Time & Work – Quick & Easy Notes
Why it matters
In the police exam you’ll see questions about how long it takes people (or machines) to finish a job together. Knowing the trick saves you time and math.
1. The Basic Idea
| What you know | What you can find |
|---|---|
| How long one person takes to finish a job | How long two or more people together will take |
| How long two people together take | How long one person would take |
| The job that’s done in a certain time | The remaining job after a part is finished |
Think of it like this:
If one man can paint a wall in 4 days, and another can paint the same wall in 8 days, together they paint it faster. They’re like two painters sharing the work.
2. The Key Formula – “Rate”
- Work = total job
(e.g., painting the whole wall) - Rate = how much work a person can do in one day
(e.g., 1/4 of the wall per day)
So:
Work = Rate × Time
or
Rate = Work ÷ Time
Everyday analogy
- Imagine a pizza that’s 1 whole.
- Person A can eat 1/4 of it in a day.
- Person B can eat 1/6 of it in a day.
Their combined “rate” is 1/4 + 1/6 = 5/12 of a pizza per day.
3. Adding Rates (People Working Together)
- Find each person’s rate
- Rate of A = 1 / (days A takes)
- Rate of B = 1 / (days B takes)
- Add the rates
- Combined Rate = RateA + RateB (+ RateC if more people)
- Find the time together
- Time together = 1 ÷ Combined Rate
Tip: Work is always “1” (the whole job). So the combined rate tells you how many whole jobs they finish each day.
4. Solving a Typical Problem
Problem
A can clean a room in 5 days. B can clean the same room in 10 days.
How long will they take if they work together?
Step‑by‑step
| Step | What to do | Numbers |
|---|---|---|
| 1 | Find A’s rate | 1 job ÷ 5 days = 0.20 job/day |
| 2 | Find B’s rate | 1 job ÷ 10 days = 0.10 job/day |
| 3 | Add the rates | 0.20 + 0.10 = 0.30 job/day |
| 4 | Find combined time | 1 ÷ 0.30 = 3.33 days (≈ 3 days 8 hours) |
Answer: They finish the room in about 3.3 days.
5. When One Person Works First
If one person works alone for some time, then the rest join:
- Compute work done by the first person
- WorkDone = Rate × TimeWorked
- Subtract from total to get remaining work
- Use the remaining work with the combined rate of the rest to find extra time
6. Quick Practice
Problem:
C can finish a task in 6 days. D can finish the same task in 12 days.
If they work together for 2 days, how many more days will they need to finish the job?
Solution (short)
- Rates
- C: 1/6 job/day
- D: 1/12 job/day
- Together’s rate
- 1/6 + 1/12 = 1/4 job/day
- Work done in 2 days
- 2 × 1/4 = 1/2 job
- Remaining work
- 1 – 1/2 = 1/2 job
- Time needed for remaining
- (1/2) ÷ (1/4) = 2 days
Answer: They need 2 more days.
7. Quick Reference Cheat‑Sheet
| Symbol | Meaning | Example |
|---|---|---|
| W | Total work (always 1 job) | 1 wall |
| R | Rate of a person | 1/5 wall per day |
| T | Time taken | 5 days |
| Rate = W ÷ T | T = W ÷ Rate | |
| Combined Rate = sum of individual rates |
8. Final Tips
- Always start with 1 job – the whole task.
- Convert all times to the same unit (days, hours, etc.).
- Check your answer – the time together must be less than the fastest person alone.
- Practice a few problems from past papers – the pattern is the same.
Good luck! Remember: treat each worker like a “pizza slice maker” – the more slices you have per day, the faster the pizza is finished.