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TIME & DISTANCE

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In this module we will deal with basic concepts of time and distance, speed, average speed, conversion from km/h to m/s and vice versa. This chapter will form the basis of further concept of relative speed which is used in train and boat problems.

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Important Formulas

- Speed=Distance/Time
- Distance=Speed × Time
- Time=Distance/Speed
- To convert Kilometers per Hour(km/hr) to Meters per Second(m/s)

x km/hr=(x×5)/18m/s - To convert Meters per Second(m/s) to Kilometers per Hour(km/hr)

x m/s=(x×18)/5 km/hr - If a car covers a certain distance at x kmph and an equal distance at y kmph, the average speed of the whole journey = 2xy/(x+y) kmph
- Speed and time are inversely proportional (when distance is constant) ⇒Speed ∝ 1/Time (when distance is constant)
- If the ratio of the speeds of A and B is a : b, then the ratio of the times taken by them to cover the same distance is 1/a:1/b or b : a

Solved Examples

Level 1

1. A person crosses a 600 m long street in 5 minutes. What is his speed in km per hour? | |

A. 8.2 | B. 4.2 |

C. 6.1 | D. 7.2 |

Answer : Option D

Explanation :

Distance = 600 meter

time = 5 minutes = 5 x 60 seconds = 300 seconds

Speed = distance/time=600/300=2m/s=(2×18)/5 km/hr=36/5 km/hr=7.2 km/hr

2. Two boys starts from the same place walking at the rate of 5 kmph and 5.5 kmph respectively in the same direction. What time will they take to be 8.5 km apart? | |

A. 17 hr | B. 14 hr |

C. 12 hr | D. 19 hr |

Answer : Option A

Explanation :

Relative speed = 5.5 – 5 = .5 kmph (because they walk in the same direction)

distance = 8.5 km

Time = distance/speed=8.5/.5=17 hr.

3. Walking 6/7 | |

A. 1 hr 42 min | B. 1 hr |

C. 2 hr | D. 1 hr 12 min |

Answer : Option D

Explanation :

New speed = 6/7 of usual speed

Speed and time are inversely proportional.

Hence new time = 7/6 of usual time

Hence, 7/6 of usual time – usual time = 12 minutes

=> 1/6 of usual time = 12 minutes => usual time = 12 x 6 = 72 minutes = 1 hour 12 minutes

4. A man goes to his office from his house at a speed of 3 km/hr and returns at a speed of 2 km/hr. If he takes 5 hours in going and coming, what is the distance between his house and office? | |

A. 3 km | B. 4 km |

C. 5 km | D. 6 km |

Answer : Option D

Explanation :

If a car covers a certain distance at x kmph and an equal distance at y kmph,the average speed of the whole journey = 2xy/(x+y) kmph

Hence, average speed = (2×3×2)/(2+3)=12/5 km/hr .

Total time taken = 5 hours

⇒Distance travelled = (12/5)×5=12 km

⇒Distance between his house and office =12/2=6 km

5. If a person walks at 14 km/hr instead of 10 km/hr, he would have walked 20 km more. What is the actual distance travelled by him? | |

A. 80 km | B. 70 km |

C. 60 km | D. 50 km |

Answer : Option D

Explanation :

Assume that the person would have covered x km if travelled at 10 km/hr

⇒Speed = Distance/Time=x/10….. (Equation1)

Give that the person would have covered (x + 20) km if travelled at 14 km/hr

⇒Speed = Distance/Time=(x+20)/14….. (Equation2)

From Equations 1 and 2,

X/10=(x+20)/14 ⇒14x=10x+200 ⇒4x=200 ⇒x=200/4=50

6. A car travels at an average of 50 miles per hour for 212 hours and then travels at a speed of 70 miles per hour for 112 hours. How far did the car travel in the entire 4 hours? | |

A. 210 miles | B. 230 miles |

C. 250 miles | D. 260 miles |

Answer : Option B

Explanation :

