Problem 289 - Elementary Algebra

Question

In the following exercises, solve. Sisters Helen and Anne live 332 miles apart. For Thanksgiving, they met at their other sister's house partway between their homes. Helen drove 3.2 hours and Anne drove 2.8 hours. Helen's average speed was four miles per hour faster than Anne's. Find Helen's average speed and Anne's average speed.

Step-by-Step Solution

Answer

Helen's average speed is 29.53 mph and Anne's average speed is 25.53 mph.

Step 1: Define Variables

Let Helen's average speed be represented as \( h \) (in miles per hour) and Anne's average speed be represented as \( a \) (in miles per hour). According to the problem, Helen's speed is 4 miles per hour faster than Anne's. Therefore, \( h = a + 4 \).

Step 2: Write Distance Formulas

Using the formula for distance, which is \( \text{Distance} = \text{Speed} \times \text{Time} \), write the expressions for the distances that Helen and Anne drove: - Helen's distance: \( D_h = h \times 3.2 \). - Anne's distance: \( D_a = a \times 2.8 \).

Step 3: Set Up the Total Distance Equation

Since the total distance between their homes is 332 miles, the sum of the distances each drove should equal half of that (because they met halfway): \( h \times 3.2 + a \times 2.8 = 332 \times 0.5 \). Simplify the total distance equation to: \( h \times 3.2 + a \times 2.8 = 166 \).

Step 4: Substitute and Solve for One Variable

Use the relation \( h = a + 4 \) and substitute \( h \) in the distance equation: \( (a + 4) \times 3.2 + a \times 2.8 = 166 \) Distribute and simplify: \( 3.2a + 12.8 + 2.8a = 166 \) Combine like terms: \( 6a + 12.8 = 166 \). Isolate \( a \) by subtracting 12.8 from both sides: \( 6a = 153.2 \), Thus, \( a = 25.53 \).

Step 5: Calculate Helen's Speed

Use \( a \) to find \( h \) using the relation \( h = a + 4 \): \( h = 25.53 + 4 = 29.53 \).

Key Concepts

Understanding the Distance-Speed-Time Relationship

In solving algebra word problems, understanding the relationship between distance, speed, and time is crucial. The fundamental formula used is:
\(\text{Distance} = \text{Speed} \times \text{Time}\).
This formula helps in calculating any one of these variables if the other two variables are known.
For example, if you know the speed and time, you can easily find the distance by multiplying those values.
Similarly, to find speed, you can rearrange the formula to \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \].
When solving word problems, always identify and isolate the variables involved. Knowing the relationship between these three variables lets you set up equations that can be solved systematically.

Solving Linear Equations

Linear equations form the backbone of most algebra word problems. The standard form of a linear equation is:
\ ax + b = c \
where \(x\) represents the variable you need to solve for, and \(a, b,\) and \(c\) are constants.
To solve a linear equation, you perform operations that isolate the variable on one side of the equation.
This often involves:

• Combining like terms
• Adding or subtracting terms from both sides
• Multiplying or dividing both sides by a constant

Once the variable is isolated, you can determine its value. In the example problem, the relationship \( h = a + 4 \) is an example of a linear equation where \(h\) and \(a\) are Helen's and Anne's speeds respectively.

Substitution of Variables

Variable substitution is a technique used to simplify or solve equations.
It's especially useful when you have more than one equation involving the same variables.
In our problem, we know from the given information that Helen's speed \(h\) is 4 miles per hour faster than Anne's speed \(a\).
Therefore, we express \(h\) as \(h = a + 4\).
This equation allows us to replace \(h\) in the main equation with \(a + 4\), reducing the number of variables in one equation.
The step-by-step process leading to solving one variable makes the problem less complex and more manageable.

Calculating Average Speed

Average speed is determined by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \].
This formula applies whether you are driving, walking, or biking.
In our example, Helen and Anne have different average speeds and cover different distances.
We use each person's speed and time to calculate their individual distances driven as part of the overall solution.
Specifically, the problem provides the total distance Helen and Anne combined should travel, which helps in setting up our equations correctly.
Calculating these individual average speeds correctly often leads to the final answer.