Time‑Speed‑Distance – Quick‑Start Study Notes
(Quantitative Aptitude – PCS‑J)
Goal: Master the basic relationship
Distance = Speed × Time
and all its common variations.
1. Simple Explanation
- Speed = how fast something moves (units: km/h, m/s, etc.).
- Time = how long it moves.
- Distance = how far it travels.
The three variables are linked by a single proportionality:
[ \boxed{\text{Distance} = \text{Speed} \times \text{Time}} ]
If any two of the variables are known, the third can be found by simple algebra.
Tip: Always keep units consistent (e.g., km with km/h, or m with m/s).
If units differ, convert first.
2. Key Points & Common Pitfalls
| # | Concept | What to Watch For | Quick Check |
|---|---|---|---|
| 1 | Unit Consistency | Speed in km/h & distance in km → time in hours. | km ÷ km/h = h |
| 2 | Relative Speed | When two objects move towards/away from each other, add/subtract speeds. | Speed₁ ± Speed₂ |
| 3 | Average Speed | If speed changes, use total distance ÷ total time. | Avg = D ÷ T |
| 4 | Speed Change | For problems with “increased by X%” or “decreased by Y%”, convert to decimal. | New = Old × (1 ± %/100) |
| 5 | Time Difference | When two objects start at different times, add the delay to the later start’s time. | T_total = T_early + T_late |
| 6 | Unit Conversion | 1 km = 1000 m, 1 h = 60 min, 1 min = 60 s. | Convert before plugging into formula. |
| 7 | Speed in m/s | 1 km/h = 1000/3600 m/s = 5/18 m/s. | Useful for short‑distance problems. |
3. Important Formulas
| Scenario | Formula | Example Variables |
|---|---|---|
| Basic | ( D = S \times T ) | (D) = distance, (S) = speed, (T) = time |
| Solve for Speed | ( S = \dfrac{D}{T} ) | (S) = distance ÷ time |
| Solve for Time | ( T = \dfrac{D}{S} ) | (T) = distance ÷ speed |
| Relative Speed (towards each other) | ( S_{\text{rel}} = S_1 + S_2 ) | Two objects approaching |
| Relative Speed (away from each other) | ( S_{\text{rel}} = S_1 - S_2 ) | Two objects moving apart |
| Average Speed | ( \text{Avg} = \dfrac{D_{\text{total}}}{T_{\text{total}}} ) | Sum of distances ÷ sum of times |
| Speed after a % change | ( S_{\text{new}} = S_{\text{old}} \times (1 \pm \frac{p}{100}) ) | Increase (+) or decrease (−) |
4. Worked Example
Problem (typical PCS‑J style)
Two cars start from the same point at the same time.
Car A travels east at 60 km/h, while Car B travels west at 45 km/h.
How long will it take before they are 180 km apart?
Step‑by‑Step Solution
-
Identify relative motion
- Cars move in opposite directions → relative speed = 60 + 45 = 105 km/h.
-
Apply basic formula
- Required distance (D = 180) km
- Relative speed (S_{\text{rel}} = 105) km/h
- Time (T = \dfrac{D}{S_{\text{rel}}} = \dfrac{180}{105}) hours.
-
Simplify
[ T = \frac{180}{105} = \frac{12}{7} = 1.7143\ \text{h} ] -
Convert to minutes
[ 0.7143\ \text{h} \times 60\ \text{min/h} \approx 42.86\ \text{min} ] -
Answer
- 1 hour 43 minutes (rounded to the nearest minute).
Check
- In 1.71 h, Car A covers (60 \times 1.71 \approx 102.6) km.
- Car B covers (45 \times 1.71 \approx 77.0) km.
- Sum ≈ 179.6 km ≈ 180 km → correct.
5. Quick Revision Summary
| Topic | Formula | Key Insight |
|---|---|---|
| Distance | ( D = S \times T ) | Multiply speed by time. |
| Speed | ( S = D \div T ) | Distance over time. |
| Time | ( T = D \div S ) | Distance over speed. |
| Relative Speed (opposite) | ( S_{\text{rel}} |