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Time & Work - Aptitude Mastery Guide

Section titled “Time & Work - Aptitude Mastery Guide”

Category: Quantitative Aptitude
Generated on: 2025-07-15 09:18:42
Source: Aptitude Mastery Guide Generator

Time & Work: A Comprehensive Guide for Competitive Exams

Section titled “Time & Work: A Comprehensive Guide for Competitive Exams”

This guide provides a comprehensive overview of the “Time & Work” topic, crucial for success in various competitive exams and placement tests. We’ll cover foundational concepts, essential formulas, powerful shortcuts, and a range of solved and practice problems, designed to build your confidence and speed.

1. Foundational Concepts

Section titled “1. Foundational Concepts”

The core concept of Time & Work revolves around the relationship between work done, time taken, and efficiency.

  • Work Done: The amount of task completed. Usually represented as a fraction or a unit.
  • Time Taken: The duration required to complete the work.
  • Efficiency: The rate at which work is done. It’s the amount of work done per unit of time.

The fundamental relationship:

  • Work Done = Efficiency x Time Taken

Understanding Efficiency:

Efficiency is inversely proportional to time. This means:

  • If A is twice as efficient as B, A will take half the time to complete the same work as B.
  • A more efficient person/machine completes more work in the same amount of time.

The ‘Why’ Behind the Formulas:

Imagine you’re building a wall. The work done is constructing the entire wall. Your efficiency is how many bricks you lay per hour. The time taken is the total number of hours you spend laying bricks. If you lay more bricks per hour (increase efficiency), you’ll finish the wall faster (decrease time taken). This simple analogy explains the core relationship.

Units:

  • Always ensure consistent units. If time is in days, efficiency should be in work/day. If time is in hours, efficiency should be in work/hour.

2. Key Tricks & Shortcuts

Section titled “2. Key Tricks & Shortcuts”

This section contains the most valuable shortcuts for solving Time & Work problems quickly.

  • Trick 1: LCM (Least Common Multiple) Method (For Efficiency Comparison):

    • When to Use: When the time taken by individuals to complete a task is given, and you need to find the time taken if they work together.
    • How it Works:
      1. Find the LCM of the time taken by each individual. This LCM represents the total work (in units).
      2. Calculate the efficiency of each individual by dividing the total work by their individual time.
      3. Add the efficiencies of all individuals to find their combined efficiency.
      4. Divide the total work by the combined efficiency to find the time taken when they work together.
    • Example: A can do a piece of work in 10 days, and B can do it in 15 days. How long will they take to complete it together?
      1. LCM(10, 15) = 30 (Total work = 30 units)
      2. A’s efficiency = 30/10 = 3 units/day
      3. B’s efficiency = 30/15 = 2 units/day
      4. Combined efficiency = 3 + 2 = 5 units/day
      5. Time taken together = 30/5 = 6 days

  • Trick 2: Work Equivalence Method (For ‘Or’ Problems):

    • When to Use: When the problem states “X men OR Y women can do a piece of work in Z days”. This indicates an equivalence between the work done by X men and Y women.
    • How it Works:
      1. Establish the equivalence: X men = Y women (This means X men do the same work as Y women).
      2. Convert everything to a common unit (either men or women).
      3. Use the formula: M1 * D1 = M2 * D2 (where M = number of men/women, D = number of days).
    • Example: 10 men or 20 women can do a work in 15 days. How many days will 5 men and 10 women take to complete the work?
      1. 10 men = 20 women => 1 man = 2 women
      2. 5 men + 10 women = 5 men + (10/2) men = 10 men
      3. M1 * D1 = M2 * D2 => 10 men * 15 days = 10 men * D2 => D2 = 15 days

  • Trick 3: Chain Rule (For Combined Work):

    • When to Use: When dealing with multiple workers, days, and hours.
    • How it Works: The basic formula is (M1 * D1 * H1) / W1 = (M2 * D2 * H2) / W2
      • M = Men/Workers
      • D = Days
      • H = Hours per day
      • W = Work done
    • Example: If 12 men working 8 hours a day can complete a work in 10 days, how many days will 16 men working 6 hours a day take to complete the same work?
      1. (12 * 10 * 8) / 1 = (16 * D2 * 6) / 1
      2. D2 = (12 * 10 * 8) / (16 * 6) = 10 days