Speed1 = 50 miles/hour

Time1 = 2*(1/2) hour=5/2 hour

⇒ Distance1 = Speed1 × Time1 = (50×5)/2=25×5=125 miles

⇒Speed2 = 70 miles/hour

Time2 = 1*1/2 hour=3/2 hour

Distance2 = Speed2 × Time2 = 70×3/2=35×3=105 miles

Total Distance = Distance1 + Distance2 =125+105=230 miles

7. Sound is said to travel in air at about 1100 feet per second. A man hears the axe striking the tree, | |

A. 1800 ft | B. 2810 ft |

C. 3020 ft | D. 2420 ft |

Answer : Option D

Explanation :

Speed of the sound = 1100 ft/s ⇒Time = ^{11}/_{5} second

Distance = Speed × Time = 1100 ×11/5=220×11=2420 ft

8. A man walking at the rate of 5 km/hr crosses a bridge in 15 minutes. What is the length of the bridge (in meters)? | |

A. 1250 | B. 1280 |

C. 1320 | D. 1340 |

Answer : Option A

Explanation :

Speed = 5 km/hr

Time = 15 minutes = 1/4 hour

Length of the bridge = Distance Travelled by the man

= Speed × Time = 5×1/4 km

=5×1/4×1000 metre=1250 metre

Level 2

1. A man takes 5 hours 45 min in walking to a certain place and riding back. He would have gained 2 hours by riding both ways. The time he would take to walk both ways is | |

A. 11 hrs | B. 8 hrs 45 min |

C. 7 hrs 45 min | D. 9 hts 20 min |

Answer : Option C

Explanation :

Given that time taken for riding both ways will be 2 hours lesser than the time needed for waking one way and riding back 2. A man complete a journey in 10 hours. He travels first half of the journey at the rate of 21 km/hr and second half at the rate of 24 km/hr. Find the total journey in km. | |

A. 121 km | B. 242 km |

C. 224 km | D. 112 km |

Answer : Option C

Explanation :

distance = speed x time

Let time taken to travel the first half = x hr

then time taken to travel the second half = (10 – x) hr

Distance covered in the first half = 21x

Distance covered in the second half = 24(10 – x)

But distance covered in the first half = Distance covered in the second half

=> 21x = 24(10 – x) => 21x = 240 – 24x => 45x = 240 => 9x = 48 => 3x = 16 ⇒x=16/3

Hence Distance covered in the first half = 21x=21×16/3=7×16=112 km. Total distance = 2×112=224 km

3. A car traveling with 5/7 of its actual speed covers 42 km in 1 hr 40 min 48 sec. What is the actual speed of the car? | |

A. 30 km/hr | B. 35 km/hr |

C. 25 km/hr | D. 40 km/hr |

Answer : Option B

Explanation :

Time = 1 hr 40 min 48 sec = 1hr +40/60hr+48/3600hr=1+2/3+1/75=126/75hr

Distance = 42 km Speed=distance/time=42(126/75) =42×75/126

⇒5/7 of the actual speed = 42×75/126

⇒actual speed = 42×75/126×7/5=42×15/18=7×15/3=7×5=35 km/hr

4. A man covered a certain distance at some speed. If he had moved 3 kmph faster, he would have taken 40 minutes less. If he had moved 2 kmph slower, he would have taken 40 minutes more. What is the the distance in km? | |

A. 36 | B. 38 |

C. 40 | D. 42 |

Answer : Option C

Explanation :

Let the distance be x km , the speed in which he moved = v kmph

Time taken when moving at normal speed – time taken when moving 3 kmph faster = 40 minutes

⇒x/v−x/(v+3)=40/60 ⇒x[1/v−1/(v+3)]=2/3 ⇒x[(v+3−v)/v(v+3)]=2/3

⇒2v(v+3)=9x…………….(Equation1)

Time taken when moving 2 kmph slower – Time taken when moving at normal speed = 40 minutes

⇒x/(v−2)−x/v=40/60 ⇒x[1/(v−2)−1/v]=2/3

⇒x[(v−v+2)/v(v−2)]=2/3 ⇒x[2/v(v−2)]=2/3

⇒x[1/v(v−2)]=1/3 ⇒v(v−2)=3x…………….(Equation2)

