Here are the steps to rearrange two identical squares to make four identical squares: 1. Cut one of the squares diagonally into two right triangles. 2. Rearrange the pieces to form four congruent right triangles. 3. Connect the hypotenuses of the four triangles to form four identical squares.

1. TRAINS

3. a km /hr =
18
5
a m/s.
a m/s =
5
18
a km / hr.

4. • Time taken by a train of length l metres to
pass a pole or a standing man or a signal post
is equal to the time taken by the train to cover
l metres.
• Time taken =
Example 1.
Find the time taken by the train of 100 m long running
at the speed of 30 km/hr to pass a man standing near
the railway line.
Lengthof thetrain
Speedof thetrain

5. • Time taken by a train of length l metres to pass a pole or a standing man
or a signal post is equal to the time taken by the train to cover l metres.
• Time taken =
Example 1.
Find the time taken by the train of 100 m long running at the speed of 30
km/hr to pass a man standing near the railway line.
Speed of train=
18
5
30 m/sec=
3
25
m/sec.
Required time taken =
100
25
3
=
3100
25
sec = 12 sec.
Lengthof thetrain
Speedof thetrain

7. Time Taken
• Time taken by a train of length l metres to pass a stationery object of
length b metres is the time taken by the train to cover (l + b) metres.
• Example 2.
• A train is moving at a speed of 132 km/hr. If the length of the train is 110
metres, how long will it take to cross a railway platform 165 metres long?
Speed of the train =
5132
18
m/sec =
110
3
m/sec.
Distance covered in passing the platform =
(110 + 165)m = 275 m.
Time taken =
3275
110
sec =
15
2
sec = 7
1
2
sec.

8. Time Taken
A train of length 300m travels at a speed of 36 kmph. In how
many seconds does it cross a bridge of length 700m?
Distance travelled = Length of the Train + Length of the Bridge
= 300 + 700 = 1000m

10. • When two trains are moving in opposite
directions, their speeds should be added to find
the relative speed. When two trains are moving
in same directions, their speeds should be
subtracted to find the relative speed.
• Suppose two trains or two bodies moving in the
same direction at u m/s and v m/s where u > v ,
then their relative speed = (u – v) m/s.
• Suppose two trains or two bodies moving in the
opposite directions at u m/s and v m/s where u >
v , then their relative speed = (u + v) m/s.

11. • If two trains of length a metres and b metres
are moving in opposite directions at u m/s and
v m/s, then time taken by the trains to cross
each other =
Required time= Sum of the lengths of two trains
Relative speed of them
a b
u v
2× Product of their speeds
Sum of their speeds

12. • Some times, we can calculate without using
any definite formula but straightly work out.
• P and Q are 300 km apart. At 8.00 am, buses X
and Y left P and Q simultaneously for Q and P
respectively. If the speeds of buses X and Y are
40 kmph and 60 kmph respectively, when do
they meet?
Time Travelled X Y Balance time travelled By X and Y
In one hour 40 60 260,240
In Two hours 80 120 220,180
In Three hours 120 180 180,120

13. • Example 3. (Refer Problem 13)
• Two trains 140m and 160m long run at the
speed of 60 km/hr and 40 km/hr respectively
in opposite directions on parallel tracks. The
time (in seconds) which they take to cross
each other is a) 9 b) 9.6 c) 10 d) 10.8
Relative speed = (60 + 40) km/hr =
Distance covered in crossing each other = sum of the lengths=
(140 + 160)m =300 m
Required time 9 54300 sec sec
5250

14. • If two trains of length a metres and b metres
are moving in same directions at u m/s and v
m/s, then time taken by the faster train to
cross the slower train =
Sum of the lengths of two trains
Relative speed of them
2× Product of their speeds
Difference of their speeds

15. • Example 4.
• Two trains of the same but with different speeds pass a
telegraph post in 8 seconds and 10 seconds respectively. In
what time will they cross each other when they are moving in:
• (i) the same direction? (ii) opposite direction?
2 8 10
10 8
2 8 10
10 8
8
98
= 80 sec.Required time =
Required time =

