# A boat traveled 210 miles downstream and back. The trip downstream took 10 hours The trip back took 70 hours. What is the speed of the boat in still water? What is the speed of the current?

Oct 16, 2016

#### Answer:

The speed of boat in still water is $12$ miles per hour and speed of current is $9$ miles per hour.

#### Explanation:

As boat travelled $210$ miles downstream in $10$ hours it's speed is $\frac{210}{10} = 21$ miles per hour.

And as it travelled $210$ miles upstream in $70$ hours it average speed upstream is $\frac{210}{70} = 3$ miles per hour.

Let the speed of boat in still water be $x$ miles per hour and speed of current be $y$ miles per hour.

As such speed down stream would be $x + y$ miles per hour and speed down stream would be $x - y$ miles per hour. As such

$x + y = 21$ and $x - y = 3$

Adding the two

$x + y + x - y = 21 + 3$ i.e.

$2 x = 24$ or $x = \frac{24}{2} = 12$ and $y = 21 - 12 = 9$.

Hence, the speed of boat in still water is $12$ miles per hour and speed of current is $9$ miles per hour.

Oct 16, 2016

#### Answer:

The speed of the boat in still water is 12 miles per hour.
The speed of the current is 9 miles per hour.

#### Explanation:

Let $x =$the speed of the boat in still water in miles/hour
Let $c =$ the speed of the current in miles/hour

When traveling downstream, the speed of the boat is "helped" by the speed of the current and the net speed is $x + c$.

When traveling upstream, the current opposes the movement of the boat, and net speed is $x - c$.

Set up a table where

$d =$distance in miles,
$r =$ rate or speed in miles/hour, and
$t =$time in hours

$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a} \mathrm{dc} o l \mathmr{and} \left(w h i t e\right) \left(a a a a a a a\right) r \textcolor{w h i t e}{a a a a a a} t$

Downstream $\textcolor{w h i t e}{a a a a a} 210 \textcolor{w h i t e}{a a a} \left(x + c\right) \textcolor{w h i t e}{a a a} 10$

Upstream $\textcolor{w h i t e}{a a a a a a a} 210 \textcolor{w h i t e}{a a a} \left(x - c\right) \textcolor{w h i t e}{{a}^{1} a a} 70$

Recall that distance = rate x time or $d = r t$

$210 = \left(x + c\right) 10 \textcolor{w h i t e}{a a a}$Equation from downstream information
$210 = \left(x - c\right) 70 \textcolor{w h i t e}{a a a}$Equation from upstream information

$210 = 10 x + 10 c \textcolor{w h i t e}{a a a}$Distribute
$210 = 70 x - 70 c$

$7 \cdot 210 = 7 \cdot 10 x + 7 \cdot 10 c \textcolor{w h i t e}{a a a}$Multiply the 1st equation by 7

$1470 = 70 x + 70 c \textcolor{w h i t e}{a a a}$Add the 1st equation to the 2nd
$\textcolor{w h i t e}{a} 210 = 70 x - 70 c$

$1680 = 140 x$

$\frac{1680}{140} = \frac{140 x}{140} \textcolor{w h i t e}{a a a}$Divide both sides by 140

$x = 12$

$210 = \left(12 + c\right) 10 \textcolor{w h i t e}{a a a}$Subsitute x=12 into one of the original $\textcolor{w h i t e}{\forall \forall \forall a a a a a a \forall \forall \forall A}$equations to find c

$\textcolor{w h i t e}{a a} 210 = 120 + 10 c$
$- 120 \textcolor{w h i t e}{a} - 120$

$\textcolor{w h i t e}{a a} 90 = 10 c$

$\textcolor{w h i t e}{a} \frac{90}{10} = \frac{10 c}{10}$

$c = 9$

The speed of the boat is 12 miles per hour.
The speed of the current is 9 miles per hour.