Problem 227 - Elementary Algebra

Question

The Jones family took a 12 mile canoe ride down the Indian River in two hours. After lunch, the return trip back up the river took three hours. Find the rate of the canoe in still water and the rate of the current.

Step-by-Step Solution

Answer

The rate of the canoe in still water is 5 mph, and the rate of the current is 1 mph.

Step 1 - Define Variables

Let the rate of the canoe in still water be represented by the variable \(c\) (in miles per hour) and let the rate of the current be represented by the variable \(r\) (in miles per hour).

Step 2 - Write Equations for Downstream and Upstream

When traveling downstream, the rate of the canoe is the rate in still water plus the rate of the current: \(c + r\). The downstream trip took 2 hours to cover 12 miles, so we have the equation: \ \[12 = (c + r) \times 2\] \ Simplify to: \ \[6 = c + r\]

Step 3 - Write Equation for Upstream

When traveling upstream, the rate of the canoe is the rate in still water minus the rate of the current: \(c - r\). The upstream trip took 3 hours to cover 12 miles, so we have the equation: \ \[12 = (c - r) \times 3\] \ Simplify to: \ \[4 = c - r\]

Step 4 - Solve the System of Equations

We now have two linear equations: \ \(6 = c + r\) \ \(4 = c - r\). \Solve these equations simultaneously. First, add the two equations: \ \[(6 = c + r) + (4 = c - r)\] \ This simplifies to: \ \[10 = 2c\] \ \[c = 5\]

Step 5 - Find the Rate of the Current

Substitute \(c = 5\) back into one of our original equations to find \(r\). Using \(6 = c + r\): \ \[6 = 5 + r\] \ \[r = 1\]

Step 6 - Verify the Solution

Check both equations with our found values \(c = 5\) and \(r = 1\): \ Downstream: \[12 = (5 + 1) \times 2 = 12\] \ Upstream: \[12 = (5 - 1) \times 3 = 12\] \ Both equations are satisfied.

Key Concepts

Rate Problems

Rate problems in algebra involve finding the speed, time, or distance traveled, and they commonly feature situations like moving vehicles or flowing water. In our exercise, the Jones family took a canoe ride down and up a river, with both the trip time and distance known. Understanding rate problems helps by applying the basic formula: \( \text{Distance} = \text{Rate} \times \text{Time} \).
Here we define the rate of the canoe in still water and the rate of the current.
To distinguish between the downstream and upstream movements, remember:

• Downstream: Canoe's rate added to the current's rate.
• Upstream: Canoe's rate minus the current's rate.

Linear Equations

Linear equations are equations of the first degree, meaning they can be written in the form \( ax + b = 0 \), where \(a\) and \(b\) are constants. They are fundamental in algebra and are used for solving various problems.
In solving the canoe problem, we set up linear equations based on the given information:

• For the downstream trip, the equation is: \[ 12 = (c + r) \times 2 \], simplified to \[ 6 = c + r \].
• For the upstream trip, the equation is: \[ 12 = (c - r) \times 3 \], simplified to \[ 4 = c - r \].

Breaking down complex problems into linear equations makes finding the solution manageable and logical.

System of Equations

A system of equations is a set of two or more equations with the same variables. The solution to the system is the values of these variables that satisfy all equations simultaneously.
For our canoe problem, solving the system of equations helps us find the rates we need:

• First equation: \(6 = c + r\).
• Second equation: \(4 = c - r\).

By adding these two equations together, we eliminate the variable \(r\):
\[ (6 = c + r) + (4 = c - r) \]
Which simplifies to: \[ 10 = 2c \], giving us \(c = 5\).
Substituting \(c = 5\) back into one of the original equations, we find \(r = 1\).

Problem-Solving Steps

Problem-solving in algebra includes clear and logical steps to reach the solution. Here are the steps taken in the exercise:

1. Define the variables: Let \(c\) be the canoe's rate in still water and \(r\) be the current's rate.
2. Write equations for each direction: Downstream and upstream.
3. Use the known values (time and distance) to form linear equations.
4. Solve the system of equations to find the values of \(c\) and \(r\).
5. Verify the solutions by substituting back into the original equations.

Following these steps ensures a systematic approach to solving algebraic problems and avoids confusion.