How to Solve: CUET Quant – Time and Work, Pipes and Cisterns

How to Solve: CUET Quant – Time and Work, Pipes and Cisterns

Introduction

"Imagine you’re managing a construction site: 3 workers, 2 pipes filling a tank, and a deadline. Mess this up, and you fail the project—or the exam. Today, we’ll turn ‘Time and Work’ from a headache into your highest-scoring topic."

What You Need To Know First

1. Unitary Method: If 1 person does 1 unit of work in 1 day, how much do 5 people do?
2. Fractions & LCM: You’ll need to add/subtract work rates (e.g., 1/4 + 1/6 = 5/12).
3. Inverse Proportion: More workers = less time (but not always linear—watch for efficiency!).

Key Vocabulary

Formulas To Know

Step-by-Step Method

Step 1: Assign Work Units

• Assume total work = LCM of individual times (simplifies fractions). Example: A takes 4 days, B takes 6 days → LCM = 12 units of work.

Step 2: Calculate Individual Rates

• Rate = Work / Time. Example: A’s rate = 12 units / 4 days = 3 units/day. B’s rate = 12 units / 6 days = 2 units/day.

Step 3: Combine Rates (If Working Together)

• Add rates for same-direction work. Example: A + B together = 3 + 2 = 5 units/day.

Step 4: Solve for Time/Work

• Use Work = Rate × Time. Example: To complete 12 units at 5 units/day → Time = 12/5 = 2.4 days.

Step 5: Adjust for Negative Work (Leaks/Pipes)

• Subtract draining rates from filling rates. Example: Pipe A fills at 4 units/hour, Pipe B drains at 1 unit/hour → Net rate = 3 units/hour.

Step 6: Check for Alternate Work

• If workers take turns, calculate work per cycle. Example: A works Day 1 (3 units), B works Day 2 (2 units) → 5 units in 2 days.

Worked Examples

Example 1 – Basic (Two Workers)

Question: A can complete a job in 8 days. B can do it in 12 days. How long will it take if they work together?

Solution: 1. Assign Work Units: LCM of 8 and 12 = 24 units. 2. Individual Rates:
- A’s rate = 24/8 = 3 units/day.
- B’s rate = 24/12 = 2 units/day. 3. Combined Rate: 3 + 2 = 5 units/day. 4. Time Together: 24 units / 5 units/day = 4.8 days.

What we did and why: - Used LCM to avoid fractions. - Added rates because they’re working together. - Final answer: 4.8 days (or 4 days and 19.2 hours).

Example 2 – Medium (Pipes + Leak)

Question: Pipe A fills a tank in 6 hours. Pipe B fills it in 8 hours. A leak empties the tank in 12 hours. If all three are open, how long to fill the tank?

Solution: 1. Assign Work Units: LCM of 6, 8, 12 = 24 units. 2. Individual Rates:
- A’s rate = 24/6 = +4 units/hour (fills).
- B’s rate = 24/8 = +3 units/hour (fills).
- Leak’s rate = 24/12 = -2 units/hour (drains). 3. Net Rate: 4 + 3 – 2 = 5 units/hour. 4. Time to Fill: 24 units / 5 units/hour = 4.8 hours.

What we did and why: - Treated the leak as negative work. - Net rate = filling – draining. - Final answer: 4.8 hours.

Example 3 – Exam Style (Efficiency + Alternate Work)

Question: A is twice as efficient as B. Together, they finish a job in 6 days. If they work on alternate days starting with A, how long will it take?

Solution: 1. Assign Efficiency: Let B’s rate = 1 unit/day → A’s rate = 2 units/day. 2. Combined Rate: 2 + 1 = 3 units/day. 3. Total Work: 3 units/day × 6 days = 18 units. 4. Alternate Work:
- Day 1 (A): 2 units.
- Day 2 (B): 1 unit.
- Cycle (2 days): 3 units. 5. Full Cycles: 18 units / 3 units per cycle = 6 cycles (12 days).
- But after 5 cycles (10 days), 15 units done.
- Day 11 (A): 2 units → 17 units (still 1 left).
- Day 12 (B): 1 unit → completes 18 units. 6. Total Time: 12 days.

What we did and why: - Used efficiency ratios to simplify rates. - Calculated work per cycle for alternate days. - Final answer: 12 days.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, last-minute cram? Here’s the cheat sheet: 1. LCM trick: Assume total work = LCM of individual times. No fractions! 2. Rates add: If they work together, add their rates. If one drains, subtract. 3. Efficiency = ratio: A is 2× B → A’s rate = 2B’s rate. 4. Alternate work: Calculate work per cycle (e.g., A+B in 2 days). 5. Watch for leaks: Negative work = draining pipes or lazy workers. Now go crush that exam—you’ve got this!