If we’re driving at 25 miles per hour, we might want to know how long it will take us to get to a location. If we are training for a race, we might want to know how fast we have to run in order to achieve a certain time. We can use the units of these rates to help us get the answers we want!
We can represent "miles per hour" as \(\frac{\text{miles}}{\text{hour}}\). We can write "minutes per mile" as \(\frac{\text{minutes}}{\text{mile}}\). When we say we want to run \(2\) miles in \(15\) minutes, we can write the fraction \(\frac{2 \text{ miles}}{15 \text{ minutes}}\). We can make these units cross cancel in order to find the units we want for our final answer.
1.
A person wants to run a 5k in 20 minutes. Their phone app is only
telling them how many minutes per mile they’re running. How many minutes
per mile should they be running if they want to run 5 kilometers in 20
minutes?
Use the conversion 1 miles \(\approx\) 1.61 km
We are told that we want to run \(5\) kilometers in \(20\) minutes. We could say "I ran 5 kilometers in 20 minutes"
\[\frac{5 \text{ kilometers}}{20 \text{ minutes}}\]
or "I spent 20 minutes running 5 kilometers"
\[\frac{20 \text{ minutes}}{5 \text{ kilometers}}\]
Either fraction works! We can decide between the two by focusing on our goal. We want to calculate "minutes per mile". Those units look like this in a fraction:
\[\frac{\text{minutes}}{\text{mile}}\]
That means time should be in the numerator. We will use the second stated fraction because time is in the numerator.
\[\frac{20 \text{ minutes}}{5 \text{ kilometers}}\]
We need to change the denominator to miles. We can use the conversion factor. 1 mile is approximately 1.61 kilometers,
\[\frac{1 \text{ mile}}{1.61 \text{ kilometers}}\]
or we could say 1.61 kilometers is 1 mile.
\[\frac{1.61 \text{ kilometers}}{1 \text{ mile}}\]
We will use the second of these two fractions so that the kilometers units will cross cancel. Multiply our given fraction by the conversion factor to make the units cross cancel.
\[\frac{20 \text{ minutes}}{5 \text{ kilometers}} \cdot \frac{1.61 \text{ kilometers}}{1 \text{ mile}} =\]
\[ \frac{20 \text{ minutes}}{5 \cancel{\text{ kilometers}}} \cdot \frac{1.61 \cancel{\text{kilometers}}}{1 \text{ mile}} =\]
\[\frac{20 \text{ minutes}}{5} \cdot \frac{1.61 }{1 \text{ mile}} =\]
\[\frac{20 \text{ minutes} \cdot 1.61}{5 \text{ miles}} =\]
\[\frac{32.2 \text{ minutes}}{5 \text{ miles}}\]
Divide 32.2 by 5 to get
\[6.44 \frac{\text{minutes}}{\text{mile}}\]
The \(0.44\) minutes might be hard to interpret. There are 60 seconds in a minute. \(0.44 \cdot 60 = 26.4\) seconds. The person must run at a pace of 6 minutes and 26.4 seconds per mile.
2.
You estimate that you’ll average 25 miles per hour when driving
across town (because you speed like a heathen, but you’ll be slowed by
lights and turns). Your friend’s house is 2.5 miles away.
How many minutes is the trip?
We are asked for "minutes" for units. We can use our given rate of 25 miles per hour
\[\frac{25 \text{ miles}}{\text{hour}}\]
but "minutes" should be in the numerator. A person could say they drove 25 miles for every hour, or they could say they spent one hour driving 25 miles. Both are equivalent!
\[\frac{\text{hour}}{25 \text{ miles}}\]
We want just time for our final units. That means the distance measurement "miles" should cancel out. We can use the fact that we drive 2.5 miles for this trip. Multiply the above fraction by 2.5 miles to make the miles units cancel out.
\[\frac{\text{hour}}{25 \text{ miles}} \cdot 2.5 \text{ miles}\]
\[\frac{\text{hour}}{25 \cancel{\text{miles}}} \cdot 2.5 \cancel{\text{ miles}}\]
\[\frac{\text{hour}}{25} \cdot 2.5\]
\[\frac{\text{hour}}{25} \cdot \frac{2.5}{1}\]
\[\frac{2.5 \text{ hours}}{25}\]
Divide 2.5 by 25
\[0.1 \text{ hours}\]
1. A person wants to run a 10k in 42 minutes. Their phone app is only telling them how many minutes per mile they’re running. How many minutes per mile should they be running if they want to run 10 kilometers in 42 minutes?
2. A person is driving 30 miles per hour. The total distance of the trip is 4 miles. How many minutes does it take to make the trip?
1. If a person runs a 5k in 22 minutes, then runs a 5k in 21 minutes, then runs a 5k in 20 minutes, their time is decreasing by the same amount each race (1 minute). Is their minutes per mile also decreasing by a constant rate? Are there example calculations you can make to support your answer?
2. If a person likes to drive the speed limit, should they always estimate their speed in city traffic to be the speed limit? Why or why not?