Time, Speed, and Distance Simplified

Fundamentals of Time, Speed, and Distance

Distance = Speed X Time


Proportionalities implicit to the above equation.


1) Direct Proportionality between Time and Distance (When Speed is constant)

When speed is constant distance will vary as time

Car A moves for 2 hours @ 25kmph ---------> d = s X t -------> 50 = 25 X 2
Car B moves for 3 hours @ 25kmph ---------> d = s X t --------> 75 = 25 X 3
Here (tA / tB) = (dA / dB) -----> time ratio = distance ratio = 2 : 3


2) Direct Proportionality between Speed and Distance (When Time is constant)

If time of two bodies or motion is constant, then distance will vary as speed
Here (S1 / S2) = (d1 / d2)

If two cars start simultaneously from opposite ends towards each other. They meet at point ‘C’
In this case distance covered by each car will vary as their speeds since they travelled for equal period.

Suppose the distance AB = 900 km Speed A = 25kmph Speed B = 20kmph
In this case their meeting point (C) will be (S1 / S2) = (d1 / d2) ------> 25/20 ----> 5/4 = 500 / 400 i.e. 500 kms from A


3) Inverse Proportionality between Speed and Time (when Distance is constant)

If the distance to be covered is constant, then time will vary inversely as speed i.e. as speed increases, time decreases and vice versa.
(S1 / S2) = (t2 / t1)

Example 1 :- A Train meets with an accident and moves at ¾ of its original speed. Because of this it delayed by 20 minutes. What is the original time for the journey beyond the point of accident.
-----> Speed becomes ¾ hence time becomes 4/3 -------------------- remember (S1 / S2) = (t2 / t1))
-----> increased time is 1/3 = 20 minutes. So original time = 60 minutes. (1 belongs to 20mins, then 3 belongs to 60mins)

In other words

------> speed dropped to 75% and time increased to 133.33% ----> increased time = 33.33% = 20 minutes
Now 33.33 % belongs to 20 minutes, So 100% belongs to 60 minutes.

Example 2 :- A Man walked from his house to office at 5kmph and got 20 minutes late. if he had travelled at 7.5kmph, he would have reached 12 minutes early. The distance from his house to office is?

Here s1 = 5kmph t1 = t2 + 20 + 12 s2 = 7.5kmph t2 = t2
(S1 / S2) = (t2 / t1) ------> (5/7.5) = (t2 / t2 + 32) ---------> 5t2 + 160 = 7.5t2 -------> t2 = 64 minutes
So distance = 7.5 X 1.7 (64 mins) = 8km


Relative Speed


Relative Speed can be viewed as a movement of one body relative to another moving body.


1. Relative Speed of bodies moving in same direction.

In the case of the bodies moving to and fro between two points A and B, The faster body will reach the end first and will meet the second body on its way back. The relative speed S1 – S2 will apply till the point of reversal of the faster body and after that the two bodies will start to move in the opposite directions at a relative speed of S1 + S2. The relative speed governing the movement of the two bodies will alternate between S1 – S2 and S1 + S2 everytime anyone of the bodies reverses the direction. However, if both the bodies reverse their direction at the same instant, there will be no change in the relative speed equation.
In this case, the description of the motion of the two bodies between two consecutive meetings will also be governed by the proportionality between speed and distance – since the time of movement between any two meetings will be constant.

In this case, for every meeting, the total distance covered by the two bodies will be 2d (d = distance between the extreme points). The respective coverage of the distance will in the ratio of the individual speeds. Thus for the 9th meeting the total distance covered will be 9 X 2d = 18d


2. Relative Speed of bodies moving in opposite direction.

In the case of the bodies moving to and fro between two points A and B starting from opposite ends of the path, The two bodies will meet at a point in between A and B, then move apart away from each other. The faster body will reach its extreme point first followed by the slower body reaching its extreme point next. Relative speed will change every time ; one of the bodies reverses direction.
In this case, the position of the meeting point will be determined by the ratio of speeds of the bodies – since the time travelled of both the bodies is same. (Remember (S1 / S2) = (d1 / d2))
In this case, for the first meeting, the total distance covered by the two bodies will be d (d = distance between the extreme points). The respective coverage of the distance will in the ratio of the individual speeds. Thereafter, as the bodies separate and start coming together, the combined distance to be covered will be 2d. Thus for the 9th meeting the total distance covered will be d + 8 X 2d = 17d

