficient than B:

Efficiency ratio A:B = (100+x):100

If B takes ‘b’ days: A takes = b × 100/(100+x) days

Example: A is 25% more efficient than B. B takes 20 days.

A takes = 20 × 100/125 = 16 days

📊 Quick Shortcuts

Shortcut 1: Same Work, Different People

If m₁ men take d₁ days, m₂ men take d₂ days:

m₁ × d₁ = m₂ × d₂

Shortcut 2: Twice as Efficient

If A is twice as efficient as B:

Time_A = Time_B / 2

Together:

Time = Time_A × 2/3 = Time_B × 1/3

Shortcut 3: Three People - Special Case

If A=B=C in efficiency:

Together they take T/3 days (where T = time for one person)

Shortcut 4: Negative Work (Destructive)

A builds, B destroys:

If A builds in ‘a’ days, B destroys in ‘b’ days:

Both working: (b-a)/ab days (if b>a)

If a>b: Building never completes!

💎 Common Patterns

Pattern 1: Started Together, One Leaves

A & B start, B leaves after x days:

Steps:

1. Work done in x days = x(1/a + 1/b)
2. Remaining = 1 - x(1/a + 1/b)
3. Time for A alone = Remaining/(1/a)
4. Total time = x + Time for A alone

Pattern 2: Work Stopped Due to Absence

A works all days, B works alternate days:

Use LCM method:

• Calculate for 2-day cycles
• Check remaining work

Pattern 3: Additional Workers Join

10 workers for 20 days. After 8 days, how many more needed to finish in total 12 days?

Work done = 10 × 8 = 80 man-days Remaining = 10 × 20 - 80 = 120 man-days Days left = 12 - 8 = 4 Workers needed = 120/4 = 30 Additional = 30 - 10 = 20

🎯 Mental Math Tricks

Trick 1: Quick Together Time

For small numbers:

• 6 & 12 together = 6×12/18 = 72/18 = 4 days
• 4 & 8 together = 4×8/12 = 32/12 = 2.67 days

Trick 2: LCM of Common Numbers

Memorize:

• LCM(4,6) = 12
• LCM(6,8) = 24
• LCM(10,15) = 30
• LCM(12,18) = 36

Trick 3: Efficiency Comparison

If A:B = 3:2

Time ratio = 2:3 (inverse!) If A takes 10 days, B takes 15 days

📝 Formula Summary Table

Scenario	Formula
A alone	1/a per day
A & B together	ab/(a+b) days
A, B, C together	abc/(bc+ac+ab) days
Man-days	M₁D₁ = M₂D₂
Inlet + Outlet	ab/(b-a) if b>a
Efficiency relation	Work = Efficiency × Time
Wage ratio	= Work ratio = Efficiency ratio

🔍 Common Mistakes

❌ Adding days directly (10 + 15 ≠ 25) ❌ Forgetting negative sign for outlet pipes ❌ Using time ratio instead of efficiency ratio ❌ Not converting to same units (days/hours) ✅ Always use LCM method for complex problems ✅ Remember: Efficiency and Time are inversely proportional

🔗 Related Resources

Practice Questions:

• Time & Work Practice Questions

Theory:

• Time & Work Theory & Concepts

Related Topics:

• Time, Speed & Distance
• Partnership
• Ratio & Proportion

Study Resources:

• All-in-One Formula Sheet

LCM Method is your secret weapon - master it! ⚡

With LCM method, solve Time & Work in under 60 seconds! 🚀