Unit Conversions (Part 1):

Introduction to Unit Conversions

The following video will show how to do unit conversions as well as explain why it works.

Introduction to Unit Conversions

Unit conversion = taking the measurement of something in one set of units and changing it to an equivalent measurement in another set of units.

The keys to doing unit conversions are the concepts that anything divided by itself is equal to 1 and that anything multiplied by 1 is equal to itself.

$\frac{\text{x}}{\text{x}}=1$

and

$1{\text{x}}={\text{x}}$

Conversion Factors

Conversion factors are fractions where the item in the numerator is equal to the item in the denominator, essentially making the fraction equal to 1.

Examples:

1 inch = 2.54 cm, therefore:

$\frac{1\text{ inch}}{2.54\text{ cm}}=1$

This is the conversion factor between inches and centimeters.

Another example is that 60 minutes = 1 hour, therefore:

$\frac{60\text{ minutes}}{1\text{ hour}}=1$

This is the conversion factor between minutes and hours.

Example 1: Inches to Centimeters

How do we convert 3 inches into the equivalent length in centimeters?

This image shows six squares stacked together in two rows of three making a rectangle that is three wide and two high. Each individual square is 1 inch by 1 inch.

Start with what we know: 3 inches.
Determine what we want to get in the end: centimeters.
Determine the conversion factor to use: 1 inch = 2.54 cm.
Multiply by 1 in the form of the conversion factor that cancels out the unwanted units. In this case, $\frac{2.54\:\text{cm}}{1\:\text{in}}$.

$ \frac{\color{DarkGreen}3 {\cancel{\text{ inches}}}}{1}\left ( \frac{{\color{DarkOrange} 2.54\:\text{cm}}}{{\color{DarkGreen} 1\:\cancel{\text{ inch}} }} \right ) = \frac{\left ( {\color{DarkGreen} 3} \right )\left ( {\color{DarkOrange} 2.54\:\text{cm}}\right )}{1} = {\color{Black} 7.62\:\text{cm}} $

Answer: 3 in = 7.62 cm

Example 2: Minutes to Hours

How do we convert 14 minutes into the equivalent time in hours?

14 minutes = ? hours

Start with what we know: 14 minutes.
Determine what we want to get in the end: hours.
Determine the conversion factor to use: 60 minutes = 1 hour.
Multiply by 1 in the form of the conversion factor that cancels out the unwanted units. In this case, $\frac{1\:\text{hour}}{60\:\text{minutes}}$.

${14\color{Magenta}\cancel{\text{ minutes}}}\times \frac{{1\color{green} \text{ hour}}}{{60\color{Magenta} \cancel{\text{ minutes}}}}=\frac{(14\times 1){\color{green} \text{hours}}}{60} $

$=0.23{\color{green}\text{ hours}} $

Answer: 14 minutes = 0.23 hours

Additional Resources

Khan Academy: Converting Units: U.S. Volume (5:16 mins; Transcript)
Khan Academy: Converting Units: Metric Distance (4:01 mins; Transcript)
Khan Academy: Same Length in Different Units (5:47 mins; Transcript)

Practice Problems

The length of a table measures 250 centimeters. There are 100 centimeters in 1 meter. Convert the length of the table to meters. Round to the nearest tenth.

Margaret has a pitcher filled with 2 liters of water. One liter is equal to approximately 4.23 cups. Convert the amount of water in Margaret’s pitcher to cups. Round to the nearest hundredth.

Kim walked 2.5 miles to the grocery store. There are 1760 yards in one mile. Convert the distance Kim walked to yards. Round to the nearest whole number.

Brent has £35 (British pounds) that he would like to convert to US dollars (US $). Assume the current conversion rate is £1 = $1.27. How much money will he have in dollars rounded to the nearest hundredth?

Max ran a half marathon which is 13.1 miles in length. There are approximately 0.62 miles in 1 kilometer. Convert the length of the half marathon to kilometers. Round to the nearest whole number.

A piano weighs 325 pounds. There are approximately 2.2 pounds in 1 kilogram. Convert the weight of the piano to kilograms. Round to the nearest whole number.

Zeezrom tried to tempt Amulek by offering him six onties of silver, which are of great worth, if he would deny the existence of a Supreme Being. (See Alma 11:21-25.) Of course, Amulek refuses the money. In the Nephite market, one onti of silver is the amount of money a judge earns after seven working days (Alma 11:3, 6-13.) How many days would a judge have to work to earn six onties of silver?

Solutions

2.5 meters
8.46 cups
4400 yd (Written Solution)
We know that Kim walked 2.5 miles to the grocery store, but we want to know how far that is in yards. To figure that out we need 3 things:
1. The original amount: 2.5 miles
2. The desired units: yards
3. The conversion rate: 1760 yards = 1 mile
The first step is to write 2.5 miles as a fraction. The equivalent fraction is: $\frac{2.5\text{ miles}}{1} $

Now we need to figure out the conversion factor that we need to use. We need it to cancel out the miles and leave us with yards. Using our conversion rate, 1760 yards = 1 mile, we know it will either be:

$\frac{1760\text{ yards}}{1\text{ mile}} $ or $\frac{1\text{ mile}}{1760\text{ yards}} $

Since miles is at the top of our original fraction, we must use the conversion factor with miles in the bottom of the fraction so that miles will cancel out when we multiply. We set it up like this:

$\frac{2.5\:\text{miles}}{1}\cdot\frac{1760\:\text{yards}}{1\:\text{mile}} $

The first step is to cancel out the miles:

$\frac{2.5\:{\color{DarkOrange} \cancel{\text{miles}}}}{1}\cdot \frac{1760\:\text{yards}}{1\:{\color{DarkOrange} \cancel{\text{mile}}}}$

Then we multiply straight across:

$\frac{2.5\:\cdot \: 1760 \text{ yards}}{1\:\cdot\: 1} $

Then simplify both the top and bottom of the fraction:

$= \frac{4400\text{ yards}}{1}$

Which equals:

= 4400 yards

So 2.5 miles is the same distance as 4400 yards.
$44.45 (Written Solution)
Brent has £35 and needs to know how much it is in US dollars. To figure that out we need three things:
1. The original amount: £35
2. The desired units: US dollars (US $)
3. The conversion rate: £1 = $1.27
The first step is to write our original amount as a fraction: $\frac{{\text{£}}35}{1}$

Using the conversion rate, £1 = $1.27, we can figure out our conversion factors. The two options are:

$\frac{£1}{\$1.27}$ or $\frac{\$1.27}{£1}$

We are changing British pounds (£) to US dollars ($) and £ is in the top (or numerator) of our original amount so we must use the conversion rate with £ in the bottom of the fraction so that we can cancel out the £.

$\frac{\$1.27}{£1}$

Next, we multiply our original amount by our conversion factor:

$\frac{£35}{1}\times\frac{\$1.27}{£1}$

Now we cancel out the £ since there is £ in the top and £ in the bottom:

$\frac{\cancel{{\color{Magenta} £}}35}{1}\times\frac{\$1.27}{\cancel{{\color{Magenta} £}}1}$

Next, we multiply straight across:

$\frac{35\:\cdot\:\$1.27}{1\:\cdot\:1}$

Then simplify top and bottom:

$=\frac{\$44.45}{1}$

Which gives us:

= $44.45

This means that £35 is the same amount of money as $44.45.
21 km

Solution Video #1 - #5 | Transcript

148 kg
42 days

Solution Video #6 -#7 | Transcript