Time & Work – Quick & Easy Study Notes
(For Mizoram TET – Quantitative Aptitude)
1. What is “Time & Work”?
- Work – The amount of job that has to be finished (e.g., painting a wall, filling a tank).
- Rate – How fast a person (or a machine) does that work.
- Time – How long it takes to finish the job.
Analogy
Think of a bucket of water that needs to be filled.
The bucket = work.
The tap = rate (how many liters per minute).
How long the tap runs = time.
2. Basic Relationship
The three quantities are linked by a simple rule:
[ \text{Work} = \text{Rate} \times \text{Time} ]
- If you know any two, you can find the third.
3. Common Ways to Talk About Work
| Phrase | Meaning | Example |
|---|---|---|
| “X does Y units of work in one hour” | X’s rate = Y units/hour | “A can paint 4 walls in 2 hours.” |
| “A and B together finish a job in Z hours” | Combined rate = 1/Z job per hour | “A & B together finish a wall in 5 hours.” |
| “A is 3 times faster than B” | A’s rate = 3 × B’s rate | “A paints 3 walls while B paints 1 in the same time.” |
4. Simple Steps to Solve a Problem
-
Read the problem carefully.
Identify what is given: rates, times, or amounts of work. -
Convert everything to the same unit.
Usually “job” is 1 unit (whole job).
If a problem talks about “walls” or “liters”, treat each wall/liter as a part of the job. -
Express each person’s rate.
[ \text{Rate} = \frac{\text{Work done}}{\text{Time taken}} ] If “A can finish 1 job in 4 hours”, rate of A = 1/4 job per hour. -
Add rates if people work together.
Combined rate = rate of A + rate of B + … -
Use the formula
[ \text{Time} = \frac{\text{Work}}{\text{Combined Rate}} ] If the whole job is 1 unit, simply divide 1 by the combined rate. -
Check your answer.
Does the time look reasonable?
If you get a fraction, it’s fine (e.g., 0.5 hours = 30 minutes).
5. Tiny Example
Problem
A can fill a tank in 8 hours. B can fill the same tank in 12 hours.
How long will they take together?
- A’s rate = 1 job / 8 h = 0.125 job/h
- B’s rate = 1 job / 12 h ≈ 0.0833 job/h
- Combined rate = 0.125 + 0.0833 = 0.2083 job/h
- Time = 1 job / 0.2083 job/h ≈ 4.8 hours
- Answer – About 4 hours 48 minutes.
6. Common Variations
| Variation | How to Handle |
|---|---|
| “One person finishes X% of the job alone, others finish the rest” | Convert % to fraction of 1 job. |
| “Work is divided into parts” | Treat each part as a separate job. |
| “Rate changes after a certain time” | Split the job into two parts; calculate each separately. |
| “Two people finish a job in 2 hours, one of them completes it in 4 hours alone” | Let the slow person’s rate be r. Then the other’s rate = 2r (since together they finish in half the time). Solve. |
7. Practice Problems (Try before checking)
| # | Problem | Hint |
|---|---|---|
| 1 | P can paint a wall in 5 hours. Q can paint the same wall in 10 hours. How long will they paint it together? | Add rates. |
| 2 | R finishes a job in 6 days. S finishes the same job in 9 days. If they work together, how many days will it take? | Use combined rate. |
| 3 | T completes 3/4 of a job in 4 hours. How many hours would it take T alone? | Rate = (3/4)/4. |
| 4 | U and V together finish a job in 7 hours. U alone would take 14 hours. How long does V take alone? | Set up equation: 1/14 + 1/x = 1/7. |
| 5 | A can fill a cistern in 2 hours. B can fill it in 3 hours. If A works for 30 minutes, how long will B need to finish the rest? | Compute work done by A first. |
8. Quick Tips
- Always start with “rate” – it’s the easiest part to write down.
- Keep the job as 1 unit – simplifies calculations.
- Use fractions – easier to add and divide than decimals.
- Round only at the end – avoid rounding errors.
- Draw a small diagram if the problem mentions “part A” and “part B”; it helps visualise.
9. Summary
| Step | What to Do | Example |
|---|---|---|
| 1 | Identify given rates or times | “1 job in 8 hrs” → rate = 1/8 |
| 2 | Convert |