Problem 39 - Algebra for College Students

Question

Nora traveled from Kansas City to Louisville, a distance of \(520 \mathrm{mi}\), in \(10 \mathrm{hr}\). Find her rate in miles per hour.

Step-by-Step Solution

Answer

Nora's rate is 52 miles per hour.

Step 1: Identify the Given Information

Nora traveled a distance of 520 miles and it took her 10 hours to cover this distance.

Step 2: Recall the Formula for Rate

The formula to find the rate is \[\text{Rate} = \frac{\text{Distance}}{\text{Time}}\].

Step 3: Substitute the Given Values

Substitute the distance (520 miles) and the time (10 hours) into the formula: \[\text{Rate} = \frac{520 \text{ miles}}{10 \text{ hours}}\].

Step 4: Simplify the Expression

Perform the division: \[\text{Rate} = 52 \text{ miles per hour}\].

Key Concepts

Rate Calculation

Understanding how to calculate the rate of travel helps in determining how fast someone or something is moving. The rate formula is a fundamental concept in algebra: \(\text{Rate} = \frac{\text{Distance}}{\text{Time}}\). In the given example, Nora traveled 520 miles in 10 hours. By applying the rate formula: \[ \text{Rate} = \frac{520 \text{ miles}}{10 \text{ hours}} \Rightarrow 52 \text{ miles per hour} \]. This means Nora was traveling at a consistent speed of 52 miles per hour.

Distance Formula

The distance formula plays a crucial role in problems dealing with movement over time. The basic formula to find the distance when rate and time are known is: \[ \text{Distance} = \text{Rate} \times \text{Time} \].
It is essential to remember that the rate and time must be in compatible units to use this formula correctly. For example, if Nora's travel rate is given in miles per hour, the time should also be in hours. In our exercise, if you wanted to find the distance Nora traveled in a specific duration, you could rearrange the formula as shown. The direct relationship between these variables helps in solving various real-life problems.

Unit Conversion

Converting units is a skill often necessary for solving algebraic problems involving distance, rate, and time. For instance, you might need to convert hours into minutes or miles into kilometers.
When converting units, use conversion factors. For example: 1 hour = 60 minutes, 1 mile ≈ 1.60934 kilometers. To convert Nora’s travel distance from miles to kilometers: \[520 \text{ miles} \times 1.60934 \frac{\text{kilometers}}{\text{mile}} \approx 837 \text{ kilometers}\].
Unit conversions ensure all measurements are in suitable units for easier and more accurate calculations. Practice is key to mastering unit conversion.

Simplifying Expressions

Simplifying expressions is a fundamental algebra skill that makes complex problems more manageable. When dealing with formulas such as rate, distance, and time, simplification often involves basic arithmetic.
In the given exercise, the expression \[ \text{Rate} = \frac{520 \text{ miles}}{10 \text{ hours}} \] can be simplified by performing division: \[ 52 \text{ miles per hour} \].
Remember to always follow the order of operations and properly handle each mathematical step to simplify effectively. Simplification not only makes results clearer but also helps you verify that calculations were done correctly.