Question

sarah and mia are out for an adventure on a river. their boat can travel 5 miles per hour in still water. on their trip, they notice that it takes the same amount of time to travel 8 miles upstream as it does to travel 12 miles downstream. sarah and mia want to figure out the speed of the river’s current. create a rational equation to determine the speed of the current. solve the equation to determine the speed of the current.

Explanation:

Step1: Define variable for current speed

Let \(x\) = speed of the river's current (mph).

Step2: Find upstream/downstream speeds

Upstream speed: \(5 - x\) mph, Downstream speed: \(5 + x\) mph.

Step3: Set time equality equation

Time = distance/speed, so $\frac{8}{5 - x} = \frac{12}{5 + x}$.

Step4: Cross-multiply to eliminate denominators

$8(5 + x) = 12(5 - x)$

Step5: Expand both sides

$40 + 8x = 60 - 12x$

Step6: Isolate \(x\) terms

$8x + 12x = 60 - 40$
$20x = 20$

Step7: Solve for \(x\)

$x = \frac{20}{20} = 1$

Answer:

Rational equation: $\boldsymbol{\frac{8}{5 - x} = \frac{12}{5 + x}}$
Speed of the current: $\boldsymbol{1}$ mile per hour