Question

A boat goes 24 km upstream and 28 km downstream in 6 hours. It goes 30 km upstream and 21 km downstream in 6 hours and 30 minutes. The speed of the boat in still water is
A. 10 km/hr
B. 4 km/hr
C. 14 km/hr
D. 6 km/hr

Answer 1

Hint: First of all, consider the speeds of the boat in upstream and downstream as variables. Split the question into two cases and solve them to get two different equations in terms of two variables.

Complete Step-by-Step solution:
Now, solve these equations by the method of substituting variables to obtain the required answer.
Let, speed of boat in still water \[ = x{\text{ km/hr}}\]
       speed of stream \[ = y{\text{ km/hr}}\]
We know that
speed of the boat in upstream = speed of the boat in still water - speed of the stream
                                                       \[ = \left( {x - y} \right){\text{ km/hr}}\]
speed of the boat in downstream = speed of the boat in still water + speed of the stream
                                                            \[ = \left( {x + y} \right){\text{ km/hr}}\]
We know that time taken for upstream \[ = \dfrac{{{\text{Distance covered in upstream}}}}{{{\text{Speed of the boat in upstream}}}}{\text{ hrs}}\]
And the time taken for down stream \[ = \dfrac{{{\text{Distance covered in downstream}}}}{{{\text{Speed of the boat in downstream}}}}{\text{ hrs}}\]
In case I:
Given that distance covered by the boat in upstream = 24 km
And distance covered by the boat in the downstream = 28 km
And the total time covered by the boat = 6 hours
We know that the total time covered = time taken for upstream + time taken for downstream
\[ \Rightarrow 6 = \dfrac{{24}}{{x - y}} + \dfrac{{28}}{{x + y}}\]
Let \[\dfrac{1}{{x - y}} = u{\text{ and }}\dfrac{1}{{x + y}} = v\]. So, the equation becomes as\[ \Rightarrow 6 = 24u + 28v......................................\left( 1 \right)\]
In case II:
Given that distance covered by the boat in upstream = 30 km
And distance covered by the boat in the downstream = 21 km
And the total time covered by the boat = 6 hours
We know that the total time covered = time taken for upstream + time taken for downstream
\[
   \Rightarrow 6\dfrac{1}{2} = \dfrac{{30}}{{x - y}} + \dfrac{{21}}{{x + y}} \\
   \Rightarrow \dfrac{{13}}{2} = \dfrac{{30}}{{x - y}} + \dfrac{{21}}{{x + y}} \\
\]
As \[\dfrac{1}{{x - y}} = u{\text{ and }}\dfrac{1}{{x + y}} = v\]. So, the equation becomes as
\[ \Rightarrow \dfrac{{13}}{2} = 30u + 21v..............................\left( 2 \right)\]
Subtracting equation (1) by multiplying with 3 from equation (2) by multiplying with 4, we get
\[
   \Rightarrow 4\left( {30u + 21v} \right) - 3\left( {24u + 28v} \right) = 4\left( {\dfrac{{13}}{2}} \right) - 3\left( 6 \right) \\
   \Rightarrow 120u + 84v - \left( {72u + 84v} \right) = 26 - 18 \\
   \Rightarrow 120u - 72u + 84v - 84v = 8 \\
   \Rightarrow 48u = 8 \\
  \therefore u = \dfrac{8}{{48}} = \dfrac{1}{6} \\
\]
Substituting \[u = \dfrac{1}{6}\] in equation (1), we have
\[
   \Rightarrow 24\left( {\dfrac{1}{6}} \right) + 28v = 6 \\
   \Rightarrow 4 + 28v = 6 \\
   \Rightarrow 28v = 6 - 4 = 2 \\
  \therefore v = \dfrac{2}{{28}} = \dfrac{1}{4} \\
\]
So, we have
\[
   \Rightarrow u = \dfrac{1}{{x - y}} = \dfrac{1}{6} \\
  \therefore x - y = 6........................................\left( 3 \right) \\
\]
And
\[
   \Rightarrow v = \dfrac{1}{{x + y}} = \dfrac{1}{{14}} \\
  \therefore x + y = 14........................................\left( 4 \right) \\
\]
Adding equations (3) and (4), we get
\[
   \Rightarrow \left( {x - y} \right) + \left( {x + y} \right) = 6 + 14 \\
   \Rightarrow 2x = 20 \\
  \therefore x = 10 \\
\]
Substituting \[x = 10\] in equation (4), we get
\[
   \Rightarrow 10 + y = 14 \\
   \Rightarrow y = 14 - 10 \\
  \therefore y = 4 \\
\]
Hence the speed of the boat in still water is 10 km/hr.
Thus, the correct option is A. 10 km/hr.

Note: Speed of downstream is always greater than the speed of the upstream. Speed of the boat in upstream is the difference of speed of the boat in still water and speed of the stream. Similarly, speed of the boat in upstream is the sum of speed of the boat in still water and speed of the stream.