Answer

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Hint: To solve the question, we have to apply the upstream and downstream speed formula and the given information to obtain equations.

Complete step-by-step Solution:

Let the speed of the stream be x km/hr.

Let the time taken to travel 24 km downstream by motor boat be t hours.

\[\Rightarrow \]The time taken to travel 24 km upstream by motor boat = (t + 1) hours.

The given speed of the motor boat in the still water is equal to 18 km/hr.

The given distance travelled by motor boat is equal to 24 km.

We know that the formulae

The upstream speed of the motor boat = Speed of the motor boat in still water – Speed of the stream

= 18 - x

The downstream speed of the motor boat = Speed of the motor boat in the still water + Speed of the stream

= 18 + x

We know that the formula for the distance travelled by a boat = Net speed of the boat \[\times \]time taken to travel

By substituting given values in the above formula for the boat travelled 25 km upstream in (t + 1) hours, we get

\[24=(18-x)(t+1)\]

\[24=18t+18-xt-x\]

\[xt+x=18t-6\]

\[x=\dfrac{18t-6}{t+1}\] …….(1)

By substituting given values in the above formula for the boat travelled 25 km downstream in t hours, we get

\[24=(18+x)t\]

By substituting the equation (1) in the above equation we get

\[24=\left( 18+\dfrac{18t-6}{t+1} \right)t\]

\[24=\left( \dfrac{18t+18+18t-6}{t+1} \right)t\]

\[24=\left( \dfrac{36t+12}{t+1} \right)t\]

\[24t+24=36{{t}^{2}}+12t\]

\[36{{t}^{2}}-12t-24=0\]

\[3{{t}^{2}}-t-2=0\]

\[3{{t}^{2}}-3t+2t-2=0\]

\[\left( 3t+2 \right)\left( t-1 \right)=0\]

\[\Rightarrow t=1,\dfrac{-3}{2}\]

The time taken to travel 24 km downstream by motor boat = 1 hour.

By substituting the t value in equation (1) we get

\[x=\dfrac{18(1)-6}{1+1}\]

\[x=\dfrac{12}{2}=6\]km/hr.

\[\therefore \] The speed of the stream = 6 km/hr.

Hence, option (b) is the right choice.

Note: The alternative procedure can be forming a quadratic equation of x instead of forming a quadratic equation of t and the options can be eliminated by substituting the values in the obtained quadratic equation of x to check whether the values satisfy the equation or not. The possibility of mistake can be the calculations since the procedure of solving has multiple calculations.

Complete step-by-step Solution:

Let the speed of the stream be x km/hr.

Let the time taken to travel 24 km downstream by motor boat be t hours.

\[\Rightarrow \]The time taken to travel 24 km upstream by motor boat = (t + 1) hours.

The given speed of the motor boat in the still water is equal to 18 km/hr.

The given distance travelled by motor boat is equal to 24 km.

We know that the formulae

The upstream speed of the motor boat = Speed of the motor boat in still water – Speed of the stream

= 18 - x

The downstream speed of the motor boat = Speed of the motor boat in the still water + Speed of the stream

= 18 + x

We know that the formula for the distance travelled by a boat = Net speed of the boat \[\times \]time taken to travel

By substituting given values in the above formula for the boat travelled 25 km upstream in (t + 1) hours, we get

\[24=(18-x)(t+1)\]

\[24=18t+18-xt-x\]

\[xt+x=18t-6\]

\[x=\dfrac{18t-6}{t+1}\] …….(1)

By substituting given values in the above formula for the boat travelled 25 km downstream in t hours, we get

\[24=(18+x)t\]

By substituting the equation (1) in the above equation we get

\[24=\left( 18+\dfrac{18t-6}{t+1} \right)t\]

\[24=\left( \dfrac{18t+18+18t-6}{t+1} \right)t\]

\[24=\left( \dfrac{36t+12}{t+1} \right)t\]

\[24t+24=36{{t}^{2}}+12t\]

\[36{{t}^{2}}-12t-24=0\]

\[3{{t}^{2}}-t-2=0\]

\[3{{t}^{2}}-3t+2t-2=0\]

\[\left( 3t+2 \right)\left( t-1 \right)=0\]

\[\Rightarrow t=1,\dfrac{-3}{2}\]

The time taken to travel 24 km downstream by motor boat = 1 hour.

By substituting the t value in equation (1) we get

\[x=\dfrac{18(1)-6}{1+1}\]

\[x=\dfrac{12}{2}=6\]km/hr.

\[\therefore \] The speed of the stream = 6 km/hr.

Hence, option (b) is the right choice.

Note: The alternative procedure can be forming a quadratic equation of x instead of forming a quadratic equation of t and the options can be eliminated by substituting the values in the obtained quadratic equation of x to check whether the values satisfy the equation or not. The possibility of mistake can be the calculations since the procedure of solving has multiple calculations.

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