Time & Work · complete guide with shortcuts
SSC · Banking Railways · Campus | ✓ LCM Method included
Quick Note
Time and Work is one of the most scoring yet tricky topics in quantitative aptitude for competitive exams like SSC, Banking, Railways, and Campus Placements. This comprehensive guide covers everything you need to know—from basic concepts to advanced shortcut tricks, formulas, and solved examples.
▣ What is Time and Work?
Time and Work is a concept that deals with the relationship between the number of people working, the time they take, and the total work completed. The fundamental principle is that Work Done is directly proportional to the number of workers and the time they put in.
In simple terms: Time and Work go together—the more time you spend on a task, the more work you can do. However, keep in mind that the efficiency of the worker also plays a crucial role.
Work Done = Time Taken × Rate of Work
▣ Key Terms You Must Know
Understanding the terminology is half the battle won:
1. Time Taken
The duration required to complete a task. Time and the Rate of Work are inversely proportional. If someone takes less time, they work faster.
2. Work Done
The total amount of work. It is often treated as a single unit (1) or represented as the LCM of given days for easier calculations.
3. Rate of Work
The amount of work done per unit of time (usually per day or per hour). Rate of Work = 1/Time Taken
▣ Time and Work Key Formulas
Here's a quick reference to the essential formulas that will help you solve questions quickly:
1. Individual Work
If a person takes 'n' days to complete a work:
- Work done in 1 day = 1/n
- Efficiency = 1/n units per day
2. Combined Work (Two Workers)
If 'A' takes 'x' days and 'B' takes 'y' days to complete a job alone, then together they take:
T = (x × y) / (x + y)
Example:
If A takes 10 days and B takes 15 days:
Together = (10 × 15) / (10 + 15) = 150/25 = 6 days
3. General Formula (Men-Days-Hours)
If M₁ men can do W₁ work in D₁ days working H₁ hours per day, and M₂ men can do W₂ work in D₂ days working H₂ hours per day:
(M₁ × D₁ × H₁) / W₁ = (M₂ × D₂ × H₂) / W₂
4. Wages and Efficiency
- Wages are distributed in the ratio of Efficiency, which is also the same as the ratio of Work Done.
- If A and B get wages in a ratio Wₐ : Wᵦ, then Wₐ : Wᵦ = Efficiencyₐ : Efficiencyᵦ
▣ Shortcut Tricks to Ace Time and Work Problems
1. The LCM Method (The Ultimate Time-Saver)
The LCM method is widely considered the fastest way to solve Time and Work problems compared to fractions.
How to do it:
- Step 1: Assume the Total Work is the LCM of the number of days taken by the individuals.
- Step 2: Calculate the efficiency (units/day) for each person by dividing the total work by their individual time.
- Step 3: Add the efficiencies to find the combined time.
Example:
A can do a work in 10 days, and B can do it in 15 days. How long will they take together?
- Total Work: LCM(10, 15) = 30 units
- A's Efficiency: 30/10 = 3 units/day
- B's Efficiency: 30/15 = 2 units/day
- Combined Efficiency: 3 + 2 = 5 units/day
- Time together: 30/5 = 6 days
2. Efficiency Ratio Shortcut
If a problem states that A is twice as good as B:
- Ratio of Efficiency A : B = 2 : 1
- Ratio of Time taken A : B = 1 : 2 (Inverse relation)
Example:
If A is 3 times as efficient as B and takes 60 days less than B, find the time they take together.
- Efficiency Ratio A:B = 3:1 → Time Ratio A:B = 1:3
- Difference in time = 3 − 1 = 2 parts = 60 days
- A takes 30 days, B takes 90 days
- Total Work (LCM 30, 90) = 90 units
- Combined Efficiency = 90/30 + 90/90 = 3 + 1 = 4
- Together = 90/4 = 22.5 days
3. The "Negative Work" Trick (Pipes and Cisterns)
- Inlet Pipes do positive work (Filling)
- Outlet Pipes do negative work (Emptying)
Example:
A fills in 6 hours, B empties in 12 hours.
- LCM(6, 12) = 12 units (Tank capacity)
- A's rate = 12/6 = +2 units/hr
- B's rate = 12/12 = −1 units/hr (Subtract)
- Net Rate = 2 − 1 = 1 unit/hr
- Time to fill = 12/1 = 12 hours
4. Work and Wages Concept
Money is always divided according to the ratio of work done, NOT the time spent.
▣ Solved Examples from Competitive Exams
Let's put these tricks into practice with questions often seen in exams.
Q1. A can do a piece of work in 10 days, B in 15 days. They work together for 3 days. How much work is left?
Solution:
- Total Work (LCM 10, 15) = 30 units
- A's Rate = 3 units/day, B's Rate = 2 units/day
- Combined in 3 days = 3 × (3+2) = 15 units
- Remaining = 30 − 15 = 15 units
- Fraction left = 15/30 = 1/2
Q2. A and B together can do a work in 12 days. A alone can do it in 20 days. How long will B take alone?
Solution:
- Total Work (LCM 12, 20) = 60 units
- Combined Rate = 60/12 = 5 units/day
- A's Rate = 60/20 = 3 units/day
- B's Rate = 5 − 3 = 2 units/day
- B alone = 60/2 = 30 days
Q3. A is 60% more efficient than B. If A can finish a work in 10 days, how long will B take?
Solution:
- If A is 60% more efficient, A/B = 160/100 = 8/5
- Time Ratio (Inverse) A : B = 5 : 8
- A takes 5 parts = 10 days → 1 part = 2 days
- B takes 8 × 2 = 16 days
Q4. 8 men can finish a job in 12 days. After working 6 days, 4 men leave. How many more days will the remaining men take?
Solution:
- Total Work = 8 × 12 = 96 Man-days
- Work done in 6 days = 8 × 6 = 48 Man-days
- Remaining Work = 96 − 48 = 48 Man-days
- Remaining Men = 8 − 4 = 4
- Days needed = 48/4 = 12 days
▣ Types of Time and Work Questions You'll Face
1. Individual/Combined Efficiencies
Finding time if A, B, or C work alone or together.
2. Departure/Joining Problems
Someone leaves the job early, or someone joins late.
3. Pipes and Cisterns
Similar to Men/Women problems but with plus/minus signs for filling/draining.
4. Men-Days-Work Equivalence
Using the MDH formula for variable resources.
▣ Frequently Asked Questions
1. What is the LCM shortcut in Time and Work?
The LCM shortcut involves taking the LCM of the days taken by individuals as the total work. This helps in calculating efficiency as whole numbers rather than fractions, significantly speeding up the calculation.
2. How are wages distributed in Time and Work problems?
Wages are distributed in proportion to the work done or efficiency, not the time spent on the job. If A's efficiency is twice B's, A gets twice the money.
3. What is 'Negative Work'?
In Pipes and Cisterns problems, an outlet pipe that empties a tank does 'negative work'. Its rate is subtracted from the filling pipes to get the net fill rate.
🏁 Conclusion
To master Time and Work, remember:
- Use the LCM Method to avoid fraction confusion.
- Memorize the core formulas for rate, time, and the MDH formula.
- Practice analyzing the pattern—is it a combined work, departure, or efficiency problem?
Pro Tip:
Consistent practice with these shortcut tricks will help you solve Time and Work questions in under a minute, giving you a significant advantage in your competitive exams.
Good luck with your preparation!
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