A man in a boat goes 12 km downstream and comes back to the starting point by rowing non-stop in a total time of 3 hours. If the speed of the stream is 3 km/h, find the speed with which the man can row the boat in still water.

Step 1: Define variables and formulate speeds

Let $v_b$ be the speed of the boat in still water, which is what we need to find. The speed of the stream, $v_s$, is given as 3 km/h. When the boat travels downstream, its speed is the sum of its speed in still water and the speed of the stream. When it travels upstream, its speed is the difference between its speed in still water and the speed of the stream.

Step 2: Formulate time equations for downstream and upstream travel

The distance traveled downstream is 12 km. So, the time taken to travel downstream, $t_d$, is the distance divided by the downstream speed. Similarly, the distance traveled upstream is also 12 km. The speed upstream is the speed of the boat in still water minus the speed of the stream. Thus, the time taken to travel upstream, $t_u$, is the distance divided by the upstream speed.

Step 3: Set up the total time equation

The problem states that the total time for the round trip (downstream and upstream) is 3 hours. We can set up an equation by adding the downstream time and the upstream time and equating it to 3.

Step 4: Solve the quadratic equation for $v_b$

To solve the equation, we first find a common denominator and combine the fractions. This leads to a quadratic equation. We simplify the quadratic equation and then factor it to find the possible values for $v_b$.

Step 5: Select the valid speed

Since speed cannot be negative, we discard the solution $v_b = -1$. Therefore, the speed of the boat in still water is 9 km/h.

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