Pipes & Cisterns
Pipes & Cisterns is Time & Work wearing a different costume: instead of people doing a job, you have pipes filling or emptying a tank. The single shift that makes the whole topic click is to treat every pipe as a signed rate of work — an inlet adds water (a positive rate), an outlet or leak removes it (a negative rate) — and to size the tank cleverly so the arithmetic stays clean. The standard CAT method is to set the tank capacity equal to the LCM of all the given times; then each pipe’s rate becomes a neat integer (units per hour) and you simply add the rates of whatever pipes are open. Net rate decides everything: a positive net fills the tank, a negative net empties it, and time = capacity ÷ net rate. CAT and XAT rarely ask a plain "two pipes fill a tank" question; they layer in a leak that wastes time, pipes opened and closed at different moments, or alternating cycles. This chapter builds the signed-rate habit across three topic areas — inlet & outlet pipes, combined filling, and leak problems — each with worked examples, the fastest LCM method, and the traps that quietly drain marks.
Topics
⚡ CAT shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Set tank capacity = LCM of all given times; every pipe’s rate becomes a clean integer (units/hour).
- Inlet = + rate, outlet/leak = − rate. Net rate = signed sum; time = capacity ÷ |net rate|.
- Two inlets a h and b h together fill in ab/(a+b) hours; inlet a with outlet b (b > a) fills in ab/(b−a).
- Leak makes an x-hour pipe take x+d hours ⇒ leak alone empties in x(x+d)/d hours. Memorise this product form.
- If the net rate is negative, an empty tank never fills — check the sign before dividing.
- For staggered pipes, work in units: each pipe pours rate × (its open hours); total must equal the capacity.
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Forgetting to subtract an outlet/leak that is left open while filling (using only the inlet rate).
- Adding times instead of rates — pipes combine by adding rates, not by averaging hours.
- Mixing up "leak empties full tank in L hours" (sets leak rate) with "tank fills in t hours despite leak" (net rate).
- Computing a fill time when the net rate is negative — the tank cannot fill from empty.
- In staggered/alternate-cycle problems, ignoring the partial last cycle and over- or under-counting the final hours.
📈 CAT exam insight & PYQ analysis
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara CAT success checklist
You have truly mastered Pipes & Cisterns when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (3 topics) | 3/3 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 12 cards |
Formula Reference Sheet
Core rates & capacity
| Tank capacity (smart units) | LCM of all the given fill/empty times |
|---|---|
| Rate of a pipe | Capacity ÷ time = units per hour |
| Pipe filling in x hours | rate = 1/x of tank per hour |
| Net rate (signed sum) | Σ inlet rates − Σ outlet rates |
| Time to fill / empty | Capacity ÷ |net rate| |
CAT power-tools
| Two inlets A (a h) & B (b h) together | time = ab/(a+b) hours |
|---|---|
| Inlet a h with outlet b h (b > a) | time = ab/(b−a) hours |
| Leak empties full tank in L h | leak rate = −1/L tank per hour |
| Inlet fills in x h, leak makes it x+d | leak alone empties in x(x+d)/d h |
| Part filled in t hours | net rate × t (as a fraction of the tank) |