Maths Lesson on Rate_Speed Time & Distance

Math Lesson on Speed, Time & Distance

Must know formulas:

Speed = Distance/time.

If, Speed is denoted by V, Distance is detonated by S and time is detonated by T, then V = S/T

Or, S = V X T. Speed is expressed in terms of per hour. So, unit of Speed is km/hr or mile/hr

Conversion between km/hr to meter/sec : 1 km/hr = 5/18 meter/sec

Conversion between meter/hr to km/sec : 1 meter/sec = 18/5 km/hr

Special Situations :

Round trip : If a car moves a certain distance at x km/hr and returns to the starting point

across the same road at y km/hr speed, then the average speed for the entire trip =

2xy/x+y km/hr

Two trains going at the same direction: Suppose, a train at the speed of x km/hr is

moving along the rail track towards north. Another train, running beside the previous

rail track is also moving towards north at the speed of y km/hr. In this situation, any

passenger sitting in the first train will feel the other train is moving at (x-y) km/hr [if

x>y]. Similarly, passenger in the second train will feel the other train is moving at (x-y)

km/hr [assuming x>y]. This phenomenon happens due to relative velocity.

Suppose, length of the faster train is M km and length of the slower train is N km, the

time required to cross each other = (M+N)/x-y hrs

Two trains coming towards each other/moving opposite direction: Suppose, a train at

the speed of x km/hr is moving along the rail track towards north. Another train is

coming from the opposite side towards the first train in another parallel rail track at the

speed of y km/hr. In this situation, any passenger sitting in the first train will feel the

other train is moving at (x+y) km/hr. Similarly, passenger in the second train will feel the

other train is moving at (x+y) km/hr. This phenomenon happens due to relative velocity.

Suppose, length of the faster train is M km and length of the slower train is N km, the

time required to cross each other = (M+N)/x+y hrs

A train crossing a human being or a signal pole: If a train crosses a human being or a

railway signal pole, it has to travel the path equal to its own length.

A train crossing a bridge/tunnel: If a train crosses a bridge/tunnel, it has to travel the

path equal to its own length + the length of the bridge. So, if the length of the train is X

km and length of the bridge or tunnel is Y km, then total distance that has to be covered

is = X+Y

Two trains crossing each other: If two trains cross each other, both have to travel the

path equal to the summation of their individual lengths. Suppose length of one train is x

meter and length of other train is y meter, then to cross each other completely, each have

to travel (x+y) meters

If the ratio of the speeds of A: B is a:b then the ratio of the times taken by them to cover

the same distance is 1/a : 1/b = b:a

Boats & Streams:

Direction along the stream is called downstream

Direction against the stream is called upstream

If the speed of the row/boat is a km/hr and the speed of stream is b km/hr, then

Speed downstream = (a+b) km/hr

Speed upstream = (a-b) km/hr

Example 01: A train normally travels 60 miles at a certain speed. On a day, due to mechanical problem, the train’s speed is reduced by 10 mph so that the journey takes 3 hours longer. What is the normal speed of the train? [IBA MBA : 2007-2008]

(A) 20 (B) 25 (C) 32 (D) 45 (E) none of these

Solution:

Traditional method: Let the normal speed is x mile/hr So, to travel 60 miles, the train takes = 60/x hr If the speed is reduced by 10 mph, the resultant speed = x-10 mph So, to travel 60 miles at this new speed, required time = 60/x-10 hr According to condition given,

(60/x-10) – (60/x) = 3

Or, 60/x-10 = 60/x +3

Or, 60/x-10 = 60+3x/x

Or, 60x = (x-10) * (60+3x) [cross multiplication]

Or, 60x = 60x + 3x2 - 600 – 30x

Or, 3x2 - 600 – 30x = 0

Or, x2 – 10x-200 = 0

Or, (x-20) (x+10) = 0

Since, x can’t be equal to -10 [as speed can’t be negative]

So, x-20 = 0 or, x = 20

So, the normal speed of the train is 20 mile/hr

shortcut technique:

In this shortcut method, we won’t calculate the solution directly rather will try to find

out the answer from the choices given by educated guess. While guessing, it is always

advisable to start with the answer choices which are multiples of 5. In that case, for this

problem, it is better to start with choice A or choice B and then try with choice D and

finally trying with choice C.

