Time & Work – SAT Quantitative Aptitude Study Notes
1. Simple Explanation
- Work (W): The amount of job or task that has to be done.
- Rate of Work (R): How fast a person/thing can do the job (units of work per unit time).
- Time (T): How long it takes to finish the job.
The fundamental relationship is
[ \boxed{W = R \times T} ]
If you know any two of the three variables (W, R, T), you can find the third.
In most SAT problems the job size is 1 unit (i.e., “the whole job”), so we often set (W = 1) and work only with rates and times.
2. Key Points & Formulas
| Concept | Formula | How to Use |
|---|---|---|
| Rate | ( R = \frac{1}{T} ) | Convert a worker’s finish‑time into a rate (work per day). |
| Combined Rate | ( R_{\text{total}} = R_1 + R_2 + \cdots ) | Sum the rates of all workers working together. |
| Time for Combined Work | ( T = \frac{1}{R_{\text{total}}} ) | Invert the combined rate to get the time. |
| Work Done by One Worker in (t) days | ( W = \frac{t}{T_{\text{worker}}} ) | Use the worker’s total finish‑time. |
| Work Remaining After Some Days | ( W_{\text{remaining}} = 1 - \frac{t}{T_{\text{worker}}} ) | Subtract the completed portion from the whole job. |
| Work Done by Two Workers for a Portion | ( W = \frac{t}{T_A} + \frac{t}{T_B} ) | Add their individual contributions. |
| Work‑Rate Diagram (Optional) | Draw a table of “Days” vs “Work Done” | Helps visualize cumulative work, especially for multi‑step problems. |
Common Tricks
- Convert to a common denominator when adding rates.
Example: ( \frac{1}{12} + \frac{1}{18} = \frac{3}{36} + \frac{2}{36} = \frac{5}{36} ). - Use “whole job = 1” to simplify.
- When one worker stops:
- Find work done until stopping (using rate * time).
- Subtract from 1 to get remaining work.
- Compute time for remaining work using the new combined rate.
- Check units (days, hours, etc.). SAT problems usually use days but sometimes use hours or minutes; keep track.
3. Worked Example
Problem (SAT‑style)
A can finish a job in 12 days, and B can finish the same job in 18 days.
They work together for 4 days.
After that, A leaves and B continues alone.
How many total days will it take to finish the job?
Step‑by‑Step Solution
| Step | Calculation | Explanation |
|---|---|---|
| 1. Find each worker’s rate | (R_A = \frac{1}{12}) work/day, (R_B = \frac{1}{18}) work/day | Rate = 1 / finish‑time |
| 2. Combined rate when both work | (R_{AB} = R_A + R_B = \frac{1}{12} + \frac{1}{18}) | Sum rates |
| Convert to common denominator: ( \frac{3}{36} + \frac{2}{36} = \frac{5}{36}) | ||
| So (R_{AB} = \frac{5}{36}) work/day | ||
| 3. Work done in first 4 days | (W_{\text{first}} = R_{AB} \times 4 = \frac{5}{36} \times 4 = \frac{20}{36} = \frac{5}{9}) | Rate × time |
| 4. Remaining work | (W_{\text{rem}} = 1 - \frac{5}{9} = \frac{4}{9}) | Whole job minus completed portion |
| 5. Work rate of B alone | (R_B = \frac{1}{18}) work/day | Same as step 1 |
| 6. Time B needs to finish remaining work | (T_{\text{rem}} = \frac{W_{\text{rem}}}{R_B} = \frac{4/9}{1/18} = \frac{4}{9} \times 18 = 8) days | Work / rate |
| 7. Total time | (T_{\text{total}} = 4 + 8 = 12) days | Sum time spent together and alone |
Answer: 12 days.
4. Quick Revision Summary
| Concept | Quick Flash‑card |
|---|---|
| Rate | Rate = 1 / Finish‑time |
| Combined Rate | Total rate = sum of individual rates |
| Time | Time = 1 / Combined rate |
| Work Done | Work = Rate × Time |
| Remaining Work | Remaining = 1 – Work_done |
| Work After a Worker Leaves | Time_left = Remaining / (Remaining workers’ rate) |
| Common Denominator Trick | Add fractions by converting to a common denominator. |
| Key Unit Check | Always keep work as a fraction of the whole job; |