Problem 61 - Elementary Algebra

Question

For Problems 61-71, answer each question with an algebraic fraction. If by jogging at a constant rate Joan can complete a race in 40 minutes, how much of the course has she completed at the end of \(m\) minutes?

Step-by-Step Solution

Answer

Joan has completed \( \frac{m}{40} \) of the course in \( m \) minutes.

Step 1: Determine the Rate

To find out how much Joan completes at a constant rate, we first determine her rate of jogging. If she completes the race in 40 minutes, then in each minute, she completes \( \frac{1}{40} \) of the race. Thus, Joan's rate is \( \frac{1}{40} \) of the course per minute.

Step 2: Calculate the Completed Fraction

To find out how much of the course Joan has completed in \( m \) minutes, we multiply her rate by the number of minutes. Thus, we take \( \frac{1}{40} \times m \), which equals \( \frac{m}{40} \). This fraction represents the part of the course completed by Joan after \( m \) minutes.

Key Concepts

Understanding the Rate of Motion

Rate of motion refers to how fast an object is moving. In the context of our exercise, it’s about how quickly Joan completes a portion of the race.

To find her rate, we divide the total task – the whole race – by the time it takes to complete it. Since Joan finishes the race in 40 minutes, her rate is \(\frac{1}{40}\) of the course per minute.

This tells us that every minute, Joan covers \(\frac{1}{40}\) of the entire course. Understanding rate is crucial because it allows us to predict progress over time and plan strategically.

Solving Word Problems in Algebra

Word problems in algebra often involve translating real-world scenarios into mathematical equations. This involves a few key steps:

• Identifying what is given and what you need to find.
• Expressing the problem as an algebraic equation.

In our case, the problem provides that Joan completes a race in 40 minutes and asks us to find out how much of the race she finishes in \(m\) minutes.

By understanding that Joan’s rate is \(\frac{1}{40}\) per minute, we can create an equation: multiplying her rate by time (\