Problem 75 - Algebra for College Students

Question

Solve each problem by using a system of equations. One day last summer, Jim went kayaking on the Little Susitna River in Alaska. Paddling upstream against the current, he traveled 20 miles in 4 hours. Then he turned around and paddled twice as fast downstream and, with the help of the current, traveled 19 miles in 1 hour. Find the rate of the current.

Step-by-Step Solution

Answer

The rate of the current is 3 mph.

Step 1: Identify Variables

Let us denote Jim's paddling speed in still water as \( p \) (in miles per hour) and the rate of the current as \( c \) (in miles per hour). We need to find the value of \( c \).

Step 2: Establish Equations for Upstream and Downstream

When Jim paddles upstream, he goes against the current, so his effective speed is \( p - c \). He paddled 20 miles in 4 hours, which gives us the equation: \( \frac{20}{4} = p - c \). Simplifying, we have: \( p - c = 5 \). For the downstream trip, where he paddles twice as fast with the current, his effective speed is \( 2p + c \). He traveled 19 miles in 1 hour, leading to the equation: \( \frac{19}{1} = 2p + c \). Simplifying, \( 2p + c = 19 \).

Step 3: Solve the System of Equations

We now have a system of equations:1. \( p - c = 5 \)2. \( 2p + c = 19 \)First, solve the first equation for \( p \):\( p = c + 5 \).Substitute \( p = c + 5 \) into the second equation:\( 2(c + 5) + c = 19 \).Simplify and solve for \( c \):\( 2c + 10 + c = 19 \)\( 3c + 10 = 19 \)\( 3c = 9 \)\( c = 3 \).

Step 4: Verification

Substitute \( c = 3 \) back into the equation for \( p \) to find the paddling speed: \( p = 3 + 5 = 8 \) mph.Check the downstream condition: \( 2(8) + 3 = 16 + 3 = 19 \) mph, which matches the downstream speed. The calculation verifies that our solution is correct.

Key Concepts

Variables in Equations

When tackling a problem involving systems of equations, the first step is identifying the variables. In math problems like these, variables are symbols that stand in place of unknown values, allowing us to work through calculations to find their specific values.

In Jim's kayaking problem, there are two unknowns:

• Jim's paddling speed in still water, which we will call \( p \) (in miles per hour).
• The rate of the river's current, called \( c \) (also in miles per hour).

These variables allow us to create equations and find a solution based on the conditions given in the problem. By setting up expressions such as \( p - c \) for upstream speed and \( 2p + c \) for downstream speed, we formulate the framework essential for solving complex motion issues in physics and real-world scenarios.

Solving Linear Equations

Once we define the variables and establish the equations based on the problem's conditions, the next step is solving these equations. Linear equations simplify this process because they represent direct relationships, such as \( p - c = 5 \) and \( 2p + c = 19 \).

The goal is to solve for the unknowns by manipulating these equations to isolate one variable at a time. Here's how:

• From the first equation, rearrange it to express \( p \) in terms of \( c \): \( p = c + 5 \).
• Substitute this expression for \( p \) in the second equation: \( 2(c + 5) + c = 19 \).
• Simplify and solve step-by-step to find \( c \):
  • Expand: \( 2c + 10 + c = 19 \)
  • Combine terms: \( 3c + 10 = 19 \)
  • Solve for \( c \): \( 3c = 9 \), leading to \( c = 3 \).

Verifying these calculations ensures the accuracy of our solution, which helps avoid errors and confirm understanding.

Upstream and Downstream Problems

Upstream and downstream problems involve the motion of an object, such as a boat or a person, in relation to a current or steady flow of water. These scenarios require considering how a current affects overall speed.

For example:

• While moving upstream, you paddle against the current, which decreases your effective speed. Jim's upstream speed is represented by \( p - c \), where he paddles 20 miles in 4 hours.
• When moving downstream, you move with the current, increasing your speed. Here, Jim's downstream speed is \( 2p + c \), covering 19 miles in 1 hour.

Mastering these concepts is essential, as such scenarios frequently appear in physics and math courses. Understanding how to construct and solve equations from these situations can greatly aid you in comprehending other related mathematical concepts, adding depth to your problem-solving skills. By representing the real-world conditions with equations and solving them correctly, you gain a clearer insight into how algebra can be applied in practical contexts.