Problem 74 - 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 7 mph.

Step 1: Understand the problem

In this problem, we have two parts of Jim's trip: paddling upstream and paddling downstream. We need to find the current's speed. Let the speed of the kayak in still water be \( k \) miles per hour and the speed of the current be \( c \) miles per hour.

Step 2: Establish equations for upstream paddling

When paddling upstream, the effective speed is the speed of the kayak minus the speed of the current, \( k - c \). Jim traveled 20 miles in 4 hours. So, we can set up the equation: \[4(k - c) = 20\]

Step 3: Solve the upstream equation for speed

Dividing both sides of the equation by 4 gives:\[k - c = 5\]This tells us that the effective speed when paddling upstream was 5 mph.

Step 4: Establish equations for downstream paddling

When paddling downstream, the effective speed is the speed of the kayak plus the speed of the current, \( k + c \). Jim traveled 19 miles in 1 hour. So, the equation is: \[1(k + c) = 19\]This simplifies to:\[k + c = 19\]

Step 5: Solve the system of equations

We have the system of equations:1. \( k - c = 5 \)2. \( k + c = 19 \)Adding these two equations, we find:\[2k = 24 \\Rightarrow k = 12\]Subtracting the first equation from the second gives:\[2c = 14 \\Rightarrow c = 7\]This tells us that the speed of the current is 7 mph.

Key Concepts

Algebraic Problem-Solving

When solving algebraic problems, it's crucial to translate the problem into mathematical expressions. This often involves setting up equations that accurately represent the situation in the problem.
A system of equations is a collection of two or more equations with a common set of variables. Solving such systems involves finding values for the variables that satisfy all equations simultaneously.

Here are some tips for successful algebraic problem-solving:

• Identify the variables: Decide what quantities the variables will represent. In this problem, variables for the kayak's speed and the current's speed in still water were used.
• Set up equations: Based on the problem description, create equations using the variables.
• Solve systematically: Use methods like substitution or elimination to solve the equations, as demonstrated in the system of equations solved for this problem.

Maintaining clarity throughout the problem-solving process helps ensure successful outcomes.

Rate Problems

Rate problems are all about understanding how speed, distance, and time relate to one another. The fundamental formula used is:\[ \text{Distance} = \text{Rate} \times \text{Time} \] This formula is usually rearranged to solve for the unknown quantity based on the problem context.

In the example of Jim kayaking, his trips were divided into upstream and downstream journeys, each with different effective rates due to the influence of the river's current. By establishing how the current affects his speed when going *with* and *against* it, Jim's kayaking times offered enough information to calculate the current's speed.

• Analyze the problem in parts, like separating upstream from downstream travels.
• Use given data, such as distance traveled and time taken, to identify rates.
• Remembering that rates can be different depending on external factors, such as the river's current in this scenario.

Understanding and setting up these rates correctly is key to solving such problems.

Paddle Upstream and Downstream

When analyzing movements in water, particularly paddling upstream and downstream, it's important to take account of how currents affect speed. In rivers, the current flows in a single direction, often affecting paddling speeds significantly.

When paddling upstream:

• The current works against you, reducing the effective speed.
• The effective speed is calculated by subtracting the current's speed from the kayak's speed: \[ \text{Upstream Effective Speed} = k - c \]

When paddling downstream:

• The current works with you, increasing the effective speed.
• You add the current's speed to the kayak's speed: \[ \text{Downstream Effective Speed} = k + c \]

The difference in effective speed results in varying travel times and can be used to calculate the speed of the current, as was shown in the solved problem. Recognizing these dynamics helps solve complex rate problems in water-related tasks.