#### Question

A motor boat whose speed in still water is 18 km/hr takes 1 hour more to go 24 km up stream that to return down stream to the same spot. Find the speed of the stream.

#### Solution

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

speed of the boat in still water = 18 km/hr.

Total Distance = 24 km.

We know that,

Speed of the boat up stream = speed of the boat in still water − speed of the stream

= (18 − *x*) km/hr

Speed of the boat down stream = speed of the boat in still water + speed of the stream

= (18 + *x*) km/hr

Time of up stream journey = *t*_{1} = \[\frac{24}{18 - x}\]

Time of down stream journey = *t*_{2} = \[\frac{24}{18 + x}\]

According to the question,

t_{1} − t_{2} = 1 hr

\[\Rightarrow \frac{24}{18 - x} - \frac{24}{18 + x} = 1\]

\[ \Rightarrow \frac{24(18 + x - 18 + x)}{(18 - x)(18 + x)} = 1\]

\[ \Rightarrow \frac{24(2x)}{(18 )^2 - x^2} = 1\]

\[ \Rightarrow 48x = 324 - x^2 \]

\[ \Rightarrow x^2 + 48x - 324 = 0\]

\[ \Rightarrow x^2 + 54x - 6x - 324 = 0\]

\[ \Rightarrow x(x + 54) - 6(x + 54) = 0\]

\[ \Rightarrow (x - 6)(x + 54) = 0\]

\[ \Rightarrow x - 6 = 0 \text { or } x + 54 = 0\]

\[ \Rightarrow x = 6 \text { or } x = - 54\]

Since, speed cannot be negative.

Thus, speed of the stream is 6 km/hr.