Section 1.5: 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 | Mass | Volume |
| 1 inch = 2.540 centimeters | 1 pound = 0.4536 kilograms | 1 fluid ounce = 29.574 milliliters |
| 1 foot = 0.3048 m | 1 ounce = 28 grams | 1 quart = 0.9464 liters |
| 1 yard = 0.9144 meters | <this space intentionally left blank> | 1 gallon = 3.785 liters |
| 1 mile = 1.6093 kilometers | <this space intentionally left blank> | <this space intentionally left blank> |
Example 13
An Olympic sprinter competes in the 400m dash. How many yards is the race?
Solution: State your givens and your goal.
Given: 400m dash
Goal: How many yards is the race?
1) Reason about your answer. A yard is less than a meter, so you expect your answer to be more than 400.
400 m = _____ yd
2) Using the factor label method, write 400 m as a fraction and use unit fractions to convert it to yards.
[latex]\frac{400~m}{1}[/latex] x [latex]\frac{1~yd}{0.9144~m}[/latex] = _____ yd
3) Cancel similar units, multiply, and simplify.
[latex]\frac{400~\bcancel{m}}{1}[/latex] x [latex]\frac{1~yd}{0.9144~\bcancel{m}}[/latex] = [latex]\frac{400~x~1~yd}{1~x~0.9144}[/latex] = 437.45 yd
Answer: A 400 m dash is 437.45 yards.
Example 14
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: State your givens and your goal.
Given: patient weighs 205 pounds
Goal: What is the patient’s weight in kilograms?
1) Reason about your answer. Pounds are smaller than kilograms, so you expect your answer to be less than 205.
205 pounds = _____ kg
2) Using the factor label method, write 205 lbs. as a fraction and use unit fractions to convert it to kilograms.
[latex]\frac{205~lbs}{1}[/latex] x [latex]\frac{1~kg}{2.2~lbs}[/latex] = _____ kg
3) Cancel similar units, multiply, and simplify.
[latex]\frac{205~\bcancel{lbs}}{1}[/latex] x [latex]\frac{1~kg}{2.2~\bcancel{lbs}}[/latex] = [latex]\frac{205~x~1~kg}{2.2}[/latex] = [latex]\frac{205~kg}{2.2}[/latex] ≈ 93.18 kg
Answer: The patient’s weight is 93.18 kg.
Example 15
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: State your givens and your goal.
Given: Need 5 gallons of lemonade. Lemonade mix says it will make 6 liters.
Goal: How many containers do you need to buy?
1) Convert 5 gallons to liters. Reason about your answer. Liters are smaller than gallons, so you expect your answer to be more than 5.
5 gallons = _____ L
2) Using the factor label method, write 5 gal as a fraction and use unit fractions to convert it to liters. Cancel similar units, multiply, and simplify.
[latex]\frac{5~\bcancel{gallons}}{1}[/latex] x [latex]\frac{3.785~L}{1~\bcancel{gallon}}[/latex] = [latex]\frac{5~x~3.785~L}{1~x~1}[/latex] = 18.925 L
3) You need 18.925 liters, and each container of mix will make 6 liters. Convert to containers.
[latex]\frac{18.925~\bcancel{L}}{1}[/latex] x [latex]\frac{1~container}{6~\bcancel{L}}[/latex] = [latex]\frac{18.925~x~1~container}{6}[/latex] = 3.15 containers
4) Since you cannot buy a partial container, and you want to have enough lemonade, you should buy 4 containers, (3 containers would not be enough)
Answer: You need to buy 4 containers of lemonade.
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°F and boils at 212°F. Another commonly used temperature scale is the Celsius scale, where water freezes to ice at 0°C and boils at 100°C. You might notice that in the Fahrenheit scale there is a 180 degree difference (212°F ─ 32°F) between the temperature where water boils and freezes, while in the Celsius scale there is a 100 degree difference (100°C ─ 0°C) between the temperature where water boils and freezes. Since neither of these two temperature scales has an absolute starting point (a lowest 21 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.
| Convert from Celsius to Fahrenheit: | F = 1.8C + 32 |
| Convert from Fahrenheit to Celsius: | C = [latex]\frac{F-32}{1.8}[/latex] |
Example 16
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°C or above 32°C.” What are the corresponding temperatures in degrees Fahrenheit?
Solution: State your givens and your goal.
Given: Do not apply in temperatures below 18°C or above 32°C
Goal: What are the corresponding temperatures in degrees Fahrenheit?
1) Both temperatures are in degrees Celsius and need to be converted into degrees Fahrenheit. Use F = 1.8C + 32
For C = 18, F = 1.8(18) + 32 = 64.4°F (low temperature)
For C = 32, F = 1.8(32) + 32 = 89.6°F (high temperature)
Answer: The weed killer should only be applied when the temperature is between 64.4°F and 89.6°F.
Example 17
When leaving the hospital with your sick child, you are told to return immediately if her temperature exceeds 101.5°F. 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: State your givens and your goal.
Given: Return immediately if her temperature exceeds 101.5°F
Goal: At what temperature, in degrees Celsius, would you need to return your child to the hospital?
1) The temperature is in degrees Fahrenheit and it needs to be converted into degrees Celsius. Use C = [latex]\frac{F-32}{1.8}[/latex]
For F = 101.5°F, C = [latex]\frac{101.5 - 32}{1.8}[/latex] = [latex]\frac{69.5}{18}[/latex] = 38.61°C
Answer: You need to return your child to the hospital if her temperature exceeds 38.61°C.
Section 1.5 You Try Problem
What was the temperature in degrees Celsius, if the evening news reports that the high temperature in Phoenix, Arizona today was 115°F?
1.5 – Answers to You Try Problems
46.1°C