Example 1: A boat goes 30 km
upstream and 44 km downstream in
10 hours. In 13 hours, it can go
40 km upstream and 55 km
down-stream. Determine the speed
of the stream and that of the boat in
still water
Answers
Step-by-step explanation:
Let the speed of the stream =x=x km/hr
Let the speed of the boat in still water =y=y km/hr
Upstream speed =y-x=y−x km/hr
Downstream speed =y+x=y+x km/hr
time=\dfrac{distance}{speed}time=
speed
distance
The boat goes 3030 km upstream and 4444 km downstream in 1010 hours.
Time taken =\dfrac{30}{y-x}+\dfrac{44}{y+x}=
y−x
30
+
y+x
44
10=\dfrac{30}{y-x}+\dfrac{44}{y+x}10=
y−x
30
+
y+x
44
................. (1)
The boat goes 4040 km upstream and 5555 km downstream in 1313 hours.
Time taken =\dfrac{40}{y-x}+\dfrac{55}{y+x}=
y−x
40
+
y+x
55
13=\dfrac{40}{y-x}+\dfrac{55}{y+x}13=
y−x
40
+
y+x
55
.................. (2)
Let \dfrac{1}{y-x}=u
y−x
1
=u
and \dfrac{1}{y+x}=v
y+x
1
=v
From (1) and (2),
30u+44v=1030u+44v=10 ...................(3)
40u+55v=1340u+55v=13 ...................(4)
Multiply equation (3) with 4 and equation (4) with 3,
120u+176v=40120u+176v=40 ........... (5)
120u+165v=39120u+165v=39 ........... (6)
subtract equation (6) from (5),
176v-165v=40-39176v−165v=40−39
11v=111v=1
v=\dfrac{1}{11}v=
11
1
\Rightarrow \dfrac{1}{y+x}=\dfrac{1}{11}⇒
y+x
1
=
11
1
y+x=11y+x=11 .......................... (7)
From equation (3),
30u=10-44v30u=10−44v
30u=10-44\times \dfrac{1}{11}30u=10−44×
11
1
30u=10-4=630u=10−4=6
u=\dfrac{1}{5}u=
5
1
\dfrac{1}{y-x}=\dfrac{1}{5}
y−x
1
=
5
1
\Rightarrow y-x=5⇒y−x=5 .................. (8)
Adding (7) and (8), we get,
2y=162y=16
y=8y=8
From equation (7),
x=11-yx=11−y
x=11-8=3x=11−8=3
Answer:
Speed of stream = 3 km / hr.
Speed of boat in still water = 8 km / hr.
Step-by-step explanation:
Let the speed of the boat in still water be a km / hr and stream be b km / hr
For upstream = a - b
For downstream = a + b
We know :
Speed = Distance / Time
Case 1 .
10 = 30 / a - b + 44 / a + b
Let 1 / a - b = x and 1 / a + b = y
30 x + 44 y = 10 ... ( i )
Case 2 .
13 = 40 / a - b + 55 / a + b
40 x + 55 y = 13 ... ( i )
Multiply by 4 in ( i ) and by 3 in ( ii )
120 x + 176 y = 40
120 x = 40 - 176 y ... ( iii )
120 x + 165 y = 39
120 = 39 - 165 y ... ( iv )
From ( iii ) and ( iv )
40 - 176 y = 39 - 165 y
11 y = 1
y = 1 / 11
120 x = 40 - 176 y
120 x = 40 - 176 / 11
x = 1 / 5
Now :
1 / a - b = 1 / 5
a - b = 5
a = 5 + b ... ( v )
1 / a + b = 1 / 11
a + b = 11
a = 11 - b ... ( vi )
From ( v ) and ( vi )
11 - b = 5 + b
2 b = 6
b = 3
a = 5 + b
a = 5 + 3
a = 8
Hence we get answer.