Speed, Time & Distance: Solved Examples- 1 | General Aptitude for GATE - Mechanical Engineering PDF Download

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Que 1: Two friends A and B simultaneously start running around a circular track. They run in the same direction. A travels at 6 m/s and B runs at b m/s. If they cross each other at exactly two points on the circular track and b is a natural number less than 30, how many values can b take? 
(A) 3
(B) 4
(C) 7
(D) 5

Correct Answer is Option (A).

• Let track length be equal to T.
• Time taken to meet for the first time = 
• Time taken for a lap for A = T/6 
• Time taken for a lap for B = T/b 
• So, time taken to meet for the first time at the starting point = LCM
  
• Number of meeting points on the track = Time taken to meet at starting point/Time taken for first meeting = Relative speed / HCF (6,b).
• So, in essence we have to find values for b such that
  
• The question is " If two people cross each other at exactly two points on the circular track and b is a natural number less than 30, how many values can b take?"
• b = 2, 10, 18 satisfy this equation. So, there are three different values that b can take.
  Hence, the answer is 3.

Que 2: Running at a speed of 60 km per hour, a train passed through a 1.5 km long tunnel in two minutes. What is the length of the train?

(A) 250 m
(B) 500 m
(C) 1000 m
(D) 1500 m

Correct Answer is Option (B).
Speed is 60 km per hour, Train passed through a 1.5 km long tunnel in two minutes
Formula used: Distance = Speed × Time
Let the length of the train be L 
According to the question, 
Total distance = 1500 m + L 
Speed = 60(5/18) 
⇒ 50/3 m/sec
Time = 2 × 60 = 120 sec 
1500 + L = (50/3)× 120 
L = 2000 - 1500 
L = 500 m
∴ The length of the train is 500 m.

Que 3: Three cars leave A for B in equal time intervals. They reach B simultaneously and then leave for Point C which is 240 km away from B. The first car arrives at C an hour after the second car. The third car, having reached C, immediately turns back and heads towards B. The first and the third car meet a point that is 80 km away from C. What is the difference between the speed of the first and the third car?
(A) 60 kmph
(B) 20 kmph
(C) 40 kmph
(D) 80 kmph

Correct Answer is Option (A).

• Let V1, V2 and V3 be the speeds of the cars.
  
  
• V₃ = 2V₁
• Condition I states that the cars leave in equal intervals of time and arrive at the same time. Or, the difference in the time taken between cars 1 and 2 should be equal to the time taken between cars 2 and 3.
• We get
  
• As the second car arrived at C an hour earlier than the first, we get a second equation
  
  
• The third car covered 240 + 80 kms when the first one covered 240 – 80 kms. 
• Therefore,
  This gives us V₃ = 2V₁
• From condition 1, we have
  
• Substituting V₃ = 2V₁, this gives us
  = 1 or V₁ = 60 kmph
  => V₂ = 80 kmph and V₃ = 120 kmph
• The question is "What is the difference between the speed of the first and the third car?"
  Hence, the answer is 60 kmph.

Que 4: Three friends A, B and C decide to run around a circular track. They start at the same time and run in the same direction. A is the quickest and when A finishes a lap, it is seen that C is as much behind B as B is behind A. When A completes 3 laps, C is the exact same position on the circular track as B was when A finished 1 lap. Find the ratio of the speeds of A, B and C?
(A) 5 : 4 : 2
(B) 4 : 3 : 2
(C) 5 : 4 : 3
(D) 3 : 2 : 1

Correct Answer is Option (C).

• Let track length be equal to T. When A completes a lap, let us assume B has run a distance of (t - d). At this time, C should have run a distance of (t - 2d).
• After 3 laps C is in the same position as B was at the end of one lap. So, the position after 3t - 6d should be the same as t - d. Or, C should be at a distance of d from the end of the lap.
• C will have completed less than 3 laps (as he is slower than A), so he could have traveled a distance of either t - d or 2t - d.
  => 3t - 6d = t - d
  => 2t = 5d
  => d = 0.4t
• The distances covered by A, B and C when A completes a lap will be t, 0.6t and 0.2t respectively. Or, the ratio of their speeds is 5 : 3 : 1.
• In the second scenario, 3t - 6d = 2t - d => t = 5d => d = 0.2t.
  The distances covered by A, B and C when A completes a lap will be t, 0.8t and 0.6t respectively. Or, the ratio of their speeds is 5 : 4 : 3.
• The question is " Find the ratio of the speeds of A, B and C?"
  The ratio of the speeds of A, B and C is either 5 : 3 : 1 or 5 : 4 : 3.
• Hence, the answer is 5 : 4 : 3

Que 5: Mr. X decides to travel from Delhi to Gurgaon at a uniform speed and decides to reach Gurgaon after T hr. After 30 km, there is some engine malfunction and the speed of the car becomes 4/5th of the original speed. So, he travels the rest of the distance at a constant speed 4/5th of the original speed and reaches Gurgaon 45 minutes late. Had the same thing happened after he travelled 48 km, he would have reached only 36 minutes late. What is the distance between Delhi and Gurgaon?
(A) 90 km
(B) 120 km
(C) 20 km
(D) 40 km

Correct Answer is Option (B).

