Time & Work, Pipes & Cisterns
Concept
Work problems are really just about rates. Think of it like this: if you can clean your room in 2 hours, your cleaning rate is 1 room per 2 hours, or ½ room per hour. That’s all “1-day work” means — how much of the job gets done in a single day.
When two workers team up, you simply add their rates. If A does 1/6 of the work per day and B does 1/3 per day, together they do 1/6 + 1/3 = 1/2 per day — meaning they finish in 2 days. Easy!
Pipes and cisterns work the same way, but here’s the twist: filling pipes are positive (+), and draining pipes are negative (–). If a pipe would fill a tank in 4 hours but another drains it in 6 hours, you add +1/4 and –1/6 to get a net rate of 1/12 per hour. The tank fills in 12 hours.
When workers leave before a job is done, or when pipes open/close at different times, you handle it in stages — calculate how much work is done in each stage, subtract from 1, then solve for what’s left.
Key Formulas
| Formula | Use |
|---|---|
| Work = Rate × Time | Core relationship |
| A’s 1-day work = 1/x | If A finishes in x days |
| Combined rate = sum of individual rates | When workers/pipes work together |
| Total Work = 1 (whole job) | Always set the full job = 1 |
| Net rate = filling rate – draining rate | Pipes and cisterns |
Worked Example
Q: Pipe A fills a tank in 10 hours. Pipe B fills it in 15 hours. Pipe C empties it in 20 hours. All pipes open together. How long to fill the tank?
Step 1: Write each rate:
- A = +1/10 per hour
- B = +1/15 per hour
- C = –1/20 per hour
Step 2: Add them: 1/10 + 1/15 – 1/20 Find common denominator (60): 6/60 + 4/60 – 3/60 = 7/60 per hour
Step 3: Time = 1 ÷ (7/60) = 60/7 = 8⅘ hours = 8 hours 34 minutes
Answer: 8 hours 34 minutes (approximately)
Common Errors
- Adding rates that work opposite directions → Always subtract draining/emptying pipes: filling (+), emptying (–)
- Forgetting the whole job = 1 → Your equation must equal 1, not the number of days
- Mixing up “3 days” with “3 times as fast” → If someone is 3× faster, their rate is 3 times bigger, not their time
📐 Diagram Reference
A detailed diagram showing a tank with two pipes: one filling at a rate that would fill the tank in 4 hours, and one emptying at a rate that would empty it in 6 hours. Label the net rate when both are open.
Diagrams are generated per-topic using AI. Support for AI-generated educational diagrams coming soon.