Time and Work
Overview
The core concept is elegantly simple: if you know how fast someone works, you can calculate how long a task takes when multiple people collaborate or when workers join and leave midway.
This topic rewards students who master the efficiency method (LCM approach) over the traditional fraction method. Once you internalize that "work = rate × time," most problems become straightforward calculations. The Clerk-level questions are direct—usually involving 2–3 workers with clean numbers—so speed and accuracy matter more than handling complex scenarios.
Mastering Time and Work also builds your foundation for Pipes and Cisterns (same logic, different context) and helps with ratio-based reasoning across other arithmetic topics.
Key Concepts
- Work as a measurable unit: Assume total work = LCM of all given time periods. This converts fractions into whole-number efficiencies, making calculations faster.
- Efficiency = Work ÷ Time: If A completes a job in 10 days and total work = 30 units, then A's efficiency = 3 units/day.
- Combined efficiency is additive: When A and B work together, their combined rate = A's rate + B's rate. This is the heart of all "working together" problems.
- Inverse relationship: Time and efficiency are inversely proportional. If A is twice as efficient as B, A takes half the time B takes for the same work.
- Work done = Efficiency × Days worked: When workers join or leave midway, calculate each phase separately and sum up the work done.
- Negative efficiency for destroyers: In problems where someone undoes work (like a leak in a tank), assign negative efficiency to that agent.
- Man-days concept: Total work can be expressed as (number of workers) × (days) = constant. If 5 men finish in 10 days, total work = 50 man-days.
Formulas / Key Facts
Primary Formula: Time taken by A and B together = (Total Work) ÷ (Efficiency of A + Efficiency of B)
If A alone takes 'a' days and B alone takes 'b' days: Time together = (a × b) ÷ (a + b) days
Efficiency ratio shortcut: If A takes 'a' days and B takes 'b' days, their efficiency ratio = b : a (inverse of time ratio)
Work equivalence: M₁ × D₁ × H₁ = M₂ × D₂ × H₂ (when work is same) M₁ × D₁ × H₁ ÷ W₁ = M₂ × D₂ × H₂ ÷ W₂ (when comparing different amounts of work)
Wages distribution: Wages are divided in ratio of work done, which equals ratio of (efficiency × time worked)
Alternate days formula: If A and B work on alternate days starting with A, find work done in 2-day cycles, then handle remaining work.
Worked Examples
Example 1: Basic Combined Work
A can complete a work in 12 days, B can complete it in 18 days. In how many days will they finish it together?
Step 1: Total work = LCM(12, 18) = 36 units
Step 2: A's efficiency = 36 ÷ 12 = 3 units/day
Step 3: B's efficiency = 36 ÷ 18 = 2 units/day
Step 4: Combined efficiency = 3 + 2 = 5 units/day
Step 5: Time together = 36 ÷ 5 = 7.2 days = 7 days and 4.8 hours (or 36/5 days)
Example 2: Worker Leaves Midway
A and B together can complete a work in 12 days. A alone can do it in 20 days. If A and B start together but B leaves after 4 days, how many more days will A take to finish?
Step 1: Total work = LCM(12, 20) = 60 units
Step 2: Combined efficiency (A+B) = 60 ÷ 12 = 5 units/day
Step 3: A's efficiency = 60 ÷ 20 = 3 units/day
Step 4: B's efficiency = 5 – 3 = 2 units/day
Step 5: Work done in first 4 days = 5 × 4 = 20 units
Step 6: Remaining work = 60 – 20 = 40 units
Step 7: Days for A to complete remaining = 40 ÷ 3 = 13⅓ days
Example 3: Efficiency Ratio Given
A is twice as efficient as B. Together they finish a work in 6 days. How long would B alone take?
Step 1: Let B's efficiency = 1 unit/day, then A's efficiency = 2 units/day
Step 2: Combined efficiency = 3 units/day
Step 3: Total work = 3 × 6 = 18 units
Step 4: Time for B alone = 18 ÷ 1 = 18 days
Common Mistakes
Mistake: Adding times instead of efficiencies Wrong thinking: "A takes 10 days, B takes 15 days, so together = 25 days" Correct fix: You add rates (efficiencies), not times. Use the LCM method or formula (a×b)÷(a+b).
Mistake: Forgetting to find remaining work in midway-exit problems Wrong thinking: Calculating total time as if one person worked throughout Correct fix: Split the problem into phases—calculate work done in each phase, subtract from total, then find time for remainder.
Mistake: Confusing efficiency ratio with time ratio Wrong thinking: "A is twice as efficient, so A takes twice the time" Correct fix: Efficiency and time are inversely related. Twice the efficiency means half the time.
Mistake: Using wrong LCM when three or more workers involved Wrong thinking: Taking LCM of only two numbers when three workers are given Correct fix: Take LCM of ALL time periods mentioned to set total work.
Mistake: Not converting days to same unit Wrong thinking: Mixing hours and days without conversion Correct fix: Standardize all time units before calculating. If one works 8 hours/day and another 6 hours/day, account for this difference.
Quick Reference
- Total Work = LCM of all individual completion times
- Efficiency = Total Work ÷ Time taken by that person
- Together time = Total Work ÷ Sum of efficiencies
- Quick formula: (a × b) ÷ (a + b) for two workers with times a and b
- Efficiency ratio = Inverse of time ratio
- For wages: divide in ratio of total work contributed (efficiency × days worked)