Problem 26 - Elementary Algebra

Question

A river tour boat averages 7 miles per hour in still water. If the total 24-mile tour downriver and 24 miles back takes 7 hours, then how fast is the river current?

Step-by-Step Solution

Answer

The river current speed is 5 miles per hour.

Step 1: Define Variables

Let the speed of the river current be \( c \) miles per hour. The speed of the boat downstream is \( 7 + c \) mph, and upstream is \( 7 - c \) mph.

Step 2: Write the Equation for Downstream Time

The time to travel downstream 24 miles is given by \( \frac{24}{7 + c} \).

Step 3: Write the Equation for Upstream Time

The time to travel upstream 24 miles is \( \frac{24}{7 - c} \).

Step 4: Set Up the Total Time Equation

The total time for the round trip is 7 hours, so we have the equation: \[ \frac{24}{7+c} + \frac{24}{7-c} = 7. \]

Step 5: Solve the Equation

Combine the terms using a common denominator: \( (7+c)(7-c) = 49 - c^2 \). The equation becomes: \[ \frac{24(7-c) + 24(7+c)}{49-c^2} = 7. \] Simplify the numerator: \( 168 \). The equation is \[ \frac{168}{49-c^2} = 7. \]

Step 6: Simplify and Solve for \( c \)

Solve the equation \( \frac{168}{49-c^2} = 7 \) by cross-multiplying to get \( 168 = 7(49 - c^2) \). This results in \( 168 = 343 - 7c^2 \). Rearrange to get \( 7c^2 = 175 \), then solve for \( c \) by dividing by 7 to find \( c^2 = 25 \), thus \( c = 5 \).

Step 7: Verify the Solution

Substitute \( c = 5 \) back into the downstream and upstream speeds to verify the total time: Downstream: \( \frac{24}{12} = 2 \) hours, Upstream: \( \frac{24}{2} = 5 \) hours. Total: \( 2 + 5 = 7 \) hours, which matches the problem statement.

Key Concepts

Algebra Word Problems

Algebra word problems like the river current speed problem help us apply mathematical concepts to real-world situations. They often require translating a descriptive scenario into mathematical equations. For instance, understanding how a boat's speed changes due to a river current involves separating the variables for downstream and upstream speed. Navigating word problems involves:

• Identifying the given information: The boat's speed in still water, the total distance, and total time available.
• Defining variables: Using symbols to represent unknown quantities, such as the river current speed, noted as \( c \).
• Constructing equations based on relationships: Here, the downstream and upstream speeds are adjusted by \( c \), leading to equations that describe the journey.

Always start by clarifying what is being asked, and consider drawing diagrams to understand the situation better.

Speed and Distance

Speed and distance are fundamental concepts in motion problems and are key to solving this river current exercise. When a boat travels downstream, its speed is increased by the current, while upstream, it decreases.Key aspects of speed and distance:

• Speed is a measure of how fast an object is moving and is often expressed as distance per unit of time.
• Distance refers to how far the object travels, and time indicates how long the travel takes. The relationship is given by the equation: \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \).
• In problems like these, the formula is manipulated to \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \) to find the travel time given speed and distance.

Understanding how to modify speed when affected by factors like current is crucial in solving similar problems.

Solving Equations

Solving equations is the process of finding the value of unknown variables. In our river current problem, an equation relates the downstream and upstream travel times to the total time. When solving:

• Formulate the equation from the problem description. For example, combine individual travel times to match the specified total.
• Use algebraic techniques like finding a common denominator to simplify complex fractions.
• Cross-multiplication helps to clear fractions and solve equations like \( \frac{168}{49-c^2} = 7 \).

These methods lead to simplified expressions that can be solved step-by-step to find the desired variable values.

Equations with Variables

Equations with variables allow us to express relationships involving unknown amounts, like the boat's speed against a river current. Tackling these equations involves systematic steps to solve for the variable.Important factors include:

• Accurate definition of variables: Assign a symbol to each unknown (e.g., \( c \) for river current speed).
• Setting up a valid equation: Use problem context to write equations that relate different speeds and distances.
• Simplifying and manipulating the equation: Rearranging terms, combining like terms, and factoring where necessary.

Ultimately, substitute back to check if solutions hold true in the original equation, confirming the variable value accurately describes the scenario.