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. \(_\square\)
