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Quantitative AptitudeBeginner5 min study

How to Solve Time and Work Questions: LCM Method, Formulas & Solved Examples

कार्य और समय (Time & Work) के प्रश्न कैसे हल करें: LCM विधि, सूत्र और ट्रिक्स

Time and Work (कार्य और समय) is one of the highest-weightage arithmetic topics in MP Police Constable, MP Sub-Inspector, MP Patwari, SSC, and Railway exams. The fundamental principle is an inverse relationship: if a person works faster (higher efficiency / कार्य क्षमता), they take fewer days to complete the job. Instead of tedious fraction addition, top scorers use the LCM Method, assuming the total work is the LCM of the days given, turning fraction problems into rapid whole-number addition.

Key Rules & Core Formulas

Understand these foundational properties before solving questions

1Total Work & Efficiency Formula

Total Work = Efficiency × Time | Efficiency = Total Work / Time

Total work done is the product of a person's rate of work per day (efficiency) and the total number of days worked. Efficiency measures the units of work done per unit time.

Efficiency and time are inversely proportional when total work is constant: Efficiency ∝ 1 / Time. If A is twice as fast as B, A takes half the time.

2Two Persons Working Together (Shortcut Formula)

Combined Time (A + B) = (A × B) / (A + B) days

When person A completes a work alone in A days and person B completes it in B days, their combined completion time is the product of their days divided by the sum of their days.

For three persons A, B, and C, the combined formula is (A × B × C) / (AB + BC + CA) days, but the LCM method is much faster.

3The LCM Method (Competitive Exam Gold Standard)

Total Work (in units) = LCM(Individual completion days)

Assume the total work is equal to the LCM of the individual completion days. Dividing the total units by each person's days directly yields their daily output in integer units.

Eliminates all cumbersome fraction additions like 1/12 + 1/18, converting calculations into rapid mental arithmetic.

4Efficiency Ratio Property

Time Ratio = 1 / Efficiency Ratio ⇒ T_A : T_B = E_B : E_A

The ratio of time taken by two workers is the exact inverse of their daily efficiency ratio.

If A is 60% more efficient than B, Efficiency A : B = 160 : 100 = 8 : 5, which means Time A : B = 5 : 8.

5The MDH Formula for Group Work / Chain Rule

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

Relates Men (M), Days (D), Hours per day (H), and Work completed or wages earned (W) across two different working scenarios.

If the work is the same (W₁ = W₂), the formula simplifies to M₁ × D₁ × H₁ = M₂ × D₂ × H₂.

6Work and Wages Distribution Rule

Wages Share Ratio = Ratio of Work Done (or Efficiency Ratio if working for same days)

Compensation is distributed strictly in proportion to the actual units of work contributed by each individual worker.

If all workers work together from start to finish for the exact same number of days, wages can be divided directly in the ratio of their efficiencies.
Standard Method

Step-by-Step Method to Solve

Follow these universal steps in order to solve any Quantitative Aptitude question accurately:

  1. 1

    Step 1: Find Total Work using the LCM of given days

    Identify the individual completion times given in the problem and calculate their Least Common Multiple (LCM). Treat this LCM value as the total work measured in 'units'.

    Pro-Tip: For example, if A takes 12 days and B takes 18 days, let Total Work = LCM(12, 18) = 36 units.
  2. 2

    Step 2: Calculate daily efficiency (1-day work) for each worker

    Divide the total units of work by each worker's individual time in days. This gives each worker's daily production rate (units/day).

    Pro-Tip: Efficiency of A = 36 / 12 = 3 units/day; Efficiency of B = 36 / 18 = 2 units/day.
  3. 3

    Step 3: Combine or adjust efficiencies according to the problem conditions

    For collaborative work, add the efficiencies together (A + B = 3 + 2 = 5 units/day). For alternate days, calculate work completed in a 2-day cycle. For leaving or joining problems, break the schedule into distinct time segments.

    Pro-Tip: Keep units/day clearly written on your scratch paper to prevent silly mistakes under exam pressure.
  4. 4

    Step 4: Compute work done in phases and find remaining units

    Multiply the daily efficiency by days worked in each phase. Subtract completed units from total units to determine the Remaining Work.

