Time & Work – Quick & Easy Study Notes
(For Lakshadweep Patwari / Lekhpal – Quantitative Aptitude)
1. What Does “Time & Work” Mean?
- Work is the job that must be finished (e.g., painting a wall, filling a tank).
- Time is how long it takes to finish that job.
- Work Rate (how fast someone works) = amount of work done per unit of time.
Analogy:
Think of a bucket that needs to be filled with water.
• One person can fill it in 4 hours (they’re the “worker”).
• Two people together can fill it faster (their combined “work rate” is higher).
2. Basic Idea – Work = Rate × Time
Work (W) = Rate (R) × Time (T)
- Rate (R) = units of work per hour.
- Time (T) = hours spent working.
- Work (W) = total units of job (usually set as 1 whole job).
3. Simple Steps to Solve a Problem
| Step | What to Do | Example |
|---|---|---|
| 1️⃣ | Find each worker’s rate. Rate = 1 / Time to finish alone. |
A can finish in 5 hrs → Rate = 1/5 job per hr |
| 2️⃣ | Add rates if workers work together. | A + B → (1/5) + (1/8) |
| 3️⃣ | Find combined rate. | (1/5)+(1/8) = 13/40 job/hr |
| 4️⃣ | Use Work = Rate × Time to find needed time. Time = Work / Rate. |
Time = 1 / (13/40) = 40/13 ≈ 3.08 hrs |
| 5️⃣ | Round or convert to minutes if needed. | 3 hrs + 0.08×60 ≈ 3 hrs 5 min |
4. Tiny Example
Question:
- A can fill a tank in 6 hrs.
- B can fill it in 12 hrs.
- How long will they take together?
Solution:
- A’s rate = 1/6 job/hr.
- B’s rate = 1/12 job/hr.
- Combined rate = 1/6 + 1/12 = 2/12 + 1/12 = 3/12 = 1/4 job/hr.
- Time = 1 job ÷ (1/4 job/hr) = 4 hrs.
Answer: 4 hours.
5. Common Variations
| Variation | How to Handle |
|---|---|
| One worker leaves early | Subtract the time they worked from total time, then solve for the remaining worker. |
| Work is split between workers | Use Work = Rate × Time for each part and add the times. |
| Work done after a break | Treat break as “no work” (time does not contribute). |
| Different units (e.g., “days” instead of “hours”) | Keep units consistent; convert if necessary. |
6. Quick Practice Problems
-
Simple:
- A finishes a job in 8 hrs.
- B finishes the same job in 12 hrs.
- Find the time if they work together.
-
One leaves early:
- A can finish a task in 10 hrs.
- B can finish it in 15 hrs.
- They start together, but B leaves after 2 hrs.
- How long does A finish the rest?
-
Work split:
- A works for 3 hrs, then B joins.
- Together they finish the job in 5 hrs total.
- If A alone would finish the job in 10 hrs, find B’s time to finish alone.
(Try solving before checking the answers in the next section.)
7. Answers to Practice Problems
-
Time together
- Rate A = 1/8, Rate B = 1/12.
- Combined rate = 1/8 + 1/12 = 5/24.
- Time = 1 ÷ (5/24) = 24/5 = 4.8 hrs (4 hrs 48 min).
-
B leaves after 2 hrs
- A’s rate = 1/10.
- B’s rate = 1/15.
- Work done in first 2 hrs by both = 2(1/10 + 1/15) = 2(5/50 + 3.33/50) ≈ 2(0.1 + 0.0667) = 0.3333.
- Remaining work = 1 – 0.3333 = 0.6667.
- A’s remaining time = 0.6667 ÷ (1/10) = 6.667 hrs (6 hrs 40 min).
-
B’s time to finish alone
- A’s rate = 1/10.
- A works 3 hrs → work done = 3 × 1/10 = 0.3.
- Remaining work = 0.7.
- Total time together = 5 hrs, so together did 5 × (1/10 + 1/?) = 0.7.
- Let B’s rate = 1/x.
- (1/10 + 1/x) × 5 = 0.7 → (0.1 + 1/x) × 5 = 0.7 → 0.5 + 5/x = 0.7 → 5/x = 0.2 → x = 25.
- B finishes alone in 25 hrs.
8. Quick Tips
- Keep work = 1 job for simplicity.
- Always convert times to the same unit (hrs or min).
- Remember “Rate” is 1 ÷ Time.
- Check your answer: if time seems too low or high, redo the rate calculation.
Final Thought
Think of “time & work” like mixing