Time and work

📘 Time & Work + Pipes and Cisterns

🔹 Time & Work

\[ \text{Work} = \text{Rate} \times \text{Time} \]

\[ \text{Work done in 1 day} = \frac{1}{\text{Total days}} \]

\[ \text{If A does work in } x \text{ days, then A's 1 day work } = \frac{1}{x} \]

Two Persons Working Together

\[ \text{Time taken together} = \frac{xy}{x+y} \text{ days (if A in x days, B in y days)} \]

Shortcut: Efficiency ∝ 1/Time

Example:
A can do a work in 10 days, B in 15 days. Time together = \( \frac{10 \times 15}{10+15} = 6 \text{ days} \)

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🔹 Pipes and Cisterns

Pipes are usually of two types: - **Inlet Pipe:** Fills the tank - **Outlet Pipe:** Empties the tank

\[ \text{Tank filled in 1 hour by a pipe} = \frac{1}{\text{Time to fill tank}} \]

\[ \text{Net rate when multiple pipes work together} = \text{Sum of inlet rates - Sum of outlet rates} \]

\[ \text{Time to fill tank together} = \frac{1}{\text{Net rate}} \]

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🔥 Exam Shortcuts & Tricks

Shortcut 1: Treat tank as 1 unit work
Work done in 1 hour = fraction of tank filled

Shortcut 2: Inlet + Outlet → net fill = Inlet rate − Outlet rate

Shortcut 3: If pipe A fills in x hrs, pipe B empties in y hrs, together time = \( \frac{xy}{y-x} \) (A fills, B empties)

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✍️ Solved Examples

Example 1:
Pipe A fills a tank in 10 hrs, Pipe B in 15 hrs. Time to fill together: \[ \frac{1}{10} + \frac{1}{15} = \frac{3+2}{30} = \frac{5}{30} \] \[ \text{Time} = \frac{30}{5} = 6 \text{ hrs} \]

Example 2:
Tank can be filled by A in 12 hrs and emptied by B in 16 hrs. Net rate = \( \frac{1}{12} - \frac{1}{16} = \frac{4-3}{48} = \frac{1}{48} \) Time to fill = 48 hrs

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