• Let the distance from Delhi to Gurgaon be ‘d’ km.
• The first 30 km he travels at his usual speed. However, the remaining ‘d-30’ km he travels at a reduced speed.
• To travel ‘d’ km he usually takes T hr. Therefore, to travel ‘d - 30’ km he should ideally take
  hr.
• However, this is only if he travels at his usual speed. It is given that he traveled only at 4/5th of his usual speed. Because of this he would have taken 5/4th of the time to travel the remaining distance, i.e., he takes  1/4th of the time extra. This is given to be 45 minutes (or 3/4th hr)
  
  
• On the other hand, had the same thing happened after he travelled 48 km, he would have reached only 36 minutes or 34 hrs late. Hence,
  
  
• Dividing (1) by (2) and solving for d, we get d = 120 km.
• The question is What is the distance between Delhi and Gurgaon? we get d = 120 km.
• Hence, the answer is d = 120 km.

Que 6: Two friends A and B leave City P and City Q simultaneously and travel towards Q and P at constant speeds. They meet at a point in between the two cities and then proceed to their respective destinations in 54 minutes and 24 minutes respectively. How long did B take to cover the entire journey between City Q and City P?
(A) 60
(B) 36
(C) 24
(D) 48

Correct Answer is Option (A).

• Let us assume Car A travels at a speed of a and Car B travels at a speed of b.
  Further, let us assume that they meet after t minutes.
• Distance traveled by car A before meeting car B = a * t. Likewise distance traveled by car B before meeting car A = b * t.
• Distance traveled by car A after meeting car B = a * 54. Distance traveled by car B after meeting car A = 24 * b.
• Distance traveled by car A after crossing car B = distance traveled by car B before crossing car A (and vice versa).
  => at = 54b ---------- (1)
  and bt = 24a -------- (2)
• Multiplying equations 1 and 2
  we have ab * t² = 54 * 24 * ab
  => t² = 54 * 24
  => t = 36
• The question is " How long did B take to cover the entire journey between City Q and City P?"
• So, both cars would have traveled 36 minutes prior to crossing each other. Or, B would have taken 36 + 24 = 60 minutes to travel the whole distance.
• Hence, the answer is 60 mins.

Que 7: Two trains of equal lengths take 13 seconds and 26 seconds, respectively, to cross a pole. If these trains are moving in the same direction, then how long will they take to cross each other?

(A) 40 seconds
(B) 50 seconds
(C) 39 seconds
(D) 52 seconds

Correct Answer is Option (D).

Given: Train A takes 13 seconds to cross a pole. Train B takes 26 seconds to cross a pole.
Concept: Speed = Distance / Time When two trains are moving in the same direction, their relative speed is the difference of their speeds.
Let the length of each train be L. 
→ Speed of train A = L/13, speed of train B = L/26. 
When the two trains cross each other, the total distance covered is 2L (length of train A + length of train B). 
Relative speed of the two trains = speed of train A - speed of train B = L/13 - L/26 = L/26.
Time taken to cross each other = total distance / relative speed = 2L / (L/26) = 52 seconds. Hence, the two trains take 52 seconds to cross each other.

Que 8: A policeman noticed a thief from 300 m. The thief started running and the policeman was chasing him. The thief and the policeman ran at the speeds of 8 km/h and 9 km/h, respectively. What was the distance between them after 3 minutes?
(A) 225 m
(B) 250 m
(C) 300 m
(D) 200 m

Correct Answer is Option (B)

Distance between policeman and thief in the starting = 300 m
Speed of policeman = 9 km/hr Speed of thief = 8 km/hr
If the speed of a policeman and thief is x km/hr and y km/hr, then Relative speed, if same directions = (x - y) km/hr Distance between them after n hrs = (x - y) × n 1 km/hr = 5/18 m/sec 1 min = 60 sec
3 min = 3 × 60 = 180 seconds 
Distance between policeman and thief in starting = 300 m 
Relative speed of policeman and thief, if same directions = (9 - 8) = 1 × (5/18) = (5/18) m/sec 
Distance covered in 180 seconds = (5/18) × 180 = 50 m 
Distance between them after 180 seconds= 300 - 50 = 250 m
∴ Distance between them after 3 min is 250 m.