# Distance, Rate, and Time

A **rate of change** is the ratio between the change in one quantity to the change in another quantity. A very common rate is **speed**, the ratio between distance and time.

## Introduction to Rates

While traveling in your car, your rate of speed might be 180 miles per 3 hours. We often simplify rates to express them as **unit rates**, which is how much of something there is per **one unit** of something else. Using the previous example, your unit rate of speed while traveling in the car is 60 miles per one hour.

If Starr walks 50 meters in 8 seconds, what is her rate of speed?

Starr's rate is 50 meters per 8 seconds, or \(50 \div 8 = 6.25\) meters per second. \(_\square\)

## Distance, Rate, and Time

￼If an object moves at a constant rate of speed, we can determine how far the object travels by multiplying its rate by the time it has been traveling:

\[\text{distance} = \text{rate} \times \text{ time}.\]

Likewise, we can find rate of speed by dividing distance by time

\[\text{rate}=\frac{\text{distance}}{\text{time}}\]

and determine time by dividing distance traveled by rate of speed

\[\text{time}=\frac{\text{distance}}{\text{rate}}.\]

An ant is traveling at an average of 8 millimeters per second. How far will the ant travel in one minute?

We can find the ant's total distance by multiplying \(\text{rate} \times \text{ time}: 8 \text{ mm/s} \times 60 \text{ seconds} = 480 \text { mm}.\ _\square\)

If a car travels at 80 kilometers per hour, how long will it take the car to travel 500 kilometers?

We can find the time by dividing the distance by the rate: \(500 \text{ kilometers} \div 80 \text{ kilometers/hour} = 6.25 \text{ hours}.\ _\square\)

It took a train \(20\) seconds to completely pass a \(500 \)-m iron bridge. When the train was passing through a \(1900 \)-m tunnel, it was invisible from outside of the tunnel for \(30\) seconds. If the speed of the train was constant, what was the length of the train (in meters)?

**Details and Assumptions:**

- The train passes the iron bridge when no part of the train is on the bridge.
- The train is invisible from outside of the tunnel when no part of the train is outside of the tunnel.

## Average Speed

In order to determine average speed, we need to divide the total distance traveled by the total time.

Charlie bikes 500 meters to work in 6 minutes and returns home in 4 minutes. What is his average speed for the day?

Charlie's total distance is 1000 meters and his total time is 10 minutes. Therefore, Charlie's average speed is \(1000 \div 10 = 100\) meters per minute. \(_\square\)

**Cite as:**Distance, Rate, and Time.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/distance-rate-and-time/