Let’s assume choice A is the answer.

So, if the normal speed is 20 mile/hr, to travel 60 mile, it will take 60/20 = 3 hr.

If the speed is reduced by 10 mile/hr, then the speed becomes 10 mile/hr and to travel

60 miles it will take = 60/10 = 6 hr

So, for second case, it is taking 6-3 = 3 hr more than the normal speed. Bingo! This is the

right choice as in the question, it has been clearly mentioned that, in second case when

speed reduced to 10 mile than the normal, 3 more hour is needed.

Example 02: Mr. Tanvir travelled to his office from his house at a speed of 4 km/hr. He returned home from his office at a speed of 6 km/hr. If Mr. Tanvir’s total travelling time is 10 hr, what is the distance of his office from house?


Solution: It is evident that, Mr. Tanvir is taking a round trip, means he took the same

path while going office from house and while returning from office to house.

So, as per round trip formula, average speed of Mr. Tanvir is = 2xy/x+y

Here, x = 4km/hr and y=6km/hr

So, the average speed in the round trip will be = 2*4*6/4+6 = 4.8 km/hr

At 4.8 km/hr speed, Mr. Tanvir travelled 10 hr. So, total path travelled = 10 *4.8 =48km

This 48 km distance is the total distance of round trip [from house to office and back to

house from office].

So, distance from house to office = 48/2 = 24 km

Example 03: A train running at 72 km/hr crosses a bridge in 60 seconds. If the length of the train is 200 meters, what is the length of the bridge in meter?


Solution:

Speed of the train is 72 km/hr = (72 X 5/18) meter/sec = 20 meter/sec

So, at this speed, in 60 seconds, the train travels = 20 * 60 = 1200 meter

This 1200 meters comprised of the train’s length + bridge’s length

So, length of the bridge is = 1200-200 = 1000 meter

N.B : Please refer to special situations : A train crossing a bridge in page 1 of your lecture sheet

Example 04: Two trains 256 meter and 264 meter long are running towards each other on parallel lines at 84 km/hr and 60 km/hr respectively. How much time will be needed to cross each other completely?

(A) 15 sec (B) 12 sec (C) 20 sec (D) 18 sec (E) 13 sec

Solution:

Here, both trains are running towards each other. So, the effective relative velocity will be = (84+60) = 144 km/hr = 144*5/18 meter/sec = 40 meter/sec

In order to cross each other completely, both trains have to travel the distance equal to the summation of their individual lengths, i.e, (256+264) meters= 520 meter.

Therefore, to cross 520 meter at the speed of 40 meter/sec, required time will be = 520/40 = 13 sec.

So, (E) is the correct answer.

N.B: Students now try to do the same problem taking both trains are going towards

same direction

Example 05: A person travels a distance of 30 km at a speed of 60 km/hr and then comes back at thrice the speed. What is his total traveling time in minutes?

(A) 15 min (B) 25 min (C) 40 min (D) 60 min (E) None of these

Solution:

To travel 30 km @ 60km/hr speed, the person will need = 30/60 = ½ hr

Again, if he returns at thrice the speed, his speed will be 180 km/hr

In that case, the person will need = 30/180 = 1/6 hr

So, total time needed = (1/2 + 1/6) = 4/6 hr = 2/3 hr = 2/3*60 minutes = 40 minutes

So, (C) is the right answer.

Example 06:

If two planes leave the same airport at 1:00 PM, how many miles apart will they be at

3:00 PM if one travels directly north at 150 mph and the other travels directly east

at 200 mph? [IBA MBA 2010-11]

(A) 50 (B) 2500 (C) 500 (D) 600 (E) None of these

Solution:

N A

S

300 300 500

B 400 C

Traditional method:

The first plane which will leave the airport at 1:00 PM, will be at point A at 3:00 PM in

the northern direction. In this 2 hours, it will fly = 2*150 = 300 miles.

