Unit Conversions (Part 2):

Perform Unit Conversions for Speeds

When we do unit conversions for speed, we have units in the numerator and denominator that we need to change to what we want. The process is exactly the same, we just need to watch 2 sets of units. The following video will explain how to do this by doing an example problem:

Video Source (03:56 mins) | Transcript

Steps for Speeds Unit Conversions

Start with what you know (On top and bottom)
Determine what you want to get in the end. (On top and bottom.)
Determine what conversion factor(s) to use. You will need more than one. (at least one for the top and one for the bottom)
Multiply by 1 in the form of the conversion factor that cancels out the unwanted units.

Additional Resources

Khan Academy: Intro to Rates (05:29 mins; Transcript)
Khan Academy: Solving Unit Rate Problem (02:30 mins; Transcript)
Khan Academy: Worked Example: Rate Problem (04:12 mins; Transcript)
Khan Academy: Multiple Rates Word Problem (06:27 mins; Transcript)
Khan Academy: Finding Average Speed or Rate (13:14 mins; Transcript)

Practice Problems

Alice was roller-skating down the street at a speed of 9 kilometers per hour (km/h). Use the fact that 1 kilometer is approximately equal to 0.6214 miles to convert this speed to miles per hour (mph). Round your answer to the nearest tenth.
1 km = 0.6214 mi
Note that mph or miles per hour is the same as miles/hour or $\frac{\text{miles}}{\text{hour}}$.

A bus is traveling at 65 miles per hour (mph). Use the fact that 1 mile is approximately equal to 1.609 kilometers to convert this speed to kilometers per hour (km/h). Round your answer to the nearest hundredth.
1 mi = 1.609 km

Kirk takes a ride on a train traveling 80 miles per hour (mph). Use the fact that 1 mile is approximately equal to 1609.344 meters and 1 hour is equal to 60 minutes to convert the speed of the train to meters per minute (m/min). Round to the nearest tenth.
1 mi = 1609.344 m
1 h = 60 min

A deep sea diver begins to move up to the surface at a rate of 20 feet per minute (ft/min). Use the following facts to convert her speed to meters per second (m/sec). Round to the nearest thousandth.
1 m = 3.2808 ft
1 min = 60 sec

Usain Bolt holds the world record for the 100-meter dash. He finished the 100 meters in 9.58 seconds. Use the following information to convert his speed to feet per minute (ft/min). Round to the nearest whole number.
1 m = 3.2808 ft
1 min = 60 sec

A horse was observed galloping at a speed of 11 meters per second (m/sec). Use the following facts to convert this speed to kilometers per hour (km/h).
Round to the nearest tenth.
1 km = 1000 m
1 min = 60 sec
1 hour = 60 min

Solutions

$\frac {9\text{km}}{1\text{h}} \times \frac {0.6214\text{mi}}{1\text{km}} = 5.6 \text{mph}$

$\frac {65\text{mi}}{1\text{h}} \times \frac {1.609\text{km}}{1\text{mi}} = 104.59\text{km/h}$

$\frac {80\text{mi}}{1\text{h}} \times \frac{1\text{h}}{60\text{min}} \times\frac {1609.344\text{m}}{1\text{mi}} = 2145.8 \frac {\text{m}}{\text{min}}$(Written Solution)

Step 1: Start with what you know is on top and bottom of the fraction (numerator and denominator, respectively)

The train is traveling at a speed of 80 miles per hour, which can be written as $\frac{80\text{mi}}{1\text{hour}}$

Step 2: Determine what you want to get in the end (on top and bottom)

We want to know how fast the train is going in $\frac{\text{m}}{\text{min}}$

Step 3: Determine what conversion factor(s) to use. You will need more than one (at least one for the top and one for the bottom).

We know that:

1 mi = 1609.344 m, so we will either use $\frac{1\text{mi}}{1609.344\text{m}} $ or $\frac{1609.344\text{m}}{1\text{mi}}$

1 hour = 60 min, so we will either use $\frac{1\text{hour}}{60\text{min}}$ or $\frac{60\text{min}}{1\text{hour}}$

Step 4: Multiply by 1 in the form of the conversion factor that cancels out the unwanted units.

We know that we need to change hours to minutes and miles to meters. We need to decide which conversion factors will allow us to do that.

$\frac{80\text{mi}}{1\text{hour}}\times\:?\:\times\:?\:=\frac{?\:\text{min}}{?\:\text{m}}$

Note that we chose the conversion factors that will enable us to cancel out the existing units and change them to the desired units (ie. hour is in the bottom of our original fraction, so we chose the conversion factor that has hour in the top so they will cancel out, etc.).

$\frac{80\text{mi}}{1\text{hour}}\times\frac{1\text{hour}} {60\text{min}}\:\times\frac{1609.344\text{m}}{1\text{mi}}$

Next, we cancel out miles:

$3 fractions multiplied together. 80 miles over 1 hour times 1 hour over 60 minutes times 1609.344 meters over 1 mile. The miles in the numerator of the first fraction and the denominator of the last fraction have been crossed out.$

Then cancel out hours:

$This is the same as the previous image of three fractions being multiplied together. This time the hours in the denominator of the first fraction and the hours in the numerator of the second fraction have been crossed out.$

Using the zig-zag method we make the calculations in a zig-zag pattern. Remember, any time we move to the denominator we divide. And any time we move to the numerator, we multiply:

$This is the same image as the previous two images except now there are arrows indicating the zig-zag method. Going from left to right, 80 in the numerator of the first fraction is divided by the 1 in the denominator. Then multiply by the 1 in the numerator of the second fraction and divide by the 60 in the denominator. Finally, multiply by the 1609.344 m in the numerator of the third fraction and divide by 1 in the denominator.$

80 ÷ 1 × 1 ÷ 60 min × 1609.344 m ÷ 1 = 2145.792

So the train is traveling at a rate of $\frac{2145.8\:\text{m}}{1\:\text{min}} = 2145.8\frac{\text{meters}}{\text{minute}}$ when rounded to the nearest tenth.