  • Trick 4: Negative Work (For Pipes and Cisterns):

    • When to Use: In problems involving pipes filling a tank (positive work) and pipes emptying the tank (negative work).
    • How it Works: Treat the emptying pipes as performing “negative work.” Calculate the net work done per unit of time by adding the rates of filling pipes and subtracting the rates of emptying pipes.
    • Example: Pipe A fills a tank in 10 hours, and Pipe B empties it in 15 hours. How long will it take to fill the tank if both are opened simultaneously?
      1. A’s filling rate = 1/10 tank/hour
      2. B’s emptying rate = -1/15 tank/hour
      3. Net filling rate = 1/10 - 1/15 = 1/30 tank/hour
      4. Time to fill the tank = 1 / (1/30) = 30 hours

  • Trick 5: Percentage to Fraction Conversion:

    • When to Use: When the problem involves percentages related to efficiency or work done.
    • How it Works: Convert the percentage directly to a fraction to simplify calculations. Remember common conversions:
      • 25% = 1/4
      • 50% = 1/2
      • 75% = 3/4
      • 20% = 1/5
      • 33.33% = 1/3
    • Example: A is 25% more efficient than B. If B can complete a work in 20 days, how long will A take?
      1. A is 25% = 1/4 more efficient than B. This means A’s efficiency is 1 + 1/4 = 5/4 times B’s efficiency.
      2. A will take 4/5 of the time B takes.
      3. A’s time = (4/5) * 20 = 16 days

  • Trick 6: Vedic Maths - Duplex Method (For Squaring Numbers):

    • When to Use: Squaring numbers can be helpful when calculating areas or volumes in work-related problems.
    • How it Works: The Duplex method is a fast way to square numbers.
      • For a 2-digit number AB: Duplex(AB) = A2 + 2AB + B2
      • For a 3-digit number ABC: Duplex(ABC) = A2 + 2AB + B2 + 2BC + C2
    • Example: Calculate 232
      • Duplex(23) = 22 + 2 * 2 * 3 + 32 = 4 + 12 + 9 = 529

  • Trick 7: Assuming a Value (For Complex Ratios):

    • When to Use: When the problem involves complex ratios or fractions, assume a convenient value for the total work to simplify calculations.
    • How it Works: Choose a value that is easily divisible by all the denominators in the problem. This will make the calculations easier.
    • Example: A does 1/3 of the work in 5 days, and B does 2/5 of the work in 10 days. How long will they take to complete the work together?
      1. Assume total work = LCM(3, 5) = 15 units.
      2. A does 1/3 * 15 = 5 units in 5 days. A’s efficiency = 1 unit/day.
      3. B does 2/5 * 15 = 6 units in 10 days. B’s efficiency = 0.6 units/day.
      4. Combined efficiency = 1 + 0.6 = 1.6 units/day.
      5. Time taken together = 15 / 1.6 = 9.375 days.

3. Essential Formulas & Rules

Section titled “3. Essential Formulas & Rules”
Formula/RuleDescription
Work Done = Efficiency x Time TakenThe fundamental relationship between work, efficiency, and time.
Efficiency ∝ 1/TimeEfficiency is inversely proportional to time.
M1 * D1 * H1 / W1 = M2 * D2 * H2 / W2Chain Rule for combined work (M = Men/Workers, D = Days, H = Hours, W = Work)
A's 1 day's work + B's 1 day's work = (A+B)'s 1 day's workIf A and B work together, their combined daily work equals the sum of their individual daily work.
If A can do a work in ‘x’ days, then A’s 1 day’s work is 1/xCalculating daily work rate.
If A is ‘x’ times as good a workman as B, then the ratio of work done by A and B is x:1Comparing the efficiency of two individuals.

