Math 20 Math 25 Math Tips davidvs.net

# Math Lecture NotesMeasurement Skills

## Unit Analysis

Remember that one-step conversions were not a nice way to do many measurement unit conversion problems. We needed a new technique.

Our new technique is a five-step process called Unit Analysis (nicknamed "Canceling Words"). It is a simple and powerful tool. Here are the steps without context, but only for later reference. To first meet the new technique it is best to see it in action in an example.

Definition

Unit Analysis uses five steps to convert measurement units.

1. write the given measurement as a fraction
2. write some "empty" rates without numbers
3. fill in the numbers for the rates
4. multiply

Here is a simple example of Unit Analysis.

Example 1

How many inches is 3.7 feet?

Our first step is to write the given measurement as a fraction. If it is not already a fraction we put the value over 1.

Our second step is to write some "empty" rates without numbers, multiplying. We do this until each unit we want to get rid of appears in a numerator-and-denominator pair so it will cancel.

Our third step is to fill in the numbers for the rates. To do this we need to memorize or look up the unit conversion rates.

Our fourth step is to multiply. This works just like any other fraction multiplication. Notice how units cancel. We do not normally reduce before multiplying.

Our fifth step is to simplify the fraction answer . Often the denominator is not 1, so we simplify by doing "numerator ÷ denominator" like old unit rate problems.

That was very methodical: if we follow the five steps we have almost no chance of making a mistake. Yet it was a one-step unit conversion, so we could have used a proportion. The proportion would not be quite so fail-safe but it would require less writing.

We really need to use Unit Analysis when the unit conversion has more than one step.

Example 2

How many miles per hour is 200 inches per minute?

Our first step is to write the given measurement as a fraction. If it is not already a fraction we put the value over 1.

Our second step is to write some "empty" rates without numbers, multiplying. We do this until each unit we want to get rid of appears in a numerator-and-denominator pair so it will cancel.

(Be alert! We want distance on top and time on bottom. So we must start that way! Starting with the fraction upside down is our biggest potential danger—besides a careless calculator button error it is the only way to mess up.)

Our third step is to fill in the numbers for the rates. To do this we need to memorize or look up the unit conversion rates.

Our fourth step is to multiply. This works just like any other fraction multiplication. Notice how units cancel. We do not normally reduce before multiplying.

Our fifth step is to simplify the fraction answer . Often the denominator is not 1, so we simplify by doing "numerator ÷ denominator" like old unit rate problems.

To summrize, Unit Analysis is an important process because it works even if we need to do more than one transition. Also, the process keeps track of when to multiply and when to divide, so we do not have to keep track in our heads. All we need to do is look up (or memorize) the unit conversion rates and line them up.

So, how does Unit Analysis fit into the six-step problem solving process? The five steps of Unit Analysis do not quite fit, because the step "Make equations" is done earlier.

1. Determine what you are looking for
write what you're given as a fraction
write "empty" rates without numbers
2. Draw pictures
3. Name things
fill in the numbers for the rates
4. Make equations
5. Solve the equations
multiply

### Lots of Examples

Now for more example problems from the textbook. We will always use Unit Analysis, even for the problems that are one-step unit conversions.

Example 3 (§8.1 # 18)

How many feet is 9.6 yards?

1. Determine what you are looking for
write what you're given as a fraction
write "empty" rates without numbers
2. Draw pictures
3. Name things
fill in the numbers for the rates
4. Make equations
5. Solve the equations
multiply

Example 4 (§8.1 # 7)

How many inches is 3 yards?

1. Determine what you are looking for
write what you're given as a fraction
write "empty" rates without numbers
2. Draw pictures
3. Name things
fill in the numbers for the rates
4. Make equations
5. Solve the equations
multiply

Example 5 (§8.1 # 3)

How many feet is 1 inch?

1. Determine what you are looking for
write what you're given as a fraction
write "empty" rates without numbers
2. Draw pictures
3. Name things
fill in the numbers for the rates
4. Make equations
5. Solve the equations
multiply

Example 6 (§8.1 # 24)

How many inches is 6 1/3 yard?

1. Determine what you are looking for
write what you're given as a fraction
write "empty" rates without numbers
2. Draw pictures
3. Name things
fill in the numbers for the rates
4. Make equations
5. Solve the equations
multiply

Example 7 (§8.5 # 21)

How many pints is 11 gallons?

1. Determine what you are looking for
write what you're given as a fraction
write "empty" rates without numbers
2. Draw pictures
3. Name things
fill in the numbers for the rates
4. Make equations
5. Solve the equations
multiply

Chapter 8 Test, Problem 1: How many inches is 4 feet?

Chapter 8 Test, Problem 2: How many feet is 4 inches?

Chapter 8 Test, Problem 11: How many ounces is 4 pounds?

Chapter 8 Test, Problem 12: How many pounds is 4.11 tons?

Chapter 8 Test, Problem 16: How many minutes is 5 hours?

Chapter 8 Test, Problem 17: How many hours is 15 days?

Chapter 8 Test, Problem 18: How many quarts is 64 pints?

Chapter 8 Test, Problem 19: How many ounces is 10 gallons?

Chapter 8 Test, Problem 20: How many ounces is 5 cups?

