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Time, Speed & Distance via Ratio

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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: Time, Speed & Distance via Ratio

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

  1. Step 1: Identify the constant.

    The distance is the same for both cars: 240 km.

  2. Step 2: Represent actual speeds using a multiplier.

    Let speedA = 3k and speedB = 4k.

  3. Step 3: Use Time = Distance ÷ Speed to find times.

    timeA = 240 ÷ (3k) = 240 / 3k.
    timeB = 240 ÷ (4k) = 240 / 4k.

  4. 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.

  5. 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.

  6. 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

  1. Step 1: Identify the constant and given ratio.

    Distance = 600 km is the same. Time ratio (fast : slow) = 5 : 6.

  2. 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.

  3. Step 3: Compute speeds using Speed = Distance ÷ Time.

    speedfast = 600 ÷ 10 = 60 km/h.
    speedslow = 600 ÷ 12 = 50 km/h.

  4. Step 4: Speed ratio (inverse of time ratio).

    Time ratio = 5 : 6 → Speed ratio = 6 : 5. Confirm: 60 : 50 reduces to 6 : 5.

  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

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.

Practice

(1/5)
1. Two cars travel equal distances at speeds 30 km/h and 40 km/h. Find the ratio of times taken.
easy
A. 3 : 4
B. 4 : 3
C. 2 : 3
D. 3 : 2

Solution

  1. Step 1: Use the inverse relation

    When distance is constant, time ∝ 1/speed.

  2. Step 2: Invert the speed ratio

    Speeds = 30 : 40 = 3 : 4 → Times = 4 : 3.

  3. Final Answer:

    4 : 3 → Option B

  4. Quick Check:

    Suppose distance = 120 → Times = 4h, 3h → 4 : 3 ✅
Hint: Take inverse ratio of speeds when distance is fixed.
Common Mistakes: Writing same ratio as speeds instead of inverting.
2. Two trains travel for the same time at speeds 45 km/h and 60 km/h. Find ratio of distances covered.
easy
A. 3 : 2
B. 2 : 3
C. 3 : 4
D. 4 : 5

Solution

  1. Step 1: Use direct proportionality

    When time is constant, distance ∝ speed.

  2. Step 2: Reduce the speed ratio

    Speeds = 45 : 60 = 3 : 4.

  3. Final Answer:

    3 : 4 → Option C

  4. Quick Check:

    Time = 1 hour → Distances = 45, 60 → 3 : 4 ✅
Hint: Use direct speed ratio when time is same.
Common Mistakes: Inverting ratio when not needed.
3. Two friends walk equal distances. Their speeds are in the ratio 5 : 7. Find ratio of times taken.
medium
A. 5 : 7
B. 7 : 5
C. 12 : 5
D. 2 : 3

Solution

  1. Step 1: Remember inverse relation for equal distances

    Distance constant → time ∝ 1/speed.

  2. Step 2: Flip the given speed ratio

    Speeds = 5 : 7 → Times = 7 : 5.

  3. Final Answer:

    7 : 5 → Option B

  4. Quick Check:

    Distance = 35 → Times = 7, 5 → 7 : 5 ✅
Hint: Flip speed ratio to get time ratio when distance is same.
Common Mistakes: Confusing which to invert.
4. A car covers distances in ratio 2 : 3 in equal times. What is the ratio of their speeds?
medium
A. 2 : 3
B. 3 : 2
C. 4 : 5
D. Cannot be determined

Solution

  1. Step 1: Relate distance and speed for equal time

    If time is same → distance ∝ speed.

  2. Step 2: Use the given distance ratio as speed ratio

    Distance ratio = 2 : 3 → Speeds = 2 : 3.

  3. Final Answer:

    2 : 3 → Option A

  4. Quick Check:

    Time = 1 hr → Distances = 2, 3 → Speeds = 2 : 3 ✅
Hint: Distances and speeds share same ratio when time is constant.
Common Mistakes: Inverting ratio incorrectly.
5. Two persons cover equal distances in 4 hours and 6 hours. Find the ratio of their speeds.
medium
A. 2 : 3
B. 3 : 2
C. 4 : 6
D. Cannot be determined

Solution

  1. Step 1: Use inverse proportionality of speed to time

    Distance constant → speed ∝ 1/time.

  2. Step 2: Simplify times and invert

    Times = 4 : 6 = 2 : 3 → Speeds = 3 : 2.

  3. Final Answer:

    3 : 2 → Option B

  4. Quick Check:

    Distance = 12 → Speeds = 3, 2 → 3 : 2 ✅
Hint: Simplify time ratio, then invert to get speeds.
Common Mistakes: Not reducing ratio before inversion.