Problem 49 - Algebra A Combined Function

Question

Two hikers are 11 miles apart and walking toward each other. They meet in 2 hours. Find the rate of each hiker if one hiker walks 1.1 mph faster than the other.

Step-by-Step Solution

Answer

The slower hiker walks at 2.2 mph, and the faster hiker walks at 3.3 mph.

Step 1: Understanding the Problem

We are given two hikers walking towards each other and meeting after 2 hours. The total distance they cover together is 11 miles. One hiker walks at a speed 1.1 mph faster than the other. Let's find their speeds.

Step 2: Formulate the Equation

Let the speed of the slower hiker be \(x\) mph. Then, the speed of the faster hiker is \(x + 1.1\) mph. Since they walk for 2 hours and cover 11 miles together, the sum of the distances covered by each hiker equals 11 miles: \[ 2x + 2(x + 1.1) = 11\]

Step 3: Simplify the Equation

From the previous step, simplify the equation: \[ 2x + 2x + 2.2 = 11 \]which simplifies to: \[ 4x + 2.2 = 11 \]

Step 4: Solve for x

To determine \(x\), first isolate \(4x\) by subtracting 2.2 from both sides: \[ 4x = 11 - 2.2 \]\[ 4x = 8.8 \]Now, divide both sides by 4: \[ x = \frac{8.8}{4} = 2.2 \]

Step 5: Finding the Speeds

The slower hiker's speed is \(x = 2.2\) mph. The faster hiker's speed is \(x + 1.1 = 2.2 + 1.1 = 3.3\) mph.

Key Concepts

Rate and Distance Problems

Rate and distance problems are math exercises that involve finding the speed of moving objects or the distance they travel. These problems often have a common theme of objects moving towards or away from each other. To solve them, we use the basic equation:
\[ \text{Distance} = \text{Rate} \times \text{Time} \]
In these problems, it's important to track how the rate and time relate to the distance traveled. Each person's speed multiplied by the time gives the distance they cover. If two people are moving towards each other, their combined distances add up to the total distance between them.
Understanding the relationship between these terms helps us create equations that describe the problem. Let's consider the original exercise: Two hikers start 11 miles apart and meet after 2 hours. This means their combined distances equal 11 miles, and this information forms the basis of our equation.

Algebraic Expressions

Algebraic expressions are key to solving rate and distance problems. They allow us to translate real-world scenarios into mathematical equations. In the hikers' problem, setting up an algebraic expression is crucial to find their speeds.
First, we assigned a variable, \(x\), to represent the unknown speed of the slower hiker. Then, knowing the faster hiker's speed is 1.1 mph more, we express it as \(x + 1.1\). Using these expressions, we can create equations based on distance and time.
For our problem, the expression \[2x + 2(x + 1.1) = 11\] describes the total distance covered. Breaking down this expression involves distributing and combining like terms, simplifying it, and then solving for \(x\). This process is critical because it allows us to unravel the relationships between the given quantities.

Problem-Solving Strategies

Problem-solving strategies in rate and distance problems often involve several steps, including understanding the problem, creating and simplifying equations, and finding solutions.
First, clarify what's given and what you need to find. This involves identifying the rates, distances, and time intervals involved. Next, use algebraic expressions to represent these relationships mathematically.
Simplifying equations can be tricky, but it's important to break down the steps:

• Translate the problem into expressions.
• Simplify these expressions by distributing and combining like terms.
• Isolate variables to find their values.

By iteratively checking each step, you can be sure you correctly capture all the relationships in the problem. For example, after finding \(x\), it's good practice to verify the solution by substituting it back into the original context to ensure it makes sense. This structured approach is vital in solving rate and distance problems efficiently and accurately.