1.4: Converting Between Systems

We have spent the last several sections learning about the U.S. customary system of measurement, and the metric system. As you might guess, there are many applications where it is useful to be able to convert between measurements of length, mass, and volume (capacity) in the two systems.

The table below gives some useful conversions between U.S. and metric measurements:

	Approximate Conversions Between Customary and Metric Systems
Length	1 inch = 2.540 centimeters
	1 foot = 0.3048 m
	1 yard = 0.9144 meters
	1 mile = 1.6093 kilometers
Mass	1 pound = 0.4536 kilograms
	1 ounce = 28 grams
Volume	1 fluid ounce = 29.574 milliliters
	1 quart = 0.9464 liters
	1 gallon = 3.785 liters

Example \(\PageIndex{1}\)

An Olympic sprinter competes in the 400m dash. How many yards is the race?

Solution

400 m = ______ y	A yard is less than a meter, so you expect your answer to be more than 400.
\(\dfrac{400 \mathrm{~m}}{1} \cdot \dfrac{1 \text { yard }}{0.9144 \mathrm{~m}}\)	Using the factor label method, write 400 m as a fraction and use unit fractions to convert it to yards.
\(\dfrac{400 m}{1} \cdot \dfrac{1 \text { yard }}{0.9144 m}=\ldots\) yards	Cancel similar units, multiply, and simplify.
\(\dfrac{400}{1} \cdot \dfrac{1 \text { yard }}{0.9144}=\dfrac{400}{0.9144}\) yards
\(\dfrac{400}{0.9144}=437.45\) yards	400 meters = 437.45 yards
The race is 437.45 yards

Example \(\PageIndex{2}\)

A patient must be weighed prior to surgery so that the proper dosage of anesthesia can be given. A nurse weighs a patient and finds that he weighs 205 pounds. The anesthesiologist prefers weights in kilograms. What is the patient’s weight in kilograms? Round to the nearest hundredth.

Solution

205 lbs = _______ kg	Pounds are smaller than kilograms, so you expect your answer to be less than 205
\(\dfrac{205 \mathrm{lbs}}{1} \cdot \dfrac{1 \mathrm{~kg}}{2.2 \mathrm{lbs}}\)	Using the factor label method, write 205 lbs as a fraction and use unit fractions to convert it to kilograms.
\(\dfrac{205 \mathrm{lbs}}{1} \cdot \dfrac{1 \mathrm{~kg}}{2.2 \mathrm{lbs}}=\)	Cancel similar units, multiply, and simplify. 205 lbs is equal to 93.18 kg The patient’s
\(\dfrac{205}{1} \cdot \dfrac{1 \mathrm{~kg}}{2.2}=\dfrac{205 \cdot 1}{1 \cdot 2.2} \mathrm{~kg}\)
\(\dfrac{205}{2.2}=93.18 \mathrm{~kg}\)	205 lbs is equal to 93.18 kg The
The patient's weight is 93.18 kg.

Example \(\PageIndex{3}\)

Your work is having its annual potluck, and you are asked to bring 5 gallons of lemonade. When you go to the store, each container of lemonade mix says it will make 6 liters. How many containers do you need to buy?

Solution

5 g = ______ liters	Liters are smaller than gallons, so you expect your answer to be more than 5.
\(\dfrac{5 \mathrm{gal}}{1} \cdot \dfrac{3.785 \mathrm{~L}}{1 \mathrm{gal}}\)	Using the factor label method, write 5 gal as a fraction and use unit fractions to convert it to liters.
\(\dfrac{5 \mathrm{gal}}{1} \cdot \dfrac{3.785 \mathrm{~L}}{1 \mathrm{gal}}\) = ______ liters	Cancel similar units, multiply, and simplify.
\(\dfrac{5}{1} \cdot \dfrac{3.785 \mathrm{~L}}{1}=\dfrac{5 \cdot 3.785}{1}\) liters
5∙3.785=18.925 liters
5 gallons = 18.925 liters
\(\dfrac{18.925 \mathrm{~L}}{1} \cdot \dfrac{1 \text { container }}{6 \mathrm{~L}}\)	You need 18.925 liters, and each container of mix will make 6 liters.
\(\dfrac{18.925}{6}=3.15\) containers	You must buy 4 containers of mix (3 is not enough)

Temperature Conversions

There are three commonly used systems for measuring temperature. One such system is usually used in science, and is called the Kelvin scale. We will focus our attention on the other two scales for measuring temperature: Fahrenheit and Celsius. The United States usually uses the Fahrenheit system. In this system, water freezes to ice at 32℉ and boils at 212℉. Another commonly used temperature scale is the Celsius scale, where water freezes to ice at 0℃ and boils at 100℃. You might notice that in the Fahrenheit scale there is a 180 degree difference (212℉ − 32℉) between the temperature where water boils and freezes, while in the Celsius scale there is a 100 degree difference (100℃ – 0℃) between the temperature where water boils and freezes. Since neither of these two temperature scales has an absolute starting point (a lowest possible temperature) we cannot meaningfully compare temperatures in these scales using conversion factors.

Instead, we have temperature conversion formulas which allow us to convert temperatures back and forth between Fahrenheit and Celsius.

To Convert Between	Conversion Formula
Celsius and Fahrenheit	F = 1.8 C + 32

Example \(\PageIndex{4}\)

Your front yard is full of weeds, so you decide to spray it with weed killer. The bottle of weed killer that you purchase says "Do not apply in temperatures below 18 ℃ or above 32 ℃. What are the corresponding temperatures in degrees Fahrenheit?

Solution

F = 1.8C + 32 Both	Both temperatures are in degrees Celsius and need to be converted into degrees Fahrenheit.
F = 1.8(18) + 32 F = 64.4℉	The lower temperature corresponds to 64.4℉.
F = 1.8(32) + 32 F = 89.6℉	The upper temperature corresponds to 89.6℉.
The weed killer should only be applied when the temperature is between 64.4 and 89.6℉.

Example \(\PageIndex{5}\)

When leaving the hospital with your sick child, you are told to return immediately if her temperature exceeds 101.5 ℉. When you get home, you discover your thermometer will only measure temperature in degrees Celsius. At what temperature, in degrees Celsius, would you need to return your child to the hospital?

Solution

F = 1.8C + 32	The temperature is in degrees Fahrenheit and it needs to be converted into degrees Celsius.
101.5 = 1.8(C) + 32 101.5 - 32 = 1.8C	Subtract 32 from both sides of the equation.
69.5 = 1.8C
\(\mathrm{C}=\dfrac{69.5}{1.8}\)	Divide both sides by 1.8
C = 38.61℃	You should return your child to the hospital if her temperature exceeds 38.61℃.

You Try It \(\PageIndex{6}\)

What was the temperature in degrees Celsius, if the evening news reports that the high temperature in Phoenix, Arizona today was 115 ℉?

Answer

46.1℃