Time & Work – Quick & Easy Study Notes
What we’ll learn
- How “work” is measured.
- How to use rates (speed of work).
- How to combine workers or machines.
- Simple steps to solve the most common “time & work” questions.
1. The Basics
| Term | Everyday Analogy | What it Means | Formula |
|---|---|---|---|
| Work (W) | Finishing a house | Total amount of job to be done | – |
| Rate (R) | Speed of a runner | How fast work can be done per unit time | R = W / T |
| Time (T) | How long the runner runs | Hours, days, minutes, etc. | T = W / R |
Think of it like this:
If we have a cake (work) and a baker (rate), the time is how long it takes the baker to finish the cake.
2. One Person (or Machine) Working Alone
- Given: Rate (e.g., 2 units/hour).
- Want: Time to finish
Wunits. - Use:
T = W / R.
Tiny Example
- A machine can paint 5 m² per hour.
- Painting a wall of 20 m².
T = 20 / 5 = 4hours → the wall is finished in 4 hours.
3. Two (or More) People Working Together
- Rates add up.
- If A paints 3 m²/hour and B paints 2 m²/hour, together they paint
3 + 2 = 5m²/hour.
- If A paints 3 m²/hour and B paints 2 m²/hour, together they paint
Step‑by‑Step
- Find each person’s rate.
- Add the rates → combined rate
R_total. - Use the single‑person formula with
R_total.
Tiny Example
- A can finish a job in 10 hours → rate =
1 job / 10 hr = 0.1 job/hr. - B can finish the same job in 15 hours → rate =
0.0667 job/hr. - Together:
0.1 + 0.0667 = 0.1667 job/hr. - Time for both together:
T = 1 / 0.1667 ≈ 6hours.
4. One Person Leaves/Stops Early
- Work done before leaving =
Rate × Time. - Remaining work =
Total – Work done. - Time for the remaining work =
Remaining / Rate of the other person.
Tiny Example
- A works 4 hours on a job that takes 10 hours alone → does
4/10 = 0.4of the job. - Remaining
1 – 0.4 = 0.6of the job. - B alone will finish it in
10 × 0.6 = 6hours.
5. One Person’s Rate Changes (e.g., Gets Faster)
- Calculate the new rate first.
- Then treat it as a new single‑person problem.
Tiny Example
- A usually paints 1 m² in 2 hours → rate =
0.5 m²/hr. - After training, he can paint 1 m² in 1 hour → new rate
1 m²/hr. - For a 10 m² wall, time =
10 / 1 = 10hours → much faster!
6. Mixed Work (Different Tasks)
If the job consists of several parts (e.g., painting, wiring, plumbing), treat each part separately:
- Find the time each person needs for each part.
- Add the times if they work sequentially.
- If they work simultaneously, use combined rates for each part.
7. Quick Checklist for Solving
| Step | What to Do |
|---|---|
| 1 | Identify all rates or times given. |
| 2 | Convert everything to the same unit (hours, minutes). |
| 3 | If rates are given as “time to finish whole job”, compute rate = 1 / time. |
| 4 | Add rates for people/machines working together. |
| 5 | Use T = W / R for the final answer. |
| 6 | Check if the answer makes sense (e.g., not negative, reasonable size). |
8. Practice Problems (Try One)
-
Problem
- A can finish a task in 12 hours.
- B can finish the same task in 18 hours.
- How long will they take together?
-
Problem
- A works alone for 3 hours on a task that takes 10 hours alone.
- B works alone for 4 hours afterward.
- How long did B take to finish the remaining part?
Answer them using the steps above!
9. Common Mistakes to Avoid
- Mixing units (hours vs minutes).
- Adding times instead of rates when people work together.
- Forgetting to convert “time to finish whole job” into a rate.
- Assuming work is always linear; it’s usually linear in these problems.
10. Takeaway
- Work = Rate × Time.
- Rates add when people/machines work together.
- Just plug numbers into the simple formulas, use clear units, and you’ll solve any time‑and‑work question.
Happy practicing!