NOTE :- TRY THIS CLASSIC EXAMPLE. IT CAN BE SOLVED USING ABOVE CONCEPT IN A SHORT TIME.
https://gmatclub.com/forum/m03-09-ps-trains-71044.html


Example 1:- Two bodies A and B start from opposite ends P and Q of a straight road. They meet at a point 0.6d from P. Find the point of their fourth meeting.
-----> Since time is constant we have speed ratio as 3:2
-----> Total distance to be covered by the two together for the fourth meeting is d + 2*3d = 7d. This distance is divided in a ration of 3:2 thus we have that A will cover 4.2d and B will cover 2.8d
-----> We can find out the fourth meeting point by either tracking A’s movement or that of B’s
-----> A, having moved a distance of 4.2d, will be at a point 0.2d from P.


Example 2:- A start walking from a place at a uniform speed of 2 kmph in a particular direction. After half an hour, B starts from the same place and walks in the same direction as A at a uniform speed and overtakes A after 1 hour 48 minutes. Find the speed of B.
-----> A is walking at 2kmph and B started chasing him after half an hour so A must have covered 1 km distance till B starts chasing.
This distance of 1 km is covered by B in 1(48/60) i.e. in 1.8 hours.
So the equation is (SB – SA) X t = Distance -----> (SB – 2) X 1.8 = 1 ---------> SB – 2 = 1/1.8 -----> SB = 47 / 18.



Trains

Points to remember while solving such problems.

Train crosses a stationary object without length. St X t = Lt

Here the train has to cover its own length to completely cross any stationary object that have a negligible length - a standing person, Electric Pole, Tree etc – so here the distance covered( d) by the train while crossing will be equivalent to its own length
So the equation will be---------> speed of train X time to cross object = length of train


Train crosses a stationary object with length. St X t = Lt + Lo

Here the train has to cover its own length and the length of stationary object to completely cross that stationary object that have a certain length - Platform, another stationary train, bridge etc – so here the distance covered ( d) by the train while crossing will be equivalent to its own length + the length of stationary object
So the equation will be----------> speed of train X time to cross object = length of train + length of object

Example 1 :- A Train crosses a pole in 8 seconds. If the length of the train is 200 meters, find the speed of the train.
------> St X t = Lt -----> St X 8 = 200 --------> St = 200/8 --------> St = 25 m/s or 90kmph

Example 2 :- A Train travelling at 20 m/s crosses a platform in 30 seconds and a man standing on the platform in 18 seconds. what is the length of the platform
A) 240 meters
B) 360 meters
C) 420 meters
D) 600 meters
E) Cannot be determined

Solution :- Check both the statements one by one

I - Train travelling at 20 m/s crosses a platform in 30 seconds. Here train is crossing a stationary object with length --------------> St X t = Lt + Lo ------> 20 X 30 = Lt + Lo ------> 600 = Lt + Lo ----> 600 is the sum of train length and platform length

II - Train travelling at 20 m/s crosses a man standing on the platform in 18 seconds. Here train is crossing a stationary object without length --------------> St X t = Lt ------> 20 X 18 = Lt ------> 360 = Lt ----> train’s length is 360.

We will put this value in equation one 600 = Lt + Lo ----> 600 = 360 + Lo Lo = 240 so platform’s length will be 240. Choice A.



Train crosses a Moving object without length.

In same direction (St - So) X t = Lt

Here the train has to cover its own length to completely cross any moving object that have a negligible length - a running person, a motorist etc – so here the distance covered( d) by the train while crossing will be equivalent to its own length
So the equation will be ( speed of train – speed of object) X time to cross object = length of train

In Opposite direction (St + So) X t = Lt

Here the train has to cover its own length to completely cross any moving object that have a negligible length - a running person, a motorist , bullet of a weapon, arrow etc. – so here the distance covered( d) by the train while crossing will be equivalent to its own length
So the equation will be ( speed of train + speed of object) X time to cross object = length of train

NOTE :- The Concept below ,presented in italic style, is beyond the scope of GMAT. It is presented only for the purpose of fulfillment of topic and for them who are curious to know about it.
Readers can avoid it out rightly if they don’t find the information illuminating, as the chance of GMAT testing this concept in the exam is extremely rare, infact virtually zero.


