Time & Work – Quick & Easy Study Notes
1. What is “Work” in this context?
- Work = the whole job you want to finish (e.g., painting a wall, filling a tank).
- Think of it as a total task that can be split into smaller pieces.
2. How fast does a person do the work?
- Rate = how much of the job a person can finish in one hour.
- Everyday analogy:
- If you can read 20 pages in an hour, your rate is 20 pages per hour.
- If you can finish the whole painting in 4 hours, your rate is 1/4 of the wall per hour.
3. The basic relationship
[ \text{Work} = \text{Rate} \times \text{Time} ]
- Work is always 1 job (the whole task).
- Rate tells you how many “parts of the job” you finish each hour.
- Time is how long you work.
Example
A person can paint the wall in 4 hours.
- Rate = ( \frac{1}{4} ) of the wall per hour.
- If they paint for 2 hours:
[ \text{Work done} = \frac{1}{4} \times 2 = \frac{1}{2} ] They finish half the wall.
4. Adding rates when people work together
When two people work together, their rates add.
- If A’s rate = ( \frac{1}{4} ) wall/hour and B’s rate = ( \frac{1}{6} ) wall/hour,
[ \text{Combined rate} = \frac{1}{4} + \frac{1}{6} = \frac{5}{12} ] - Time to finish the wall together: [ \text{Time} = \frac{\text{Work}}{\text{Combined rate}} = \frac{1}{5/12} = \frac{12}{5} \text{ hours} = 2.4 \text{ hours} ]
5. Common Question Types
| Question Type | What to Look For | Quick Trick |
|---|---|---|
| Two people together | Find each person’s rate, add them, then use (\text{Time} = 1 / \text{combined rate}). | Add the fractions first. |
| One person works, then the other | Split the job into two parts: part 1 done together, part 2 done by one person. | Use “remaining work” = 1 – (work done together). |
| Percentage of work done | Convert percentages to fractions of the whole job. | 30% = 0.3 of the job. |
| Work left after a certain time | Subtract the work done from the whole job. | Work left = 1 – (rate × time). |
6. Tiny Practice Problem
Problem:
- A can fill a bucket in 8 hours.
- B can fill the same bucket in 12 hours.
- They work together for 2 hours.
- Then B leaves, and A works alone until the bucket is full.
- How long did A work after B left?
Solution Steps (short sentences)
- Find rates
- A’s rate = ( \frac{1}{8} ) bucket/hr
- B’s rate = ( \frac{1}{12} ) bucket/hr
- Combined rate
- ( \frac{1}{8} + \frac{1}{12} = \frac{5}{24} ) bucket/hr
- Work done in first 2 hrs
- ( \frac{5}{24} \times 2 = \frac{5}{12} ) bucket
- Work left
- 1 – ( \frac{5}{12} = \frac{7}{12} ) bucket
- A works alone
- Time = remaining work ÷ A’s rate
- ( \frac{7}{12} \div \frac{1}{8} = \frac{7}{12} \times 8 = \frac{56}{12} = 4 \frac{2}{3} ) hours
Answer: A worked 4 ⅔ hours after B left.
7. Quick Tips for the SAT
- Read the problem carefully – note who works, for how long, and who leaves.
- Write down each rate – even if it’s a fraction.
- Add rates when working together – always use fractions.
- Use the formula
[ \text{Time} = \frac{1}{\text{Combined rate}} \quad\text{or}\quad \text{Work done} = \text{Rate} \times \text{Time} ] - Check units – hours, fractions of the job, etc.
8. Final Mini‑Recap
- Work = 1 whole job.
- Rate = part of job finished each hour.
- Time = hours worked.
- Formula: Work = Rate × Time.
- Together: add rates.
- Practice: use tiny numbers and short sentences to keep it clear.
Keep practicing with simple numbers, and the SAT “Time & Work” questions will feel like a