Similarly, the second plane, leaving the airport from point B and flying towards eastern

direction will be at point C at 3 : 00 PM. So, in this 2 hour, it will fly = 2*200 = 400 miles

It is clear from the picture that, the linear distance between the two planes at 3 PM is

the AC of the right angled triangle ABC. So, as per Pythagorean theory,

AC2 = AB2 + BC2

Or, AC2 = 3002 + 4002

Or, AC2 = 90000 + 160000 = 250000

Or, AC = 500 [taking square root of both sides]

shortcut technique:

There are two Pythagorean triplets which we should remember—

3:4:5 and 5:12:13

In this problem ratio of two sides of the triangle is 300 & 400. So, obviously, without any

calculation, it can be said that, other hand will be 500. So, correct answer is (C)

Example 07:

Two cyclists start biking from the starting point of a trail 3 hours apart. The second cyclist

travels at 10 miles per hour and starts 3 hours after the first cyclist who is travelling at 6 miles

per hour. How much time in hours will pass before the second cyclist catches up with the first

from the second cyclist started biking? [IBA MBA 2010-11]

(A) 2 (B) 4.5 (C) 0.75 (D) 6 (E) None of these

Solution :

Here, both the cyclists are traveling in the same direction. So relative speed is (10-6) = 4

mile/hour

As the first cyclist starts 3 hours ago, in this time, s/he will travel = 3*6 = 18 miles

We know, S = V X T

Here, S = 18 miles, V = 4 mile/hour

So, T = S/V = 18/4 = 4.5 hour

So, (B) is the correct answer.

Example 08:

The speed of a boat is 40 km/hr and the speed of stream is 8 km/hr. moving with the stream,

the boat went 240 km. What distance in KM will the boat cover during the same time going

against the stream?


Solution:

Here, when the boat was moving along with the stream, the effective speed = 40+8 = 48 km/hr

At this speed, to cross 240 km path, it will take = 240/48 = 5 hr time.

When, the boat will be returning against the stream, the effective speed will be = 40-8 = 32

km/hr

So, at 32 km/hr, in 5 hours, the boat can travel = 32*5 = 160 km

So, the correct answer is (B)

Example 09:

Tanvir is 1/3 of the way across a bridge when he hears a train whistle behind him. A huge

locomotive and tons of boxcars are coming at him at 45 mph. Tanvir knows that he can make it

to the far edge of the bridge at the exact same instant as the train, but he also knows that he

can run toward the train and reach the near end of the bridge just as the train gets there. How

fast does Tanvir run?

(A) 8 (B) 10 (C) 12 (D) 14 (E) 15

shortcut technique:

Tanvir has twice the time to run to point B as he has to run to point A. So the relative speed of Tanvir and the train when Tanvir runs towards the train is twice the relative speed as when he runs away from the train. If Tanvir’s speed is u mph, This means 45 + u = 2(45 - u) 45 + u = 90 - 2u Or,3u = 45 So, u = 15 . So correct answer is (E)

Problems for practice in the class

1. A train covers a distance of 80 km at a speed of 40 km/hr for the first 60 km and the

remaining distance at a speed of 20 km/hr. What is the average speed (in km/hr) of the

train on the journey? [IBA MBA 2008-2009]


2. Sonia travels to IBA from Uttara by car at a speed of 40 km/hr and returns to Uttara at a

speed of 30 km/hour by a CNG. What is her average speed in the entire journey in

Km/hour? [IBA 2009-2010]

(B) 35 (B) 34.3 (C) 37.5 (D) 35.3 (E) 36

3. Two cars start towards each other from 200 km apart. One car travels at 40 km/hr and

the other travels at 35 km/hr. How far apart will be the two cars after 4 hours?


4. A thief is spotted by a policeman from a distance of 400 meter. When the policeman

starts chase, the thief starts running to escape. If speed of the thief is 20 km/hr and that

of the policeman is 24 km/hr, then how far the thief will have run before he is caught?

(A)1 km (B) 2 km (C) 3 km (D) 4 km (E) None of these

5. Sharif ran a 2 mile race at an average speed of 8 miles per hour. If Arif ran the same race

at an average speed of 6 miles per hour, how many minutes longer than Sharif did Arif

take to complete the race? [IBA MBA 2001-02]

(A)15 (B) 10 (C) 8 (D) 5 (E) None of these

6. Arif travels 1/3rd of the distance at an average speed of 5 km/hr, 2/5th of the distance at

an average speed of 4 km/hr and the rest 12 miles in 2 hours. What is the total distance

traveled by Arif? [IBA MBA 2001-02]


7. Anwar usually walks to his house from his office at a speed of 8 km/hr. It takes him 10

minutes longer to walk the same distance at 6 km/hr. What is the distance (in km)

between his house and office? [IBA MBA 2002-03]


8. A train running at 25 km/hr takes 18 seconds to pass a platform. Next it takes 12

seconds to pass a man walking at 5 km/hr in the same direction. Find the length of the

platform.

