Q. 4.44 - Intermediate Algebra

Question

Translate to a system of equations and then solve:

Jason paddled his canoe $24$miles upstream for $4$hours. It took him $3$hours to paddle back. Find the speed of the canoe in still water and the speed of the river current.

Step-by-Step Solution

Answer

The speed of the canoe in still water is $7$ mph and the speed of the river current is $1$ mph.

Step 1. Given

A canoe was paddled upstream for $4$ hours and paddled back for $3$ hours. It was a total distance of $24$ miles.

Step 2. Translating into system of equations.

We know that distance is the product of rate and time.

So we will calculate the distance for the canoe by assuming the speed of canoe and river current as $s\&c$ respectively.

1.  When travelling upstream, time taken is $4$ hours and the rate will be $(s-c)$. So the distance comes out to be $4(s-c)$.
2. When travelling downstream, time taken is $3$ hours and the rate will be $(s+c)$. So the distance comes out to be $3(s+c)$.

We know the total distance is $24$ miles.

Step 3. Finding speed of canoe in still water.

We have two equations, i.e.,
$$\begin{gathered} 3(s+c)=24 \\ 4(s-c)=24 \end{gathered}$$

Taking the common factors out give us the equations as-
$$\begin{gathered} s+c=8 \\ s-c=6 \end{gathered}$$

Adding both the equations give us $2s=14$ which also means
$s=7$
This shows that the speed of the canoe in still water is $7$ mph.

Step 4. Finding the speed of river current.

We know that the speed of river boat in still water, i.e., $s=7$.

Putting this value in $s-c=6$ gives us,

$$\begin{gathered} 7-c=6 \\ c=1 \end{gathered}$$

This shows that speed of river current is $1$ mph.

Step 5. Check the solution.

We will calculate the downstream and upstream rate by adding or subtracting the river current and the speed of canoe and then calculate the distance covered. If the equation gives us the given value, our answers are right. 

1. Downstream rate will become $7+1=8$ mph.
   In $3$ hours, the distance covered will be $3\times 8=24$ miles.
2. Upstream rate will become $7-1=6$ mph.
   In $4$ hours, the distance covered will be $4\times 6=24$ miles.

Thus, we get a given distance which shows our answer is right.