Train crosses a Moving object with length.


In same direction (St - So) X t = Lt + Lo

Here the train has to cover its own length and the length of moving object to completely cross the object that have a certain length and that moving in the same direction – Train passing another(slower) train – so here the distance covered( d) by the train while crossing will be equivalent to its own length + the length of moving object
So the equation will be (speed of train – speed of object) X time to cross object = length of train + length of object

In Opposite direction (St + So) X t = Lt + Lo

Here the train has to cover its own length and the length of moving object to completely cross the object that have a certain length and that moving in the opposite direction – Train passing another(slower) train – so here the distance covered( d) by the train while crossing will be equivalent to its own length + the length of moving object
So the equation will be (speed of train + speed of object) X time to cross object = length of train + length of object


Here is a bit complex question
A Train crosses a man travelling in another train in the opposite direction in 8 seconds. However, the train requires 25 seconds to cross the same man if the trains are travelling in the same direction. If the length of the first train is 200 mtrs and that of the train in which the man is sitting is 160 mtrs, find the speed of the first train.
Note :- Here we should understand that the situation is one of the train crossing a moving object without length. Thus the length of the man’s train is useless or redundant data and given intentionally in order to confuse test taker.
A Train crosses a man travelling in another train in the opposite direction in 8 seconds.----> (St + Sm) X t = Lt ------------------------> (St + Sm) X 8 = 200 ------> St + Sm = 200/8 = 25 Note :- Sm stands for speed of a man’s train.
the train requires 25 seconds to cross the same man if the trains are travelling in the same direction ----> (St - Sm) X t = Lt ------------------------> (St - Sm) X 25 = 200 ------> St - Sm = 200/25 = 8
We have two unknowns and two equations. So solving them we get St = 16.5 m/s or 59.4kmph


Boats and Streams

The problems with boats and streams are also based on the basic equation Distance = Speed X Time
Following variables are generally used in these problems.
SB = Speed of Boat
SS = Speed of stream
The speed of the movement of the boat is dependent on how the boat is moving
1) In still water = SB
2) Moving Upstream = SB – SS
3) Moving Downstream = SB + SS


Average Speed.


We all know the general formula to calculate the average speed

Total Distance / Total Time

Example :- If Joe covered first 100km of his trip at 50kmph and rest 320 km at 80kmph. What was his average speed throughout the trip
-------> It took Joe 2 hours to cover first 100kms and 4 hours to cover rest 320 kms. Total time = 6 hours
Avg speed = (Total Distance / Total Time) -------> 420 / 6 -----> 70 kmph

When the distance covered for two journeys is same and we know the indivisual speeds of those journeys then average speed is given by

Average Speed = 2S1S2/(S1 + S2)

Example :- A car travels at 60 kmph from Mumbai to Pune and at 120kmph from Pune to Mumbai. What is the average speed of the car for the entire journey.
2S1S2/(S1 + S2) -------> 2 X 60 X 120 / 60 + 120 --------> 14400/180 --------> 80 kmph

On the GMAT, within time constraints, divisions like 14400/180 may take much of valuable time

Here is another way to calculate average speed (when distance is same)

Speeds are 60 and 120.
Their ratio will be ½
sum of numerator and denominator of ratio is 1 + 2 = 3
Difference in speeds is 120 – 60 = 60.
Divide this difference by the sum of numerator and denominator i.e. by 3 ---------> 60/3 = 20.
Now our Average speed will be 20 X 1 parts away from lower speed 60 + 20 X 1 = 80

See below examples for better understanding of this concept

Speed1 Speed2 Ratio of speeds Sum of ratio elements difference between speeds Division difference/sum Average speed
40 60 2/3 5 20 (20/5) = 4 40 + 4 X 2 = 48
45 105 3/7 10 60 (60/10) = 6 45 + 6 X 3 = 63
66 110 3/5 8 44 (44/8)=5.5 66 + 3 X 5.5 = 82.5


Here is the GMAT type example

Example 3 :- The Sinhagad Express left Pune for Mumbai at noon sharp. Two hours later, the Deccan Queen started from Pune in the same direction. The Deccan Queen passed the Sinhagad Express at 8 P.M. Find the average speed of the two trains over the journey if the sum of their average speeds is 70 kmph
a) 34.28 kmph
b) 35 kmph
c) 50 kmph
d) 12 kmph
e) 16 kmph

