Introduction
Many problems give speed, time or distance in the form of ratios. The basic relation is Distance = Speed × Time. Use direct or inverse relations depending on whether distance or time is held constant.
Learn to spot what stays the same (distance or time), convert ratios into actual numbers using a multiplier, perform the calculation, and then verify with a quick check.
Pattern: Time, Speed & Distance via Ratio
Pattern
Key rules:
• If distance is same, time ∝ 1/speed (for same distance, time ratio = inverse of speed ratio).
• If time is same, distance ∝ speed (for same time, distance ratio = speed ratio).
Always identify which quantity is constant before applying direct or inverse proportion.
Step-by-Step Example
Question
Two cars A and B travel the same route. Their speeds are in the ratio 3 : 4. If the distance is 240 km, find the time taken by each car and the time ratio.
Solution
-
Step 1: Identify the constant.
The distance is the same for both cars: 240 km. -
Step 2: Represent actual speeds using a multiplier.
Let speedA = 3k and speedB = 4k. -
Step 3: Use Time = Distance ÷ Speed to find times.
timeA = 240 ÷ (3k) = 240 / 3k.
timeB = 240 ÷ (4k) = 240 / 4k. -
Step 4: Simplify the ratio of times.
timeA : timeB = (240 / 3k) : (240 / 4k).
Cancel 240 and k → (1/3) : (1/4).
Multiply both terms by 12 → 4 : 3. -
Step 5: Final Answer.
Time ratio A : B = 4 : 3. Actual times can be calculated by substituting k = 20 → A = 4 hours, B = 3 hours. -
Step 6: Quick Check.
Check distances: (240/3k) × 3k = 240 and (240/4k) × 4k = 240 ✅. Inverse relation check: speeds 3:4 → times 4:3 ✅.
Question
Two trains cover the same 600 km route. Their times are in the ratio 5 : 6. If the slower train takes 12 hours, find the speeds of both trains and the speed ratio.
Solution
-
Step 1: Identify the constant and given ratio.
Distance = 600 km is the same. Time ratio (fast : slow) = 5 : 6. -
Step 2: Find actual times using the ratio.
6 parts = 12 hours → 1 part = 2 hours. Fast train = 5 parts = 10 hours. Slow train = 6 parts = 12 hours. -
Step 3: Compute speeds using Speed = Distance ÷ Time.
speedfast = 600 ÷ 10 = 60 km/h.
speedslow = 600 ÷ 12 = 50 km/h. -
Step 4: Speed ratio (inverse of time ratio).
Time ratio = 5 : 6 → Speed ratio = 6 : 5. Confirm: 60 : 50 reduces to 6 : 5. -
Step 5: Final Answer & Quick Check.
Speeds - Fast = 60 km/h, Slow = 50 km/h; Speed ratio = 6 : 5.
Quick check: 60 × 10 = 600 km and 50 × 12 = 600 km → both distances match ✅.
Quick Variations
Same time, different speeds: If time is same, distance ratio = speed ratio. Example: speeds 5 : 7 for same time → distances covered are 5 : 7.
Multiple legs: For journeys with several legs at different speeds, compute each leg separately and add total time or distance as required.
Relative speed (meeting/overtaking): Use relative speed = sum/difference of speeds when objects move in opposite/same directions (this is usually combined with ratio ideas).
Trick to Always Use
- Step 1: Decide which quantity is constant: same distance or same time.
- Step 2: Use direct proportion when time is same (distance ∝ speed) and inverse proportion when distance is same (time ∝ 1/speed).
- Step 3: Represent ratios with a multiplier (k) to obtain actual numbers when needed.
- Step 4: Always verify by plugging values into Distance = Speed × Time.
Summary
Summary
For ratio-style Time-Speed-Distance problems:
- Same distance: time ratio = inverse of speed ratio.
- Same time: distance ratio = speed ratio.
- Use multipliers (k): convert ratios to numbers when a total or one value is given.
- Quick check: verify using Distance = Speed × Time.
Mastering these relations makes most ratio-based TSD problems fast and reliable.
Hint: Take inverse ratio of speeds when distance is fixed.
Common Mistakes: Writing same ratio as speeds instead of inverting.