4. Detailed Solved Examples

Section titled “4. Detailed Solved Examples”

Example 1: Basic LCM Method

A can complete a piece of work in 20 days and B can complete the same work in 30 days. How long will they take to complete the work together?

Solution:

  1. Identify the method: LCM Method
  2. Find the LCM: LCM(20, 30) = 60. Let the total work be 60 units.
  3. Calculate Efficiencies:
    • A’s efficiency = 60/20 = 3 units/day
    • B’s efficiency = 60/30 = 2 units/day
  4. Combined Efficiency: 3 + 2 = 5 units/day
  5. Time Taken Together: 60/5 = 12 days.

Example 2: Work Equivalence (‘Or’ Problem)

5 men or 8 women can do a piece of work in 12 days. How many days will 2 men and 4 women take to complete the same work?

Solution:

  1. Identify the method: Work Equivalence Method
  2. Establish Equivalence: 5 men = 8 women => 1 man = 8/5 women
  3. Convert to a Common Unit (Women): 2 men + 4 women = 2 * (8/5) women + 4 women = 16/5 women + 20/5 women = 36/5 women
  4. Use M1 * D1 = M2 * D2 (in terms of women): 8 women * 12 days = (36/5) women * D2
  5. Solve for D2: D2 = (8 * 12 * 5) / 36 = 40/3 = 13.33 days (approximately)

Example 3: Chain Rule (Combined Work)

If 15 men working 6 hours a day can complete a work in 20 days, how many days will 12 men working 8 hours a day take to complete the same work?

Solution:

  1. Identify the method: Chain Rule
  2. Apply the formula: (M1 * D1 * H1) / W1 = (M2 * D2 * H2) / W2. Since the work is the same, W1 = W2.
  3. Substitute the values: (15 * 20 * 6) = (12 * D2 * 8)
  4. Solve for D2: D2 = (15 * 20 * 6) / (12 * 8) = 18.75 days

Example 4: Pipes and Cisterns (Negative Work)

Pipe A can fill a tank in 15 hours, and Pipe B can empty the same tank in 20 hours. If both pipes are opened simultaneously, how long will it take to fill the tank?

Solution:

  1. Identify the Method: Negative Work
  2. Calculate Individual Rates:
    • A’s filling rate = 1/15 tank/hour
    • B’s emptying rate = -1/20 tank/hour
  3. Calculate Net Filling Rate: 1/15 - 1/20 = (4 - 3) / 60 = 1/60 tank/hour
  4. Time to Fill the Tank: 1 / (1/60) = 60 hours

5. Practice Problems (Graded Difficulty)

Section titled “5. Practice Problems (Graded Difficulty)”

[Easy]

A can do a piece of work in 10 days, and B can do it in 12 days. How many days will they take to complete the work if they work together?

[Easy]

A can do a piece of work in 15 days. B is 50% more efficient than A. How long will B take to do the same work?

[Medium]

10 men can complete a piece of work in 12 days. How many days will 15 men take to complete the same work?

[Medium]

A and B can together complete a work in 8 days. A alone can complete it in 12 days. How many days will B alone take to complete the work?

[Hard]

A, B, and C can complete a piece of work in 10, 12, and 15 days respectively. If they work together, how long will they take to complete the work?

[Hard]

12 men or 18 women can do a piece of work in 14 days. How long will 8 men and 16 women take to complete the same work?

[Hard]

Pipe A can fill a tank in 20 hours, Pipe B can fill the same tank in 30 hours, and Pipe C can empty the tank in 40 hours. If all three pipes are opened simultaneously, how long will it take to fill the tank?

6. Advanced/Case-Based Question

Section titled “6. Advanced/Case-Based Question”

A contractor undertakes a project to build a road in 60 days. He employs 100 workers initially. After 20 days, he finds that only 1/4 of the road has been built. He then decides to increase the number of workers by ‘x’ to complete the work on time.

  1. Find the value of ‘x’.
  2. If the contractor had not increased the number of workers, how many days beyond the initial deadline would it have taken to complete the road?
  3. If each worker is paid Rs. 500 per day, what is the total cost incurred by the contractor for the project?