    Pro-Tip: Remaining Time = Remaining Units / Efficiency of the active worker(s).
  5. 5

    Step 5: Solve for the required time and perform a sanity check

    Divide the relevant units of work by the active efficiency to find the required days. Double-check whether the question asks for 'remaining days' or 'total days taken to complete the entire work'.

    Pro-Tip: Sanity check: When two people work together, the combined time must ALWAYS be strictly less than the time taken by the faster person alone.

Worked Examples & Walkthroughs

Real questions solved step-by-step showing every calculation

1Example 1Method: LCM Unit Method & Formula Shortcut

Question

A can complete a piece of work in 12 days and B can complete the same work in 18 days. If they work together, in how many days will they finish the entire work?

Step-by-Step Solution:

Step 1: Determine Total Work via LCM:Find the LCM of 12 and 18: LCM(12, 18) = 36 units. Let Total Work = 36 units.
Step 2: Calculate individual 1-day efficiencies:Efficiency of A = 36 / 12 = 3 units/day. Efficiency of B = 36 / 18 = 2 units/day.
Step 3: Calculate combined daily efficiency:Together, (A + B) accomplish: 3 + 2 = 5 units/day.
Step 4: Compute total days required:Time taken = Total Work / Combined Efficiency = 36 / 5 = 7.2 days = 7 1/5 days.
Final Result:
7 1/5 days (or 7.2 days)
Exam Shortcut / Verification: Formula shortcut: (A × B) / (A + B) = (12 × 18) / (12 + 18) = 216 / 30 = 36 / 5 = 7.2 days. Takes under 10 seconds!
2Example 2Method: LCM Remaining Work Method

Question

A can complete a task in 15 days, and B can complete it in 20 days. They worked together for 4 days, after which A left the work. How many more days will B take to complete the remaining work alone?

Step-by-Step Solution:

Step 1: Set up Total Work:Total Work = LCM(15, 20) = 60 units.
Step 2: Find daily efficiencies:Efficiency of A = 60 / 15 = 4 units/day. Efficiency of B = 60 / 20 = 3 units/day.
Step 3: Compute work done together in 4 days:Combined rate = 4 + 3 = 7 units/day. Work completed in 4 days = 4 × 7 = 28 units.
Step 4: Calculate Remaining Work:Remaining Work = Total Work - Completed Work = 60 - 28 = 32 units.
Step 5: Calculate time taken by B alone:B's efficiency is 3 units/day. Time required by B = Remaining Work / B's Efficiency = 32 / 3 = 10 2/3 days.
Final Result:
10 2/3 days (approx. 10.67 days)
Exam Shortcut / Verification: Pay close attention to what the question asks: If asked for 'more days taken by B alone', the answer is 10 2/3 days. If asked for 'total days for the whole project', the answer is 4 + 10 2/3 = 14 2/3 days.
3Example 3Method: Cyclic 2-Day Block Method

Question

A and B can complete a work individually in 20 days and 30 days respectively. If they work on alternate days starting with A on the first day, in how many days will the entire work be completed?

Step-by-Step Solution:

Step 1: Determine Total Work:Total Work = LCM(20, 30) = 60 units.
Step 2: Find daily rates:Efficiency of A = 60 / 20 = 3 units/day. Efficiency of B = 60 / 30 = 2 units/day.
Step 3: Calculate work done in one 2-day cycle:Day 1: A works alone and completes 3 units. Day 2: B works alone and completes 2 units. Work done in 1 cycle (2 days) = 3 + 2 = 5 units.
Step 4: Find total cycles needed:Number of full 2-day cycles = 60 units / 5 units per cycle = 12 complete cycles.
Step 5: Convert cycles to total days:Total days = 12 cycles × 2 days/cycle = 24 days.
Final Result:
24 days
Exam Shortcut / Verification: Since 60 is an exact multiple of 5 (60 ÷ 5 = 12), the work finishes exactly at the end of B's turn on Day 24 without any leftover fraction.
4Example 4Method: MDH Chain Rule Formula

Question

12 men can complete a project in 18 days working 8 hours a day. In how many days can 16 men complete twice the amount of work if they work 9 hours a day?