16. • If two trains (or bodies) start at the same time
from points A and B towards each other and
after crossing they take a and b sec in reaching
B and A respectively, then
• (A’s speed) : (B’s speed) =

17. • Example 5.
• Two trains, one from Howrah to Patna and the other from
Patna to Howrah, start simultaneously. After they meet, the
trains reach their destinations after 9 hours and 16 hours
respectively. The ratio of their speeds is:
• a) 2:3 b) 4:3 c) 6:7 d) 9:16.
• Let us name the trains as A and B. Then, (A’s speed) : (B’s
speed) =

18. Example 6.
Two trains start at the same time from Cuttack and Delhi and move toward each other at
the rate of 70 km/h and 80 km/h respectively. When they meet, it is found that one train
has travelled 120 km more than the other. Find the distance between Delhi and Cuttack.
Required distance = 80 70120
80 70

19. • Time of rest per hour = Difference in average speed
Speed without stoppage
Example 7.
Without stoppage a train travels at an average speed of 90 km/h
and with stoppage it covers the same distance at an average
speed of 72 km/h. How many minutes per hour does the train
stop?
Time of rest per hour = = 12 minutes.
Eg Pg 24 Excluding stoppages, the speed of a bus is 54 kmph and
including stoppages, it is 45 kmph. For how many minutes does
the bus stop per hour?

20. • Length of the train =
Speed of the train = Product of speed- Product of times
Difference in times
Example 8.
A train passes two persons who are walking in the direction opposite which the
train is moving at the rate of 6 m/s and 9 m/s in 7 seconds and 6 seconds
respectively. Find the length of the train and speed of the train.
Length of the train = =126 m.
9 6 7 6
7 6
Speed of the train =
9 6 7 6
7 6
=12 m/sec.

21. • Length of the train = Timetopassapole×Length of platform
Difference in time to cross a pole and platform
Example 9.
A train passes a pole in 12 seconds and passes a platform 120m long in 20 seconds.
Find its length.
Required length = 12 120 180 .
20 12
m
18. A train speeds past a pole in 15 seconds and a
platform 100m long in 25 seconds. Its length is

22. Speed of the 1st train=Speed of the 2nd train x
Timetaken by firsttrain after meeting
Timetaken by secondtrain after meetingExample 11.
Two trains A and B start from Kolkata and Patna towards Patna and Kolkata respectively.
After passing each other they take 8 hours and 2 hours to reach Patna and Kolkata
respectively. If the train from Kolkata is moving at 50 km/h, then find the speed of the
other train.
Required speed = 850
2
= 100 km/h.

23. Example 12.
Two places A and B are 180 km apart. A train leaves P for Q and at the same time
another train leaves B to A. Both the trains meet 4 hours after they start moving. If the
train travelling from A to B travels 5 km/h faster than the other train, find the speed of
the two trains.
Speeds of the trains = = 25 and 20.
180 4 5 180 4 5
2 4 2 4
and

24. • A car moves at the speed of 80 km/hr. what is
the speed of car in metres per second? a)
8m/s b) m/s c) m/s d) None f these

25. • Sound is said to travel in air at about 1100 feet
per second. A man hears the axe striking the
tree, seconds after he sees it strike the tree.
How far is the man from the woodchopper?
a)2197ft b) 2420 ft c) 2500 ft d) 2629
ft

26. • A salesman travels a distance of 50 km in 2
hours 30 minutes. How much faster, in
kilometres per hour, on an average, must he
travel to make such a trip in hour less time?
a)10 b)20 c)30 d)None

27. • A person travels from P to Q at a speed of 40
kmph and returns by increasing his speed by
50%. What is his average speed for both trips?
a)36kmph b)45kmph c)48kmph d)50 kmph

28. • 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. The speed of the car is
• a)100kmph b)110kmph c)120kmph d)130kmph.

29. Rearrange two to make four identical
squares