TIME & DISTANCE
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.
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^{th} of his usual speed, a man is 12 minutes too late. What is the usual time taken by him to cover that distance? 

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, ^{11}/_{5} seconds after he sees it strike the tree. How far is the man from the wood chopper? 

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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