# A boat can go 12 mph in calm water. If the boat goes down a river 45 miles and back up the river 45 miles it takes him 8 hours. What is the current of the river?

Oct 22, 2015

Let's call the speed of the current $c$

#### Explanation:

Downstream he will have a speed of $12 + c$
and upstream his speed will be $12 - c$

Now the downstream journey will take $\frac{45}{12 + c}$ hours

And the upstream journey will take $\frac{45}{12 - c}$ hours

So we get to the equation:

$\frac{45}{12 + c} + \frac{45}{12 - c} = 8 \to$ make equal denominators

$\frac{45 \left(12 - c\right)}{\left(12 + c\right) \left(12 - c\right)} + \frac{45 \left(12 + c\right)}{\left(12 + c\right) \left(12 - c\right)} = 8 \to$

$\frac{45 \left(12 - c + 12 + c\right)}{144 - {c}^{2}} = 8 \to \frac{1080}{144 - {c}^{2}} = 8 \to$

$8 \cdot \left(144 - {c}^{2}\right) = 1080 \to 1152 - 8 {c}^{2} = 1080 \to$

$8 {c}^{2} = 72 \to {c}^{2} = 9 \to c = 3$ mph

Downstream: $\frac{45}{12 + 3} = 3$ hrs
Upstream: $\frac{45}{12 - 3} = 5$ hrs