Equation1/Equation2

⇒2(v+3)/(v−2)=3 ⇒2v+6=3v−6⇒v=12

Substituting this value of v inEquation1⇒2×12×15=9x

=>x= (2×12×15)/9= (2×4×15)/3=2×4×5=40. Hence distance = 40 km

5. In covering a distance of 30 km, Arun takes 2 hours more than Anil. If Arun doubles his speed, then he would take 1 hour less than Anil. What is Arun’s speed? | |

A. 8 kmph | B. 5 kmph |

C. 4 kmph | D. 7 kmph |

Answer : Option B

Explanation :

Let the speed of Arun = x kmph and the speed of Anil = y kmph

distance = 30 km

We know that distance/speed = time. Hence, 30/x−30/y=2………..(Equation1)

30/y−30/2x=1………..(Equation2)

Equation1 + Equation2⇒30/x−30/2x=3 ⇒30/2x=3 ⇒15/x=3 ⇒5/x=1⇒x=5. Hence Arun’s speed = 5 kmph

6. A car travels first 160 km at 64 km/hr and the next 160 km at 80 km/hr. What is the average speed for the first 320 km of the tour? | |

A. 70.24 km/hr | B. 74. 24 km/hr |

C. 71.11 km/hr | D. 72.21 km/hr |

Answer : Option C

Explanation :

If a car covers a certain distance at x kmph and an equal distance at y kmph,the average speed of the whole journey = 2xy/(x+y) kmph.

By using the same formula, we can find out the average speed quickly average speed = (2×64×80)/(64+80)=(2×64×80)/144 ⇒ (2×32×40)/36 = (2×32×10)/9 ⇒ (64×10)/9=71.11 kmph

7. A man rides his bicycle 10 km at an average speed of 12 km/hr and again travels 12 km at an average speed of 10 km/hr. What is his average speed for the entire trip approximately? | |

A. 11.2 kmph | B. 10 kmph |

C. 10.2 kmph | D. 10.8 kmph |

Answer : Option D

Explanation :

Total distance travelled = 10 + 12 = 22 km

Time taken to travel 10 km at an average speed of 12 km/hr = distance/speed=10/12 hr

Time taken to travel 12 km at an average speed of 10 km/hr = distance/speed=12/10 hr

Total time taken =10/12+12/10 hr

Average speed = distance/time=22/(10/12+12/10)=(22×120)/{(10×10)+(12×12)}

(22×120)/244=(11×120)/122=(11×60)/61=660/61≈10.8 kmph

8. An airplane covers a certain distance at a speed of 240 kmph in 5 hours. To cover the same distance in 123 hours, it must travel at a speed of: | |

A. 660 km/hr | B. 680 km/hr |

C. 700 km/hr | D. 720 km/hr |

Answer : Option D

Explanation :

Speed and time are inversely proportional ⇒Speed ∝ 1/Time (when distance is constant)

Here distance is constant and Speed and time are inversely proportional

Speed ∝ 1/Time⇒Speed1/Speed2=Time2/Time1

⇒240/Speed2=(1*2/3)5⇒240/Speed2=(5/3)/5⇒240/Speed2=1/3 ⇒Speed2=240×3=720 km/hr

9. A train can travel 50% faster than a car. Both start from point A at the same time and reach point B 75 kms away from A at the same time. On the way, however, the train lost about 12.5 minutes while stopping at the stations. What is the speed of the car? | |

A. 80 kmph | B. 102 kmph |

C. 120 kmph | D. 140 kmph |

Answer : Option C

Explanation :

Let speed of the car = x kmph

Then speed of the train = x *(100+50)/100=150 x /100=3 x /2 kmph

Time taken by the car to travel from A to B=75/x hours

Time taken by the train to travel from A to B=75/(3 x /2)+12.5/60 hours

Since both start from A at the same time and reach point B at the same time

75/x=75/(3 x /2)+12.5/60 ⇒25/x=12.5/60 ⇒x=(25×60)/12.5=2×60=120

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