Step-by-Step Solution:

Step 1: Identify given variables for Scenario 1:M₁ = 12 men, D₁ = 18 days, H₁ = 8 hours/day, W₁ = 1 unit of work.
Step 2: Identify given variables for Scenario 2:M₂ = 16 men, D₂ = ?, H₂ = 9 hours/day, W₂ = 2 units of work.
Step 3: Set up the MDH equation:(M₁ × D₁ × H₁) / W₁ = (M₂ × D₂ × H₂) / W₂ ⇒ (12 × 18 × 8) / 1 = (16 × D₂ × 9) / 2.
Step 4: Simplify both sides:Left Side = 12 × 18 × 8 = 1,728. Right Side = (144 × D₂) / 2 = 72 × D₂.
Step 5: Solve for D₂:D₂ = 1,728 / 72 = 24 days.
Final Result:
24 days
Exam Shortcut / Verification: Quick cancellation trick: (12 × 18 × 8) / 1 = (16 × D₂ × 9) / 2 ⇒ cancel 18 by 9 to get 2; cancel 16 by 2 to get 8. The equation becomes 12 × 2 × 8 = 8 × D₂. Cancel 8 on both sides: D₂ = 12 × 2 = 24 days!

Common Mistakes to Avoid

Avoid these frequent traps to protect yourself from negative marking

1

Directly adding the number of days taken by each person

Why it happens: Students intuitively think: 'If A takes 10 days and B takes 15 days, together they take 10 + 15 = 25 days'.
Correct approach: Remember that two workers working together must complete the job in LESS time than either person alone. Never add days directly; always add their efficiencies or rates of work.
2

Confusing 'Remaining Days' with 'Total Project Days'

Why it happens: In problems where one worker leaves or joins midway, students often calculate the remaining days and immediately mark that as the final answer without re-reading the question prompt.
Correct approach: Clearly circle the required target during exam scratchwork: note whether the question asks for 'more days taken by the remaining person' or 'total days taken from beginning to end'.
3

Inverting the relationship between efficiency and time taken

Why it happens: Assuming that if worker A is twice as efficient as worker B, A takes twice as many days.
Correct approach: Efficiency is work done per day (speed). Higher efficiency means FEWER days required. Efficiency ratio E_A : E_B = 2 : 1 implies Time ratio T_A : T_B = 1 : 2.
4

Misallocating odd remainder units during Alternate Days problems

Why it happens: Dividing the leftover units by the combined 2-day cycle efficiency rather than assigning them to the specific person whose turn it is next.
Correct approach: Always calculate full cycles first. If leftover units remain, check whose turn it is (e.g., A). If remainder is less than A's daily rate, add (Remainder / A's rate) days.
Self-Assessment

Try It Yourself (Practice Questions)

Test your understanding on your own first, then reveal the step-by-step solution to check your work:

Practice #1

A, B, and C can complete a piece of work individually in 12, 15, and 20 days respectively. If all three work together, in how many days will the work be completed?

Practice #2

A is 50% more efficient than B. If B alone can complete a piece of work in 30 days, in how many days can A and B working together complete the same work?

Practice #3

A and B undertook to do a piece of work for ₹1,200. A alone can do it in 8 days while B alone can do it in 12 days. With the assistance of C, they finish the work in 4 days. What is C's share of the wages?

Quick Revision Summary

  • Total Work is best represented as the LCM of individual completion days to avoid fractional math.
  • Efficiency = Total Work / Time. Efficiency and time are inversely proportional (Higher efficiency = Fewer days).
  • Two persons working together formula: Time = (A × B) / (A + B) days.
  • For group work with varying men, hours, days, and output, use the MDH rule: (M₁ × D₁ × H₁) / W₁ = (M₂ × D₂ × H₂) / W₂.
  • Wages are distributed strictly according to the proportion of total work done by each individual.
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