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Introduction to Unit Conversions
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Introduction

In this lesson, you will learn how to convert from one unit to another using what are called unit conversions (simple equations to help you change from unit to unit). Unit conversions is just a matter of taking the measurement of something in one set of units and changing it to an equivalent measurement but in a different set of units. This lesson covers the conversion of hours (hr) and minutes (min) as well as inches (in) and centimeters (cm).


This video illustrates the lesson material below. Watching the video is optional.


Unit Conversions


A unit conversion is taking the measurement of something in one set of units and changing it to an equivalent measurement in another set of units.

Example 1
A rectangle includes six, equally sized boxes, three in the top row, and three in the bottom row. Each box represents an inch. The base of the rectangle is three inches. What is that length in centimeters?

A rectangle made of 6 equally-sized boxes, 3 in the top row and 3 in the bottom row. Each box is one inch wide. 

Figure 1

The calculation to do this is a unit conversion because you are converting from inches to an equivalent length in centimeters.

\begin{align*} inches \rightarrow centimeters \end{align*}

The key to unit conversions is the concept that:

  • anything divided by itself is equal to one such as \(\large\frac{x}{x} = 1\), and
  • one multiplied by anything is equal to itself as well \(1x =x\).

Fractions used in unit conversions are called conversion factors. A conversion factor is a key that shows something in the numerator and denominator that are equal to each other. For example:
\begin{align*} 1 \;inch &= 2.54\; centimeters &\color{red}\small\text{Unit equivalents}\\\\ &\frac{1\;inch}{2.54\;centimeters} &\color{red}\small\text{Conversion factor}\\\\ &\frac{2.54\;centimeters}{1\;inch} &\color{red}\small\text{Conversion factor} \end{align*}

Both fractions are equal to one. They don’t look the same, but they represent the same length. They both represent the same conversion factor.

Take what you know and multiply it by the conversion factor. Remember that the conversion factor has an interchangeable numerator and denominator, meaning that you can put either number in the top or bottom and it will not change the conversion factor. Here you put inches in the denominator because you are trying to convert to centimeters.

\begin{align*} &\frac{3in}{1}\times \frac{2.54cm}{1in} &\color{red}\small\text{Multiply by the conversion factor} \end{align*}

The units that are the same in opposite positions (numerator and denominator), will cancel out.

\begin{align*} &\frac {3 \color{red}\cancel{in}}{1} \times \frac{2.54cm}{1 \color{red}\cancel{in}} &\color{red}\small\text{Inches cancel out} \end{align*}

In this case, the inches cancel out. You are left with \(3\times2.54cm\) over 1.

\begin{align*} &\frac{3 \times 2.54cm}{1} = 7.62cm &\color{red}\small\text{Simplify the fraction} \end{align*}

After simplifying, you are left with 7.62cm. Therefore,

\begin{align*} 3in &= 7.62cm\\ &or \\
7.62cm &= 3in \end{align*}

Example 2
How many hours is 14 minutes?

\begin{align*}14\; minutes = ?\; hours \end{align*}

Note this conversion factor:

\begin{align*} 60 \;minutes&= 1\; hour&\color{red}\small\text{Unit equivalents}\\\\ &\frac{1\;hour}{60\;minutes}&\color{red}\small\text{Conversion factor}\\\\ &\frac{60\;minutes}{1\;hour}&\color{red}\small\text{Conversion factor}\\\\ \end{align*}

Both fractions are equal to one. They don’t look the same, but they represent the same length of time. They both represent the same conversion factor.

Now, multiply what you know by the conversion factor.

\begin{align*} &\frac{14\;min}{1}\times \frac{1\;hr}{60\;min} &\color{red}\small\text{Multiply by the conversion factor}\\\\ & \frac {14 \color{red}\cancel{min}}{1} \times \frac{1hr}{60 \color{red}\cancel{min}} &\color{red}\small\text{Minutes cancel out}\\\\
&\frac{14 hr}{60} &\color{red}\small\text{Multiply fractions across}\\\\
& 0.23hr &\color{red}\small\text{Simplify the fraction} \end{align*}

14 minutes is 0.23 hours.

In review, start with what you know and determine what you want in the end. Next, find a conversion factor that changes the units. Multiply what you know by the conversion factor in such a way that the undesired units cancel each other out. Simplify the fraction as needed.


Things to Remember


  • Start with what you know and determine what you want in the end.
  • Anything divided by itself always equals one.
  • Units that are opposite (top to bottom) cancel each other out.
  • The numerator and denominator of a conversion factor are interchangeable.

Practice Problems

Click Here for a Video Solution for Problems 1 - 5
| Transcript
1. 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. (
Solution
x
Solution: \(2.5\) meters
)
2. 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. (
Solution
x
Solution: \(8.46\) cups
)
3.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. (
Solution
x
Solution: 4400 yd
Details:
You know that Kim walked \(2.5\) miles to the grocery store, but you want to know how far that is in yards. To figure that out you 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 the following:

\(\dfrac{2.5\text{ miles}}{1}\)

Now you need to figure out the conversion factor that you need to use. You need it to cancel out the miles and leave you with yards. Using the conversion rate, 1760 yards = 1 mile, you know it will be one of the following:

\(\dfrac{1760\text{ yards}}{1\text{ mile}}\) or \(\dfrac{1\text{ mile}}{1760\text{ yards}}\)

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

\(\displaystyle\frac{2.5\:\text{miles}}{1}\cdot\frac{1760\:\text{yards}}{1\:\text{mile}}\)

The first step is to cancel out the miles:

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

Then multiply straight across:

\(\displaystyle\frac{2.5\:\cdot \: 1760 \text{ yards}}{1\:\cdot\: 1}\)

Then simplify both the top and bottom of the fraction:

\(= \dfrac{4400\text{ yards}}{1}\)

Which equals to the following:

= 4400 yards

\(2.5\) miles is the same distance as 4400 yards.
)
4. 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? (
Solution
x
Solution: \($44.45\)
Details:
Brent has \(£35\) and needs to know how much it is in US dollars. To figure that out you 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 the original amount as a fraction:

\(\dfrac{{\text{£}}35}{1}\)

Using the conversion rate, \(£1 = $1.27\), you can figure out the conversion factors. The two options are the following:

\(\dfrac{£1}{\$1.27}\) or \(\dfrac{\$1.27}{£1}\)

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

\(\dfrac{\$1.27}{£1}\)

Next, multiply the original amount by the conversion factor:

\(\displaystyle\frac{£35}{1}\times\frac{\$1.27}{£1}\)

Now cancel out the \(£\) since there is \(£\) in the top and \(£\) in the bottom:

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

Next, multiply straight across:

\(\dfrac{35\:\cdot\:\$1.27}{1\:\cdot\:1}\)

Then simplify top and bottom:

\(=\dfrac{\$44.45}{1}\)

Which gives you the following:

\(= $44.45\)

This means that \(£35\) is the same amount of money as \($44.45\).
)
5. 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. (
Solution
x
Solution: \(21\) km
)

Click Here for a Video Solution for Problems 6 - 7
| Transcript
6. 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. (
Solution
x
Solution: \(148\) kg
)
7. 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? (
Solution
x
Solution: \(42\) days
)

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