$\frac{20\text{ft}}{1\text {min}}\times\frac{1\text{min}}{60\text{sec}}\times\frac{1\text{m}}{3.2808\text{ft}} = 0.102 \frac{\text{m}}{\text{sec}}$

$\frac{100\text{m}}{9.58\text{sec}}\times\frac{3.2808\text{ft}}{1\text{m}}\times\frac{60\text{sec}}{1\text{min}} = 2055 \frac{\text{ft}}{\text{min}}$

$\frac{11\text{m}}{1\text{sec}}\times\frac{1\text{km}}{1000\text{m}}\times\frac{60\text{sec}}{1\text{min}}\times\frac{60\text{min}}{1\text{h}} = 39.6 \frac{\text{km}}{\text{h}}$ (Written Solution)

Step 1: Start with what you know (on top and bottom)

The horse is galloping at a speed of 11 meters per second, which can be written as $\frac{11\text{m}}{1\text{sec}}$

Step 2: Determine what you want to get in the end. (on top and bottom)

We want to know how fast the horse is going in $\frac{\text{km}}{\text{h}}$

Step 3: Determine what conversion factor(s) to use. You will need more than one: at least one for the top (numerator) and one for the bottom (denominator).

We know that:

1 km = 1000 m, so we will either use $\frac{1\text{km}}{1000\text{m}}$ or $\frac{1000\text{m}}{1\text{km}}$

1 min = 60 sec, so we will either use $\frac{1\text{min}}{60\text{sec}}$ or $\frac{60\text{sec}}{1\text{min}}$

1 hour = 60 min, so we will either use $\frac{1\text{hour}}{60\text{min}}$ or $\frac{60\text{min}}{1\text{hour}}$

Step 4: Multiply by 1 in the form of the conversion factor that cancels out the unwanted units.

We know that we need to change seconds to minutes to hours and meters to kilometers. We need to decide which conversion factors will allow us to do that.

$\frac{11\:\text{m}}{1\:\text{sec}}\:\times\:?\:\times\:?\:\times\:?\:=\frac{?\:\text{km}}{?\:\text{hour}}$

Note that we chose the conversion factors that will enable us to cancel out the existing units and change them to the desired units (ie. seconds is in the bottom of our original fraction, so we chose the conversion factor with seconds in the top so they will cancel out, etc.).

$\frac{11\:\text{m}}{1\:\text{sec}}\times\frac{60\:\text{sec}}{1\:\text{min}}\times\frac{60\:\text{min}}{1\:\text{hour}}\times\frac{1\:\text{km}}{1000\:\text{m}}$

Next, we cancel out seconds:

$\frac{11\:\text{m}}{1\:\cancel{{\color{Red} \text{sec}}}}\times\frac{60\:\cancel{{\color{Red} \text{sec}}}}{1\:\text{min}}\times\frac{60\:\text{min}}{1\:\text{hour}}\times\frac{1\:\text{km}}{1000\:\text{m}}$

Then cancel out meters:

$\frac{11\:\cancel{{\color{Purple} \text{m}}}}{1\:\cancel{{\color{Red} \text{sec}}}}\times\frac{60\:\cancel{{\color{Red} \text{sec}}}}{1\:\text{min}}\times\frac{60\:\text{min}}{1\:\text{hour}}\times\frac{1\:\text{km}}{1000\:\cancel{{\color{Purple} \text{m}}}}$

Then cancel minutes:

$\frac{11\:\cancel{{\color{Blue} \text{m}}}}{1\:\cancel{{\color{Red} \text{sec}}}}\times\frac{60\:\cancel{{\color{Red} \text{sec}}}}{1\:\cancel{{\color{Teal} \text{min}}}}\times\frac{60\:\cancel{{\color{Teal} \text{min}}}}{1\:\text{hour}}\times\frac{1\:\text{km}}{1000\:\cancel{{\color{Blue} \text{m}}}}$

Using the zig-zag method we make the calculations in a zig-zag pattern. Remember, any time we move to the denominator we divide. Any time we move to the numerator, we multiply:

$Four fractions multiplied together with arrows indicating the zig-zag method. 11 in the numerator of the first fraction divided by the 1 in its denominator. Next multiply by 60 in the numerator of the second fraction and divide by the 1 in its denominator. Next, multiply by 60 in the numerator of the third fraction and divide by the 1 hour in its denominator. Finally, multiply by the 1 km in the numerator of the last fraction and divide by the 1000 in its denominator.$

11 ÷ 1 × 60 ÷ 1 × 60 ÷ 1 hour × 1 km ÷ 1000 = 39.6 kilometers per hour

So the horse is traveling at a rate of $\frac{39.6\:\text{km}}{\text{hr}}$ or 39.6 km/h.