### A Timely Warning

Except for inches to centimeters, the unit conversion rates that move between Standard and SI are rounded. Because they are rounding that happens before the problem is finished they introduce some error. Let's do the same problem in two different ways to see this happen.

Example 8

3.171 quarts is how many liters? (Use 1 liter = 1.057 quarts.)

Example 9

3.171 quarts is how many liters? (Use 1 quart = 0.946 liters.)

Is the answer slightly less than three liters or not? It is actually slightly more! But because our original number had quite a few decimal places the unit conversion rates we used did not have enough decimal places to produce answers with enough accuracy.

If your original numbers have many decimal places (thus asking your answers to also have many decimal places) you will need unit conversion rates with lots of decimal places.

Chapter 8 Test, Problem 5: How many meters is 200 yards?

Chapter 8 Test, Problem 6: How many miles is 2,400 kilometers?

## Temperature Conversions

Because of a strange bit of history we cannot use proportions or Unit Analysis for temperature conversion between Celsius and Fahrenheit. Instead we need to use (but not memorize) formulas.

First we need a formula to switch from Celsius to Fahrenheit. Here are three equivalent and equally workable options. Pick your favorite and ignore the other two.

• F = 1.8 × C + 32
• F = 9/5 × C + 32
• F = (C + 40) × 9/5 − 40

We also need a formula to switch from Fahrenheit to Celsius. Again, here are three equivalent and equally workable options. Pick your favorite and ignore the other two.

• C = (F − 32) ÷ 1.8
• C = (F − 32) × 5/9
• C = (F + 40) × 5/9 − 40

The last formula of each group was created in 2005 by Robert Warren. He thinks they are easier to remember. They are based on the coincidence that -40 °C is also -40 °F.

Example 10 (§8.6 # 25)

A hot tub is 40 °C. What is this temperature in degrees Fahrenheit?

Example 11 (§8.6 # 45)

The temperature on a March afternoon was 68 °F. What is this temperature in degrees Celsius

Chapter 8 Test, Problem 22: Convert 95°F to Celsius.

Chapter 8 Test, Problem 23: Convert 59°C to Fahrenheit.

We have already seem some examples of metric prefixes:

• a kilogram is 1 gram × 1,000
• a milliliter is 1 liter ÷ 1,000
• a millimeter is 1 meter ÷ 1,000
• a centimeter is 1 meter ÷ 100

These examples are part of a general pattern.

Here are the metric prefixes used in these math lectures:

• kilo × 1,000
• (unit)
• centi ÷ 100
• milli ÷ 1,000
• micro ÷ 1,000,000

In this list (unit) is a placeholder for the plain unit, whether meters, grams, or liters.

These are prefixes are part of a longer list.

kilo    hecto    deca    (unit)    deci    centi    milli    •    •    micro

Strangely, there are no prefixes for the two spots between milli and micro (for ten-thousandths and hundred-thousandths of a unit).

Even though we do not use hecto-, deca-, or deci- it is still useful to memorize this longer list. Why? Because of a shortcut!

We can use Unit Analysis to convert between metric prefixes. We would use one unit conversion rate to remove the prefix and a second unit conversion rate to put on the desired one.

Example 12

How many millimeters is 2.8 centimeters? (Use Unit Analysis.)

But it is quicker to use a shortcut.

Shortcut

To convert between metric prefixes, first write the list of all SI prefixes. Then count decimal point scoots: for every "place" you move right or left along the list of prefixes, move the decimal point the same way.

Example 13

How many millimeters is 2.8 centimeters? (Use the shortcut.)

Notice that we do not actually need to write the whole list. Just the initials are sufficient.

k    h    d    unit    d    c    m    •    •    micro

Many math students memorize a cute saying (try making your own!) and remember the acronym for the first seven spots. For example, "King Henry does usually drink chocolate milk" or "Killer hobos dance under dazzling crystal mobiles."

Writing the acronym at the top of your homework and your test scratch paper is a great idea. Then you can use the shortcut instead of Unit Analysis. But remember that the shortcut is only for doing a metric-to-metric unit conversion.

Let's do more examples.

Example 14 (§8.2 # 39)

A calculator is 18 centimeters long. Change this to meters and millimeters.

Example 15 (§8.4 # 33)

A backpack weighs 8,492 grams. Change this to kilograms.

Example 16 (§8.5 # 35)

A shampoo bottle contains 355 cubic centimeters. Change this to milliliters and liters.

Chapter 8 Test, Problem 3: How many meters is 6 kilometers?

Chapter 8 Test, Problem 4: How many centimeters is 8.7 millimeters?

Chapter 8 Test, Problem 7: Something is 0.5 cm wide. Express this width in meters and millimeters.

Chapter 8 Test, Problem 8: Marvin L. Bittinger is 1.8542 meters tall. Express this height in centimeters and millimeters.

Chapter 8 Test, Problem 9: How many liters is 3,080 milliliters?

Chapter 8 Test, Problem 10: How many milliliters is 0.25 liters?

Chapter 8 Test, Problem 13: How many grams is 3.8 kilograms?

Chapter 8 Test, Problem 15: How many grams is 2,200 milligrams?

Chapter 8 Test, Problem 21: How many micrograms is 0.37 milligrams?