(A)100 (B) 50 (C) 90 (D) 35 (E) 25

9. If a boat goes 7 km upstream in 42 minutes and the speed of the stream is 3 km/hr, then

the speed of the boat is?

(A)10 (B) 5 (C) 3 (D) 8 (E) 2

10. Sujon and Sumon agreed to race a 50 feet pool and back again. They started together

but Sujon finished 10 feet ahead of Sumon. If Sujon finishes the race in 27 seconds, how

long Sumon will need to finish?

(A)10 (B) 50 (C) 30 (D) 18 (E) 27

11. A man started at 8 am from his home, walked at the rate of 3 km/hr and reached his

office 45 minutes late. The next day he started at the same time and walked at the rate

of 5 km/hr and reached his office 15 minutes earlier than the scheduled time. What was

the distance between his office and home?

(A)6 (B) 7.5 (C) 9 (D) 12 (E) 18

12. Two cars started from the same point, at 5 am, traveling in opposite directions at 40 and

50 mph respectively. At what time will they be 450 miles apart?

(A)6 am (B) 7 am (C) 9 am (D) 10 am (E) 12 pm

13. Two trains, traveling towards each other, left from two stations that are 900 miles apart,

at 4 pm. If the rate of the first train is 72 mph and the rate of the second train is 78 mph,

after how many hours will they pass each other?

(A)4 (B) 5 (C) 6 (D) 10 (E) 12

14. A motor cycle stunt man belonging to a fair rides over the vertical walls of a circular wall

at an average speed of 54 km/hour for 5 minutes. If the radius of the wall is 5 meters, then

the distance traveled is? [IBA MBA 2009-2010]

(A)2.5 km (B) 3.5 km (C) 5 km (D) 4.5 km (E) None

15. Two trains started from the same point. At 8:00 am the first train traveled East at the

rate of 80 mph. At 9:00 am, the second train traveled West at the rate of 100 mph. At what

time were they 530 miles apart?

(A)9:30 am (B) 10:45 am (C) 11 am (D) 11:15 am (E) 11:30 am

16. There is a dolphin swimming at 250 km per hour and the water flows at 150 km per hour and It is trying to get to his destination, which is 500 miles away. How long would it take him to get to it’s destination?

(A)8 hr (B) 10 hr (C) 11 hr (D) 12 hr (E) 13 hr

17. A passenger train was traveling at a speed of 72km/hr. A man on the passenger train observed a goods train traveling at a speed of 54km/hr in the opposite direction. If the goods train passed him in 8 seconds, what is the length of the goods train in meter? [EMBA- 11th Batch, Oct-2006]

(A) 180 (B) 220 (C) 240 (D) 280 (E) 300

18. Mr. Nader drove from Mymenshingh to Dhaka at 60 miles/hr. Returning on the same route, There was a lot of traffic and he was only able to drive at 40 miles/hr. If the return trip took 1 hr longer, what is the distance between Dhaka & Mymenshingh? [Basic Bank Assistant Manger’12]

(A) 100 mile (B) 104 mile (C) 106 mile (D) 108 mile (E) 120 mile

19. A’s house is 5 km east of the city center point and B’s house is 12km south of the said city center point. If the taxi fare is tk 5 per km and there is a triangular road communication System between B’s house, A’s house and the city center point, how much taka will cost of the triangular trip? [EMBA, Jan-2012]

(A) 125 (B) 85 (C) 75 (D) 150 (E) 140

Answer Keys

1. D 2. B 3.A 4.B 5.D 6.A 7.D 8.E 9.A 10.C 11.B 12.D

13. C 14. D 15.E 16.A 17. D 18. E 19. D

Maths Lesson on Rate_Speed Time & Distance

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