Write each formula as an English phrase using the word varies or proportional. r = d/t, where r is the speed when traveling d miles in t hours.

Verified step by step guidance

1. Identify the formula given: $r=\frac{d}{t}$.
2. Recognize that $r$ represents speed, $d$ represents distance, and $t$ represents time.
3. Understand that the formula $r=\frac{d}{t}$ expresses a relationship between speed, distance, and time.
4. Translate the formula into an English phrase: 'The speed $r$ is directly proportional to the distance $d$ and inversely proportional to the time $t$.'

Key Concepts

Direct Variation

Direct variation describes a relationship between two variables where one variable is a constant multiple of the other. In the context of the formula r = d/t, speed (r) varies directly with distance (d) when time (t) is held constant. This means that as distance increases, speed increases proportionally, assuming the time taken remains unchanged.

Proportional Relationships

A proportional relationship indicates that two quantities maintain a constant ratio. In the equation r = d/t, speed is proportional to distance when time is constant. This implies that if you double the distance, the speed will also double, illustrating how one quantity varies in relation to another.

Rate of Change

The rate of change refers to how one quantity changes in relation to another. In the formula r = d/t, the rate of change is represented by speed (r), which changes as distance (d) changes over a fixed time (t). Understanding this concept is crucial for interpreting how variations in distance affect speed in real-world scenarios.