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Time, Speed & Distance via Ratio

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Introduction

कई problems में speed, time या distance ratios के रूप में दिए जाते हैं। Basic relation है: Distance = Speed × Time. जब distance constant हो या time constant हो, तब direct या inverse relation apply किया जाता है।

यह पहचानना ज़रूरी है कि कौन-सी quantity same रहती है (distance या time), फिर ratio को multiplier के साथ actual numbers में बदलो, calculation करो, और Quick Check से verify करो।

Pattern: Time, Speed & Distance via Ratio

Pattern: Time, Speed & Distance via Ratio

Key rules:

• अगर distance same हो, तो time ∝ 1/speed (same distance के लिए time ratio = speed ratio का उल्टा)।
• अगर time same हो, तो distance ∝ speed (same time के लिए distance ratio = speed ratio)।

हमेशा पहले ये तय करो कि कौन-सी quantity constant है। फिर direct या inverse proportion लगाओ।

Step-by-Step Example

Question

दो कारें A और B एक ही route पर चलती हैं। उनकी speed का ratio 3 : 4 है। Distance 240 km है। दोनों कारों का समय और time ratio निकालो।

Solution

  1. Step 1: Constant पहचानो।

    Distance दोनों के लिए same है: 240 km.

  2. Step 2: Actual speeds को multiplier से लिखो।

    speedA = 3k, speedB = 4k.

  3. Step 3: Time = Distance ÷ Speed से times निकालो।

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

  4. Step 4: Time ratio simplify करो।

    timeA : timeB = (240/3k) : (240/4k)
    240 और k cancel → (1/3) : (1/4)
    दोनों को 12 से multiply → 4 : 3

  5. Step 5: Final Answer.

    Time ratio A : B = 4 : 3 k = 20 लेने पर: A = 4 hours, B = 3 hours

  6. Step 6: Quick Check.

    Distance check: (240/3k) × 3k = 240, (240/4k) × 4k = 240 ✅ Inverse check: speed ratio 3:4 → time ratio 4:3 ✅

Question

दो trains एक ही 600 km route को cover करती हैं। उनकी time ratio 5 : 6 है। Slower train को 12 hours लगते हैं। दोनों trains की speeds और speed ratio निकालो।

Solution

  1. Step 1: Constant और ratio पहचानो।

    Distance = 600 km same है। Time ratio (fast : slow) = 5 : 6.

  2. Step 2: Time ratio से actual times निकालो।

    6 parts = 12 hours → 1 part = 2 hours Fast train = 5 parts = 10 hours Slow train = 6 parts = 12 hours

  3. Step 3: Speed = Distance ÷ Time से speeds निकालो।

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

  4. Step 4: Speed ratio लिखो।

    Time ratio = 5 : 6 → Speed ratio = 6 : 5 Check: 60 : 50 → reduce → 6 : 5

  5. Step 5: Final Answer & Quick Check.

    Fast = 60 km/h, Slow = 50 km/h; Speed ratio = 6 : 5
    Quick check: 60 × 10 = 600 और 50 × 12 = 600 → दोनों distances match करते हैं ✅

Quick Variations

Same time, different speeds: अगर time same हो, तो distance ratio = speed ratio। Example: speeds 5 : 7 → distances 5 : 7.

Multiple legs: अगर यात्रा कई parts में हो और हर part की speed अलग हो, तो हर leg का time/distance अलग निकालकर add करो।

Relative speed (meeting/overtaking): Opposite direction में relative speed = sum, same direction में = difference (ये ratio के साथ combine होता है)।

Trick to Always Use

  • Step 1: पहचानो कि same क्या है - distance या time।
  • Step 2: Same time → direct proportion, Same distance → inverse proportion।
  • Step 3: Ratios को multiplier (k) से actual numbers में बदलो जब ज़रूरत हो।
  • Step 4: Distance = Speed × Time से हमेशा verify करो।

Summary

Ratio-based Time-Speed-Distance problems में:

  • Same distance: time ratio = speed ratio का inverse
  • Same time: distance ratio = speed ratio
  • Multipliers (k): totals या किसी एक value से actual numbers निकालने के लिए उपयोग
  • Quick check: हमेशा Distance = Speed × Time से verify करें

इन relations को समझने पर ratio-based TSD problems बहुत जल्दी solve हो जाते हैं।

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.