🕒 Time & Work – Complete Guide (Concepts, Shortcuts, Solved Examples)
Aptitude Time & Work Pipes & Cisterns
1) Basic Concept
[Formula: Work = Rate × Time]
If a person completes a job in N days, then in one day the person completes 1/N of the work. This “1/N” is called efficiency or rate of work.
- More efficiency → less time. [Time ∝ 1/Efficiency]
- Combined work = sum of individual works (rates add up).
2) Key Formulas & Relations
- One-day work: If A finishes in x days → A’s 1-day work = [1/x].
- Two persons together: A in x days, B in y days → Time together = [xy/(x+y)].
- n times efficiency: If A is n times as efficient as B → [A’s time : B’s time = 1 : n].
- Three persons: A in x, B in y, C in z → Together = [xyz/(xy + yz + zx)].
- Part work: Work done = [Rate × Time] (apply in phases if people join/leave).
3) LCM Method (Shortcut)
Step-1: Take LCM of given days → assume it as Total Work (units).
Step-2: Efficiency = Total Work ÷ individual days (units/day).
Step-3: Add/subtract rates; Time = Total Work ÷ combined rate.
| Person | Alone Days | Rate (units/day) |
|---|---|---|
| A | 12 | Total/12 |
| B | 18 | Total/18 |
| LCM | LCM(12,18) = 36 ⇒ set Total Work = 36 units | |
Then A = 3 units/day, B = 2 units/day, Together = 5 units/day → Time = 36/5 = 7.2 days.
4) Solved Examples (Step by Step + Shortcuts)
Example 1: Two persons working together
Q. A completes a job in 12 days; B in 18 days. How long together?
Method-1 (Fractions):
A’s 1-day work = 1/12, B’s = 1/18 ⇒ Sum = (3+2)/36 = 5/36.
Time = 36/5 = 7.2 days (7 days 4.8 hours).
Method-2 (LCM Shortcut):
LCM(12,18)=36 ⇒ Total=36 units. A=3 u/d, B=2 u/d ⇒ Together=5 u/d ⇒ Time=36/5= 7.2 days.
[Shortcut: Together = xy/(x+y) days]
Example 2: Relative efficiency
Q. A is twice as efficient as B. Together they finish in 18 days. Find A’s alone time.
Step-1: Let B’s rate = 1 unit/day ⇒ A’s rate = 2 u/d ⇒ Together = 3 u/d.
Step-2: Total Work = 18 × 3 = 54 units.
Step-3: A alone time = 54 ÷ 2 = 27 days. [Time ∝ 1/Efficiency]
Example 3: Joining/Leaving scenario
Q. A can finish in 15 days; B in 20 days. They work together for 4 days, then A leaves. Remaining days for B?
LCM: LCM(15,20)=60 ⇒ Total=60 units.
A=4 u/d, B=3 u/d ⇒ Together 7 u/d for 4 days = 28 units done.
Left = 60 − 28 = 32 units. B alone rate = 3 u/d ⇒ Time = 32/3 = 10⅔ days.
Example 4: Group comparison (inverse rule)
Q. If 8 men can do a job in 15 days, how long will 12 men take (same efficiency)?
Men × Days = Constant ⇒ 8 × 15 = 12 × D ⇒ D = 120/12 = 10 days.
[Inverse Proportion: Time ∝ 1/Number of Men]
5) Work & Wages
Wages split in proportion to work done (efficiency × time worked).
Q. A and B earn ₹1200. A is 2× as efficient as B and both worked equal days. Find shares.
Efficiency ratio = 2:1 ⇒ Work ratio = 2:1 ⇒ Wages = ₹800 : ₹400.
Tip: If working days differ, multiply efficiency by respective days to get the work ratio.
6) Pipes & Cisterns (Application)
Treat tank capacity as Total Work. Inlet = positive rate; Outlet = negative rate.
Q. Inlet fills in 12 h, outlet empties in 18 h. Both opened. Time to fill?
Net rate = 1/12 − 1/18 = (3 − 2)/36 = 1/36 ⇒ Time = 36 h.
[Net Rate = Sum of (±) rates]
7) Practice Questions (Try Yourself)
- A can do a work in 15 days, B in 20 days. In how many days working together?
- A is thrice as efficient as B. Together they finish in 12 days. Find B’s alone time.
- Two inlets fill a tank in 24 h and 36 h; an outlet empties in 12 h. All opened together. Time?
- A, B, C finish in 12, 18, 36 days respectively. Find combined time.
- 16 women finish a job in 25 days. How long will 20 women take (same efficiency)?
- (1) LCM(15,20)=60 ⇒ Rates 4 & 3 ⇒ Together 7 ⇒ 60/7 days.
- (2) Let B=1 u/d, A=3 u/d ⇒ Together 4 ⇒ Total=12×4=48 ⇒ B alone=48/1=48 days.
- (3) LCM(24,36,12)=72 ⇒ +3 +2 −6 = −1 ⇒ Tank never fills (outlet dominates). If outlet closed after t hours, split phases.
- (4) xyz/(xy+yz+zx) = 12×18×36 / (12×18 + 18×36 + 36×12) = 7776 / (216 + 648 + 432) = 7776 / 1296 = 6 days.
- (5) Men-days = constant ⇒ 16×25 = 20×D ⇒ D = 20 days.