# Time and Work (with Pipes and Cisterns) — The LCM Method

Source: https://skills101.in/learn/aptitude/time-and-work — Skills101 free tutorial (Quantitative Aptitude). Updated 2026-11-09.

Time and work questions ask: *if A takes 10 days and B takes 15 days, how long together?* The fraction method (1/10 + 1/15 …) works, but there is a much faster way.

## 1. The LCM method

Pretend the job has a fixed number of **units** — the LCM of the days. Then everyone's daily work is a whole number.

![A takes 10 days and B takes 15: total work = LCM = 30 units, A does 3 units/day, B does 2, together 5 units/day so 6 days](https://skills101.in/learn-img/aptitude/work-units.svg)

### Example 1
*A can finish a job in 10 days and B in 15 days. How long will they take together?*

- Total work = LCM(10, 15) = 30 units
- A = 30/10 = 3 units/day, B = 30/15 = 2 units/day
- Together = 5 units/day → 30 ÷ 5 = **6 days**

## 2. Someone leaves or joins midway

### Example 2
*A can do a job in 12 days and B in 18 days. They work together for 4 days, then A leaves. How many more days does B take?*

- Total = LCM(12, 18) = 36 units. A = 3/day, B = 2/day.
- 4 days together = 4 × 5 = 20 units. Left = 16 units.
- B alone: 16 ÷ 2 = **8 more days**.

### Example 3
*A and B together finish a job in 8 days. A alone takes 12 days. How long does B alone take?*

Total = LCM(8, 12) = 24. A + B = 3/day, A = 2/day → B = 1/day → **24 days**.

## 3. Efficiency

If A is **twice as efficient** as B, A does 2 units for every 1 of B — so A takes **half the time**.

### Example 4
*A is 50% more efficient than B. B alone takes 30 days. How long does A take?*

Efficiency A : B = 150 : 100 = 3 : 2. Time is the inverse: 2 : 3. A takes 30 × 2/3 = **20 days**.

## 4. The men-days formula

For groups of workers:

> **M₁ × D₁ × H₁ ÷ W₁ = M₂ × D₂ × H₂ ÷ W₂**
> (M = men, D = days, H = hours per day, W = work done)

### Example 5
*12 men can build a wall in 10 days. How many men are needed to build it in 8 days?*

12 × 10 = M × 8 → M = **15 men**.

### Example 6
*10 men working 6 hours a day finish 1 km of road in 12 days. How many days will 15 men working 8 hours a day take for 2 km?*

(10 × 12 × 6)/1 = (15 × D × 8)/2 → 720 = 60D → D = **12 days**.

## 5. Pipes and cisterns

The same idea, but a pipe that **empties** the tank does **negative** work.

### Example 7
*Pipe A fills a tank in 20 minutes, pipe B in 30 minutes, and pipe C empties it in 15 minutes. All three are opened together. How long to fill the tank?*

- Tank = LCM(20, 30, 15) = 60 units
- A = +3, B = +2, C = −4 per minute
- Net = +1 unit/minute → **60 minutes**

### Example 8
*A tap fills a tank in 6 hours. Because of a leak, it takes 8 hours. How long would the leak take to empty the full tank?*

Tank = 24 units. Tap = 4/hour. With the leak, 3/hour → leak = −1/hour → **24 hours**.

## 6. Wages are shared by work done

### Example 9
*A and B finish a job together for ₹3,000. A alone takes 6 days, B alone 12 days. How much should A get?*

A's rate : B's rate = 12/6 : 12/12 = 2 : 1 → A gets ⅔ × 3000 = **₹2,000**.

## Common mistakes

- Adding **days** (10 + 15 = 25) instead of adding **work per day**.
- Forgetting that a leak or emptying pipe is **negative**.
- In the men-days formula, forgetting to divide by the amount of work when it changes.

## Quick recap

- Total work = LCM of the days. Find units per day. Add or subtract rates.
- Efficiency up → time down (inverse).
- M₁D₁H₁/W₁ = M₂D₂H₂/W₂.
- Pay is shared in the ratio of work done.


Practice quiz and free certificate: https://skills101.in/learn/aptitude/time-and-work