Question

A motor boat can travel 30 km upstream and 28 km downstream in 7 hours. It can travel 21 km upstream and return in 5 hours. Find the speed of the boat in still water at km/h.
A. 3
B. 15
C. 10
D. 11

Answer 1

Hint: When a boat goes upstream then its speed reduces by the speed of the stream. And when a boat goes downstream then its speed increases by the speed of the stream. So, in this question, we will use the concept of linear equation in two variables to find the speed of the boat in still water.

Complete step-by-step answer:
Now, here in this question, a boat travels in water going upstream and downstream. Let the speed of boat in still water be x km/hr and speed of stream be y km/hr.
Now, when the boat is travelling upstream, it means the boat is going against the flow of water in the stream. So, its speed reduces and is equal to (x – y) km/hr. When the boat goes downstream, it goes in the direction of the stream, so its speed increases and is equal to (x + y) km/hr.
Also, time = distance/speed.

Now, according to first case,
Let T₁ = time taken in going 30 km upstream, so
T₁ = \[\dfrac{{30}}{{({\text{x - y)}}}}\]
Let T₂ = time taken in going 28 km downstream, so
T₂ = \[\dfrac{{28}}{{({\text{x + y)}}}}\]
According to question,
T₁ + T₂ = 7
Therefore, $\dfrac{{30}}{{({\text{x - y)}}}}{\text{ + }}\dfrac{{28}}{{({\text{x + y)}}}}{\text{ = 7}}$ …… (1)
Similarly, in the second case T₁ is the time to go 21 km upstream and T₂ is the time to go 21 km downstream.
So, according to question,
T₁ + T₂ = 5
Therefore, $\dfrac{{21}}{{({\text{x - y)}}}}{\text{ + }}\dfrac{{21}}{{({\text{x + y)}}}}{\text{ = 5}}$ …… (2)

Now, to solve equation (1) and equation (2), let $\dfrac{1}{{({\text{x + y)}}}}{\text{ = u}}$ and \[\dfrac{1}{{({\text{x - y)}}}}{\text{ = v}}\]
So, equation (1) becomes,
28u + 30v = 7 …… (3)
And equation (2) becomes,
21u + 21v = 5 …… (4)
We will find the value of u and v with the help of substitution method,
From equation (3), \[{\text{v = }}\dfrac{{7{\text{ - 28u}}}}{{30}}\] ……. (5)

Now, putting this value of v in equation (4), we get
$21{\text{ (u + }}\dfrac{{7{\text{ - 28u}}}}{{30}}){\text{ = 5}}$
On solving the above equation, we get, u = $\dfrac{1}{{14}}$
Putting value of u in the equation (5), we get ${\text{v = }}\dfrac{1}{6}$
Now, as $\dfrac{1}{{({\text{x + y)}}}}{\text{ = u}}$ and \[\dfrac{1}{{({\text{x - y)}}}}{\text{ = v}}\]. So, we can write
\[\dfrac{1}{{({\text{x + y)}}}}{\text{ = }}\dfrac{1}{{14}}\] $ \Rightarrow $ x + y = 14 …… (6)
\[\dfrac{1}{{({\text{x - y)}}}}{\text{ = }}\dfrac{1}{6}\] $ \Rightarrow $ x – y = 6 …… (7)
Adding both equation (6) and equation (7), we get
2x = 20
$ \Rightarrow $x = 10
So, the speed of the boat in still water = x km/hr = 10 km/hr.
So, option (C) is correct.

Note: whenever we come up with such a type of problem, we will proceed by following a few steps. First, we will let the value be found as a variable. Then we will make an equation according to the cases given in question. Then solve the equations formed accordingly to the question to find the value of the required variable. Such questions can be solved with the help of one variable or with the help of two variables according to the requirement of the question.