# A boat can travel 8 mph in still water. If it can travel 15 miles downstream in the same time that it can travel 9 miles up the stream, what is the rate of the stream?

Jul 29, 2016

$9 \frac{1}{7} \text{ miles per hour}$

#### Explanation:

Let the time of travel in each direction be $t$

Let the velocity of the water be $v$

Let distance be $s$

Given that the boat velocity is 8 mph

Downstream $\to s = \left(8 + v\right) t = 15$.....................Equation(1)

Upstream $\to s = \left(8 - v\right) t = 9$...........................Equation(2)
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Write Eqn(1) as: $8 t + v t = 15 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . \left({1}_{a}\right)$
Write Eqn(2) as: $8 t - v t = 9. \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left({2}_{a}\right)$

$E q n \left({1}_{a}\right) + E q n \left({2}_{a}\right)$ gives:

$16 t = 14 \implies t = \frac{14}{16} \equiv \frac{7}{8}$.............................(3)

Using $E q n \left(3\right)$ substitute for $t$ in $E q n \left({1}_{a}\right)$

(This should work no matter which equation you choose)

color(brown)(8t+vt=15)color(blue)(" "->" "8(7/8)+v(7/8)=15

v=8/7(15-7) = 64/7" "("miles")/("hour")