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### 2004-A: Conversion Factors 1: One-Step Conversions

• Topic Cluster: Pure Mathematics
• Topic: Unit Conversion and Dimensional Analysis
• Objective: Convert a quantity to a different unit using a single conversion factor; properly write conversion factors to show how dimensional analysis is used
• Content:
• Level: 1

#### BACK to Ladder Unit Conversion and Dimensional Analysis

In this unit, we will learn a method for completing conversion problems called the conversion factor method. Even if you find conversion problems very easy, or even if you can just use Google for conversions, mastering the conversion factor method is still crucial because it is based upon the mathematical principles of dimensional analysis. Dimensional analysis lays at the heart of all physics formulas. In this pod, we will learn to use the conversion factor method for simple conversion problems, those involving intuitive, well known unit and only one conversion factor. By conscientiously completing conversion problems, even very easy ones, with the conversion factor method, you will begin to understand how unit and formulas work. Beyond that, once you start doing much harder conversion problems, like those related to electromagnetic wave quantities, the conversion factor method is the way to ensure that you will never make a mistake.

### Conversion Factor List

• Length Conversion Factors
• 1 foot = 12 inches
• 1 yard = 3 feet
• 1 football field = 120 yards
• 1 mile = 5280 feet
• 1 meters = 1000 millimeters
• 1 meter = 100 centimeters
• 1 kilometer = 1000 meters
• 1 inch = 2.54 centimeters
• 1 meter = 3.28 feet
• 1 mile = 1.61 kilometers
• Time Conversion Factors
• 1 minute = 60 seconds
• 1 hour = 60 minutes
• 1 day = 24 hours
• 1 normal year = 365 days
• 1 leap year = 366 days
• 1 full rotation of sun = 365.25 days
• 1 century = 100 full rotations of sun
• Note that "months" are not a standard unit, because not all months have the same number of days.
• Mass Conversion Factors
• 1 kilogram = 1000 grams
• 1 gram = 1000 milligrams
• 1 slug = 14.6 kilograms
• 1 kilogram = 2.2046 pounds
• 1 metric ton (also called tonne) = 1000 kilograms
• Force Conversion Factors
• 1 pound-force = 4.45 Newtons
• 1 Newton = 100,000 dynes
• 1 ton = 2000 pounds

#### Example Questions

##### Example Question 1

"How many feet are in 36 inches?"

Obviously, the answer is 3. I know that you know that. However, the goal here is not to learn an answer, but to learn how to use a new method. It can be helpful to practice by using a problem that is pretty simple, where you aren't stretching your mind to figure out what is going on. When writing answers in this pod and on the accompanying miniquiz, you need to explicitly indicate all of the steps below.

1. Write the original value as a fraction with 1 in the denominator: $$36 \, \text{inches} = \frac{36 \, \text{inches}}{1}$$
2. Multiply by a conversion factor that will allow you to cancel out the unit you don't want and give you the unit you do want. A conversion factor is a fraction with equal values in the numerator and the denominator: $$\frac{36 \, \text{inches}}{1} \left( \frac{1 \, \text{foot}}{12 \, \text{inches}} \right)$$
3. Cancel out he unit that are in both the numerator and the denominator, and ignore all of the 1s: $$\require{enclose} \frac{36 \, \enclose{horizontalstrike}{\text{inches}}}{1} \left( \frac{1 \, \text{foot}}{12 \, \enclose{horizontalstrike}{\text{inches}}} \right) = \frac{36 \, \text{feet}}{12}$$
4. Multiply or divide to get the right answer: $$\frac{36 \, \text{feet} }{12} = 3 \, \text{feet}$$

Here is the answer written out fully: $$36 \, \text{inches} = \frac{36 \, \enclose{horizontalstrike}{\text{inches}}}{1} \left( \frac{1 \, \text{foot}}{12 \, \enclose{horizontalstrike}{\text{inches}}} \right) = \frac{36 \, \text{feet}}{12} = 3 \, \text{feet}$$

##### Example Question #2

"How many inches are in 6 feet?"

1. Write the original value as a fraction with 1 in the denominator: $$6 \, \text{feet} = \frac{6 \, \text{feet}}{1}$$
2. Multiply by a conversion factor that will allow you to cancel out the unit you don't want and give you the unit you do want: $$\frac{6 \, \text{feet}}{1} \left( \frac{12 \, \text{inches}}{1 \, \text{foot}} \right)$$
3. Cancel out he unit that are in both the numerator and the denominator, and ignore all of the 1s. $$\frac{6 \, \enclose{horizontalstrike}{\text{feet}}}{1} \left( \frac{12 \, \text{inches}}{1 \, \enclose{horizontalstrike}{\text{foot}}} \right) = \frac{6 \cdot 12 \, \text{inches}}{1}$$
4. Multiply or divide to get the right answer: $$\frac{6 \cdot 12 \, \text{inches}}{1} = 72 \, \text{inches}$$
##### Requirements

For full credit on a conversion factor problem, you must complete the following steps, as dictated in the style guide. For reference, the answer section of this pod is written according to these rules.

Whenever completing a non-metric unit conversion, explicitly write out conversion factors. When writing conversion factors:
• Quantities should be represented as fractions.
• Units should be represented as words in the correct part of a fraction.
• Conversion factors should be written as fractions with a different unit in the numerator and denominator and value equal to one.
• Units that cancel should be crossed out.
• Equal signs must be included in appropriate places.
• You should explicitly write the final answer of the conversion.

Once upon a time, I had a very smart class of students, and they all said that conversions were easy. So I gave them a quiz that required them to use conversion factors with very non-intuitive unit, unit they could not easily imagine or understand. Their answers were off by a billion billion. And that's not two billion, that's a billion billion! The next day in class, I told them they were terrible at conversions, and we learned about feet an inches. Of course, they could easily figure out conversion factors when it was presented with feet and inches, which they clearly understood. When they redid the same quiz, I told them they would not get full credit unless they used the conversion factor method correctly and completely. They did much, much better.

#### Questions

Use proper conversion factors, as described above, to complete each of the following problems. If you cannot give a precise answer, please round to three significant figures.

1. Convert 48 inches to feet.
(I know this question is easy, but uou still need to answer it, using conversion factors. See above.)

2. Convert 5 feet to inches.

3. Convert 29 feet to meters.

4. Convert 6 meters to feet.

5. Convert 785 hours to days.

6. Convert 294 hours to days.

7. Convert 13 days to hours.

8. Convert 24 kilograms to slugs.

9. Convert 146 slugs to kilograms.

10. Convert 21,419 feet to miles.

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