Time and Work
This guide simplifies the Time and Work topic from your PDF, including core concepts, key formulas, solved examples, a cheatsheet, practice questions (with answers), and visual aids for clarity.
1. Core Concepts
- Work: Any task to be completed (e.g., building a wall, filling a tank).
- Time: Duration taken to finish the work.
- Efficiency: Amount of work done per unit time (higher efficiency = less time).
2. Key Formulas (Cheatsheet)
| Formula/Rule | Use Case |
|---|---|
| If A completes work in n days, A’s 1 day work = 1/n | Basic calculation |
| Work = Time × Rate of Work | Find total work done |
| Rate of Work = 1 / Time | Work done per unit time |
| Time = 1 / Rate of Work | Find time from rate |
| Total Work = Efficiency × Number of Days | Work done by a person/group |
| If A is x times as efficient as B, time ratio = 1:x | Comparing times for same work |
| If A & B together finish in T days: 1/T = 1/A + 1/B | Combined work |
| If A, B, C together: 1/T = 1/A + 1/B + 1/C | Three people together |
| If x men do W1 work in D1 days, x men do W2 in D2 days: | Work/men/days relationship |
| M₁D₁/W₁ = M₂D₂/W₂ | |
| If A takes x days more than (A+B) and B takes y days more | (A+B) finish in √(xy) days |
3. Visual: Work-Time-Efficiency Relationship
Work
/ \
Time Efficiency
\ /
(Inverse)- More efficiency → less time for same work.
- More people → less time for same work.
4. Solved Examples
Example 1: Two People Working Together
Q: A does a work in 10 days, B in 15 days. How long together?
- A’s 1 day work = 1/10
- B’s 1 day work = 1/15
- Combined 1 day work = 1/10 + 1/15 = (3+2)/30 = 5/30 = 1/6
- Together: 6 days
Example 2: Finding Individual Time
Q: A alone: 40 days. A+B together: 8 days. How long for B alone?
- A’s 1 day work = 1/40
- Together = 1/8
- B’s 1 day work = 1/8 – 1/40 = (5–1)/40 = 4/40 = 1/10
- B alone: 10 days
Example 3: Three People Together
Q: A+B: 3 days, B+C: 4 days, A+C: 6 days. How long all together?
- A+B: 1/3, B+C: 1/4, A+C: 1/6
- Add: (1/3 + 1/4 + 1/6) = (4+3+2)/12 = 9/12 = 3/4
- All together: 1/2 × 3/4 = 3/8 per day ⇒ 8/3 days
Example 4: Efficiency Ratio
Q: A is thrice as efficient as B. Together finish in 12 days. B alone?
- Let B’s time = x, A’s = x/3
- 1/x + 1/(x/3) = 1/12 ⇒ (1 + 3)/x = 1/12 ⇒ 4/x = 1/12 ⇒ x = 48
- B alone: 48 days
Example 5: Alternate Days
Q: A: 10 days, B: 15 days, work alternately starting with A. Days to finish?
- A’s 1 day: 1/10, B’s: 1/15, 2 days: 1/10 + 1/15 = 1/6
- 6 cycles (12 days) = 1 work done
- Total: 12 days
5. Practice Questions with Answers
- A can do a work in 12 days, B in 16 days. Together? A’s 1 day: 1/12, B’s: 1/16, Together: 1/12+1/16=7/48 → 6.86 days
- A+B: 20 days, B alone: 30 days. A alone? 1/20 – 1/30 = 1/60 → 60 days
- A is twice as efficient as B. Together: 18 days. A alone? Let B = x, A = x/2; 1/x + 2/x = 1/18 → x = 54, A = 27 days
- 15 men do a job in 8 days. After 3 days, 3 more join. Remaining days? Total work: 120 units. Done: 45 units. Left: 75 units. New men: 18. 75/18 = 4.17 days
- 1200 soldiers have food for 28 days. After 4 days, some leave, food lasts 32 more days. How many left? x = 300 soldiers left
6. Tricks & Shortcuts
- If A is x times as efficient as B, A’s time : B’s time = 1:x
- Combined work for n people: 1/Total days = 1/x₁ + 1/x₂ + … + 1/xₙ
- Work and wages: Wages are divided in the ratio of work done (or efficiency).
7. Images & Graphics
A. Work Distribution Bar
|----------|----------|----------|
A (10d) B (15d) Together (6d)B. Pie Chart: Work Done Per Day
If A = 1/10 per day, B = 1/15 per day:
- A: 60%
- B: 40%
8. Summary Table
| Situation | Formula/Approach |
|---|---|
| A alone: n days | 1/n per day |
| A+B together: t days | 1/t per day |
| A’s share in work/wages | Efficiency ratio |
| More men, less days | M₁D₁ = M₂D₂ |
| Men, Days, Hours, Work | M₁D₁H₁/W₁ = M₂D₂H₂/W₂ |
9. Final Tips
- Always express work in “per day” or “per hour” units.
- Use ratios to compare efficiency and time.
- For alternate day problems, sum cycles and add remaining work if needed.
- Practice with a variety of problems for speed and accuracy.
With these explanations, formulas, examples, cheatsheets, solved and practice questions, and visuals, you’ll be ready to solve any Time and Work problem in exams!
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