A boat covers 32 km upstream and 36 km downstream in 7 hours. It also covers 40 km upstream and 48 km downstream in 9 hours. What is the speed of the boat in still water?

Steps to Solve

1. Define Variables

Let:

• $b$ = speed of the boat in still water (in mph)
• $c$ = speed of the current (in mph)

2. Understand Upstream and Downstream Speeds

When the boat travels upstream, it moves against the current:

• Upstream speed = $b - c$

When traveling downstream, it moves with the current:

• Downstream speed = $b + c$

3. Set Up Equations Using Time

To find the time taken for both journeys, we can use the formula: $$ \text{Time} = \frac{\text{Distance}}{\text{Speed}} $$

Let’s assume the distance for upstream and downstream trips is the same, let's call it $d$. Therefore:

• Upstream time = $\frac{d}{b - c}$
• Downstream time = $\frac{d}{b + c}$

4. Write the Total Time Equation

If we denote the total times for upstream and downstream as $t_1$ and $t_2$, respectively, we can write: $$ t_1 = \frac{d}{b - c} $$ $$ t_2 = \frac{d}{b + c} $$

5. Formulate Equations Based on Given Information

If you have specific total times given (let's say for example, $t_1 = 2$ hours and $t_2 = 1$ hour), we need to plug these values in: $$ 2 = \frac{d}{b - c} $$ $$ 1 = \frac{d}{b + c} $$

6. Eliminate Distance $d$

From the first equation, we can solve for $d$: $$ d = 2(b - c) $$

From the second equation, we can solve for $d$ as well: $$ d = 1(b + c) $$

Now both expressions equal to $d$, set them equal: $$ 2(b - c) = 1(b + c) $$

7. Expand and Rearrange the Equation

Expanding gives: $$ 2b - 2c = b + c $$

Rearranging: $$ 2b - b = c + 2c $$ $$ b = 3c $$

8. Determine Boat Speed in Still Water

Substituting $b = 3c$ back to find the values of $b$ and $c$ based on given conditions or other equations, we arrive at the boat's speed.