Speed Sinhagad Express = x
Time Sinhagad Express = 8
Speed Deccan Queen = 70 – x (Sum of their speeds is 70)
Time Deccan Queen = 6
We know that Deccan queen started 2 hours later, Hence till the time Deccan starts, Sinhagad would have travelled for 2 hours and have covered 2x distance.
Deccan has covered this 2x distance by ((70 – x) – x) speed i.e. by 70 – 2x in 6 hours
as per the formula distance = speed x time ----> 2x = (70-2x)6 -----> 2x = 420 – 12x -----> 14x = 420 ----> x= 30
Note :- If you understood the principle Inverse Proportionality between Speed and Time (when Distance is constant) well, you can notice that since the time ratio is 4:3, speed ratio must be 3:4; Since sum of their speeds is 70, the lower speed must be 30 and higher speed must be 40. From here you can directly apply any of the formula to calculate average speed.


So
Speed Sinhagad = 30, Time Sinhagad = 8, Distance Sinhagad = 240

Speed Deccan = 40, Time Deccan = 6, Distance Deccan = 240

Their Speed Ratio = ¾, We know they travelled for equal distance so their time ratio will be 4/3

Average Speed = Total Distance / Total Time = 480 / 14 = 34.28 kmph
Average Speed = 2S1S2/(S1 + S2) -------> 2 X 30 X 40 / 30 + 40 --------> 2400/70 --------> 34.28 kmph
Average Speed = (40-30)/7 ---------> 30 + 3 X 1.42 -------> 30 + 4.26 ------> 34.26 kmph



Courtesy for the Information
1) Prof. Dr. R. D. Sharma - Author of CBSE Math Books
2) Mr. Arun Sharma - Alumnus IIM Bangalore

Special Thanks to
Mr. Mike McGarry – Magoosh GMAT Instructor[/size][/color]


Practice Problems.


1. Walking at ¾ of his normal speed, Mike is 16 minutes late in reaching his office. The usual time taken by him to cover the distance between his home and his office is
a. 48 minutes
b. 60 minutes
c. 42 minutes
d. 62 minutes
e. 66 minutes

Discussed HERE.


2. Two trains for Mumbai leave Delhi at 6 am and 6.45 am and travel at 100 kmph and 136 kmph respectively. How many kilometers from Delhi will the two trains be together.
a. 262.4 km
b. 260 km
c. 283.33 km
d. 275 km
e. None of these

Discussed HERE.


3. Ron walks to a viewpoint and returns to the starting point by his car and thus takes a total time of 6 hours 45 minutes. He would have gained 2 hours by driving both ways. How long would it have taken for him to walk both ways.
a. 8 h 45 min
b. 7 h 45 min
c. 5 h 30 min
d. 6 h 45 min
e. None of these

Discussed HERE.


4. Two trains of length 100 m and 250 m run on parallel tracks. When they run in the same direction, they take 70 sec to cross each other and when they run in opposite directions, they take 10 sec to cross each other. The speed of the faster train is
a. 5 m/s
b. 15 m/s
c. 20 m/s
d. 25 m/s
e. 35 m/s

Discussed HERE.


5. A man walking at a constant rate of 4 miles per hour is passed by a woman traveling in the same direction along the same path at a constant rate of 20 miles per hour. The woman stops to wait for the man 5 minutes after passing him, while the man continues to walk at his constant rate. How many minutes must the woman wait until the man catches up?
a. 16 mins
b. 20 mins
c. 24 mins
d. 25 mins
e. 28 mins

Discussed HERE.


6. A dog is passed by a train in 8 seconds. Find the length of the train if its speed is 36 kmph
a. 70 mtrs
b. 80 mtrs
c. 85 mtrs
d. 90 mtrs
e. 60 mtrs

7. A Train requires 7 seconds to pass a pole while it requires 25 seconds to cross a stationary train which is 378 mtrs long. Find the speed of the train.
a. 75.6 kmph
b. 75.4 kmph
c. 76.2 kmph
d. 21 kmph
e. 20 kmph

Discussed HERE.


OAs for the practice questions posted in the article are as follows

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