Time and Work
Overview
Time and Work is a high-scoring topic in HSSC CET Quantitative Ability. Questions test your ability to calculate how long workers take to complete jobs, how efficient different workers are, and how work gets divided among groups. This topic also covers pipes-and-cisterns problems, which are essentially the same concept applied to filling or emptying tanks.
The good news: once you master the core concept that work = rate × time, every problem becomes a variation of the same logic. Most questions are direct formula applications, though some require careful handling of workers joining or leaving midway.
The key skill here is converting "A can do a job in X days" into "A does 1/X of the job per day." This fractional thinking is the foundation of every solution.
Key Concepts
- Work as a unit: Treat total work as 1 (or 100%, or LCM of days for easier calculation). Each worker completes a fraction of this daily.
- Efficiency = 1/Time: If A finishes work in 10 days, A's daily efficiency is 1/10. More days means lower efficiency.
- Combined work: When workers work together, add their efficiencies. If A does 1/10 and B does 1/15 per day, together they do 1/10 + 1/15 per day.
- LCM method: Take LCM of individual times as total work units. This converts fractions to whole numbers, making calculations faster.
- Pipes-and-cisterns: Inlet pipes add water (positive work), outlet pipes remove water (negative work). Net rate = sum of inlet rates − sum of outlet rates.
- Work and wages: Wages are distributed in ratio of work done, which equals ratio of efficiencies when time is same.
- Negative work: When someone destroys work or a pipe empties a tank, treat their rate as negative in calculations.
- Man-days concept: Total work = Number of workers × Number of days. If 5 men finish in 10 days, total work = 50 man-days.
Formulas / Key Facts
Basic Formula: Work = Rate × Time, or equivalently, Time = Work / Rate
If A completes work in 'a' days: A's one day work = 1/a
If A and B complete work in 'a' and 'b' days respectively: Combined one day work = 1/a + 1/b Time to complete together = ab/(a+b) days
If A is 'n' times as efficient as B: Ratio of time taken = 1 : n (A takes less time)
Man-Days Formula: M₁ × D₁ = M₂ × D₂ (when work is constant) M₁ × D₁ × H₁ = M₂ × D₂ × H₂ (including hours)
Pipes-and-Cisterns: If pipe fills in 'a' hours and empties in 'b' hours: Net time to fill = ab/(b−a) hours (when b > a)
Work and Wages: Wage of A : Wage of B = Efficiency of A : Efficiency of B
Worked Examples
Example 1: Basic Combined Work A can complete a job in 12 days, B can complete it in 18 days. In how many days will they finish working together?
Solution: A's one day work = 1/12 B's one day work = 1/18 Combined one day work = 1/12 + 1/18 = (3+2)/36 = 5/36 Time taken together = 36/5 = 7.2 days or 7 days 4 hours 48 minutes
Using LCM method: LCM of 12 and 18 = 36 units (total work) A's daily work = 36/12 = 3 units B's daily work = 36/18 = 2 units Combined = 5 units/day Time = 36/5 = 7.2 days
Example 2: Pipes and Cisterns Pipe A fills a tank in 20 minutes, Pipe B fills it in 30 minutes, and Pipe C empties it in 15 minutes. If all three are opened together, how long to fill the tank?
Solution: Take LCM of 20, 30, 15 = 60 units (tank capacity) A fills = 60/20 = 3 units/min B fills = 60/30 = 2 units/min C empties = 60/15 = 4 units/min
Net rate = 3 + 2 − 4 = 1 unit/min Time to fill = 60/1 = 60 minutes
Example 3: Workers Joining Midway A can do a piece of work in 30 days. After working for 10 days, B joins and together they finish the remaining work in 10 more days. In how many days can B alone finish the work?
Solution: Total work = 30 units (A does 1 unit/day) A works alone for 10 days = 10 units done Remaining work = 30 − 10 = 20 units
A and B together complete 20 units in 10 days Combined rate = 20/10 = 2 units/day A's rate = 1 unit/day B's rate = 2 − 1 = 1 unit/day
B alone can finish 30 units at 1 unit/day = 30 days
Common Mistakes
- Adding times instead of rates: Students calculate 12 + 18 = 30 days for combined work. Wrong! Add the rates (1/12 + 1/18), not the days.
- Forgetting negative sign for outlet pipes: When a pipe empties a tank, its rate must be subtracted, not added. Always mark outlet/leak as negative.
- Confusing efficiency ratio with time ratio: If A is twice as efficient as B, A takes half the time, not double. Efficiency and time are inversely related.
- Not accounting for partial work: When someone leaves or joins midway, calculate work done in each phase separately, then add.
- Ignoring units in man-days problems: If question gives hours per day, use M×D×H formula. Missing the hours component gives wrong answers.
Quick Reference
- A's one day work = 1/(days A takes alone)
- Together time = (a×b)/(a+b) when individuals take 'a' and 'b' days
- LCM method: Total work = LCM, individual rates = LCM ÷ individual days
- Outlet pipe or leak = negative rate
- More efficient = fewer days; Efficiency ∝ 1/Time
- Wages distributed in ratio of efficiencies, not time