Problem 43 - Prealgebra

Question

The formula \(d=r t\) can be rewritten as \(\frac{d}{t}=r .\) How is the rate affected if the time \(t\) increases and the distance \(d\) remains the same? A It increases. B It decreases. C It remains the same. D There is not enough information.

Step-by-Step Solution

Answer

B It decreases.

Step 1: Introducing the Formula

We start with the formula for rate, which is given as \( r = \frac{d}{t} \). This formula states that the rate is equal to the distance \(d\) divided by the time \(t\).

Step 2: Understanding the Variables

In the context of the question, \(d\) represents distance and \(t\) represents time. The exercise tells us to consider how the rate \(r\) changes when \(t\) increases, while \(d\) stays constant.

Step 3: Analyzing the Relationship

Since \(r = \frac{d}{t}\), as \(t\) (the denominator) increases and \(d\) remains constant, the value of \(r\) must decrease. This is because dividing a constant \(d\) by a larger \(t\) results in a smaller quotient.

Step 4: Conclusion

Based on the formula \(r = \frac{d}{t}\), increasing \(t\) while keeping \(d\) constant will cause \(r\) to decrease. Therefore, the rate decreases when time increases and distance remains unchanged.

Key Concepts

Understanding the Distance Formula

The distance formula can be written as \( d = rt \), where \(d\) represents distance, \(r\) is the rate (or speed), and \(t\) stands for time. This formula is a fundamental concept in physics and mathematics. It helps us calculate one of these three variables when we have the other two. Understanding this formula is crucial because it forms the basis for analyzing motion and various practical problems, such as traveling or scheduling.
A simple way to look at it is to think of distance as the product of how fast someone is moving (rate) and how long they have been moving (time). If you know the speed and the time traveled, you can easily find the distance. For example:

• If a car travels at 60 miles per hour for 2 hours, the distance covered is \(d = 60 \times 2 = 120\) miles.
• If a runner runs at 10 km/h for 3 hours, the distance is \(d = 10 \times 3 = 30\) kilometers.

Using this formula, you can rearrange it to solve for different variables, making it a versatile tool in problem-solving.

The Relationship between Rate, Time, and Distance

In the formula \( r = \frac{d}{t} \), rate is calculated as the distance divided by time. This means the rate indicates how much distance is covered per unit of time. Let's break this down with an example:

• If a cyclist covers 50 kilometers in 2 hours, their rate is \( \frac{50}{2} = 25 \) km/h. This means the cyclist moves 25 kilometers every hour.
• If a plane covers 1000 miles in 4 hours, its rate is \( \frac{1000}{4} = 250 \) miles per hour.

Understanding the rate and time relationship is important because it directly affects how we calculate speed and plan travel.
This direct relationship tells us that if the time taken to travel increases while the distance remains constant, the rate must decrease. For instance, if it takes you more hours to cover the same distance, your average speed or rate decreases.

Inverse Proportionality in Motion

Inverse proportionality is a mathematical concept where an increase in one variable results in a decrease in another, and vice versa, provided a constant product. In our scenario, the rate and time relationship \( r = \frac{d}{t} \), as \(t\) increases and \(d\) remains constant, demonstrates inverse proportionality.

• As time \(t\) becomes larger, the rate \(r\) must become smaller because dividing by a larger number gives a smaller result. This inverse relationship ensures that the formula holds true and balance is maintained.
• Conversely, if time decreases while keeping distance constant, the rate will increase, showing the reciprocal effect.

Understanding this concept of inverse proportionality can clarify why, for a constant journey, increasing the time it takes will decrease the speed. It also helps us comprehend various real-world phenomena, such as traffic flow, where slower speeds result often from longer travel times over the same distances. Grasping this idea can thus aid in strategic planning for optimizing travel and resources.