CONVERSION FACTORS AND DIMENSIONAL
ANALYSIS

Quick!
Tell me how many seconds are in
a year!

Most of us can't do that off the top of our heads. So let's see how
we can do it mathematically.

RULES FOR UNIT CONVERSIONS USING
DIMENSIONAL ANALYSIS
The best way to understand the rules for converting is to see them in action. Let's start with an example problem and go through the rules as we solve the problem. 
You will want to have a pencil, a calculator, and a piece of paper ready to go. 
Example Problem:
A roll of Christmas gift wrap contains
2.5 meters of paper. What is the length
of the paper in yards?

Step 1:
Write down the known
values and the unknown values.

Known
:
length of paper in meters:
2.5 meters
Unknown
:
length of paper in yards

Step 2: Find the conversion factor that gives a ratio between meters and yards.
You can use the reference
table (Conversion Chart)
included with this lesson.

Conversion factor
1 meter = 1.09 yards

Step 3:
Write the conversion factor
as a fraction with the unit you want to
convert to on top and set the
problem up as an equation.

2.5
2.725 yards
You can see that the unit
meters will cancel.


Problem:
Convert 325
millimeters to meters.


Step 1:
Write down the known
values and the unknown values.

Known
:
length in mm = 325
Unknown
:
length in meters

Step 2:
Find the conversion factor that
gives a ratio between meters and
millimeters.
You can use the reference table
(Conversion Chart) included with this
lesson.

Conversion factor
(use reference)
1 meter = 1000 mm 
Step 3:
Write the conversion factor
as a fraction with the unit you want to
convert to on top and set the
problem up as an equation.

325
1000
.325 meters
The unit millimeters will cancel.

So far, the conversions solved have only required one conversion
factor. Let's solve a problem with several conversion factors.

Problem:
How many centuries are in 237,439 days?

Step 1:
Write down the known
values and the unknown values.

Known
: number of days =
237,439 days
Unknown
: number of
centuries

Step 2:
Find the
conversion factor
that gives a ratio
between days and
centuries.
You can use the
reference table
(Conversion Chart)
included with this
lesson.

Conversion factor
: We do not have a
conversion factor for centuries/days. So
we need to go from days to the nearest unit
we know. We know that there are 365
days in a year. We know that there are 100
years in a century.

Step 3:
Write the conversion
factor as a fraction with the
unit you want to convert to on
top and set the problem up
as an equation.

Equation:
237,439
237,439 =
36500
6.51 centuries
The units days and years will
cancel.

Conversion Factors and Dimensional Analysis
are mathematical problems. Here are a
couple of things that you must remember:

It's time to practice solving unit conversions using dimensional analysis. Try Basic Conversion Problems first to give you some practice. 
Practice makes perfect!
You would never play in a
football game or take on a
champion cross country runner
without a lot of practice yourself.
You should approach math the
same way. Practice, practice,
practice!

PRACTICE PROBLEMS
DIRECTIONS: Solve the following problems using dimensional analysis. Show all work and don't forget to label all conversion factors (m, L, s, etc.). If you need the conversion chart for reference, click on the link below. 1. Convert 2,378 feet into meters. 2. How many seconds are in 768 days? 3. A car has traveled 7.2 miles. How many inches did the car travel? 4. How many centuries have you been alive? 5. If a truck has a mass of 23,456 kg, what is its mass in milligrams? 6. How many centiliters are in 3.7 gallons? 7. Write 2,465 milliseconds as seconds. 8. How many milligrams are in a 22 ounce steak? 9. How many hours are there in 6.5 centuries? 10. Convert 34,823 centimeters into miles. 
CAN YOU APPLY IT?
DIRECTIONS: Look at the situation below and see if
you can use your new conversion skills to solve the
problems.
PROBLEM #1: 
You have received an email from your cousin Madeline
in France, who wants to make your family's famous
Pistachio Ice Cream as a surprise for her parents
anniversary party.
Unfortunately, Madeline can not find her mother's recipe for the ice cream and wants you to email the recipe to her. She also asks you to send it with metric measurements instead of American measurements. You need to do some converting! 

The recipe below is in English American units. Now get out your paper,
pencil, calculator, and conversion chart and convert the measurements to
metric. Show your work using dimensional analysis and then rewrite the
recipe using only metric measurements. Remember that all fractions
need to be changed to decimals since the metric system uses decimals.
PISTACHIO ICE CREAM
You will need the following items to pack the ice cream freezer:
Shell and blanch:
Pound them in a mortar with:
Add to them:
Stir these ingredients until the sugar is dissolved.
Heat, but do not boil:
Add and stir until dissolved:
Chill these ingredients. Add the pistachio mixture and:
Put mixture into the canister for the ice cream churn. Pack ice and salt
around the canister. Allow the pack to stand about 3 minutes before you
start churning.
While waiting for the ice cream to harden, prepare the chocolate syrup
topping.
TOPPING:
Melt in a double boiler:
Stir and blend well, then add:
Stir well and add:
Let the sauce boil readily over direct heat for about 3 minutes. Reduce
heat and cook for about 2 minutes more without stirring. Add just before
serving:
The ice cream should be ready in about 20 minutes. ENJOY!

CONVERSION CHART
BRITISH AMERICAN UNITS


Length:

Volume:

1mile (mi) = 1760 yards = 5280 feet

Liquid:

1 yard (yd) = 3 feet

16 fluid ounces (fl oz) = 1 pint

1 foot (ft) = 12 inches

2 cups = 1 pint

1 inch (in) = 1/12 foot = .083 feet

2 pints = 1 quart (qt)

4 quarts = 1 gallon (gal)
 
Dry:
 
2 pints = 1 quart (qt)
 
8 quarts = 1 peck (pk)
 
4 pecks = 1 bushel (bu)

Mass:

16 ounces (oz) = 1 pound (lb)

2000 pounds = 1 ton (t)

Time:

60 seconds (s) = 1 minute

60 minutes = 1 hour (hr)

24 hours = 1 day

7 days = 1 week

365 days = 1 year (yr)

10 years = 1 decade

100 years = 1 century

METRIC UNITS:

Length:

1 kilometer (km) = 1000 meters (m)

1 hectometer (hm) = 100 meters

1 dekameter (dam) = 10 meters

Base Unit for Length: 1 meter = 1m

1 decimeter (dm) = .10 meters (1m = 10 dm)

1 centimeter (cm) = .01 meters ( 1m = 100 cm)

1 millimeter (mm) = .001 meters (1m = 1000mm)

Volume:

1 kiloliter (kL) = 1000 liters (L)

1 hectoliter (hL) = 100 liters

1 dekaliter (daL) = 10 liters

Base Unit for Volume: 1 liter = 1L

1 deciliter (dL) = .10L (1 liter = 10 dL)

1 centiliter (cL) = .01L (1liter = 100cL)

1 milliliter (mL) = .001L (1liter = 1000mL)

Mass:

1 kilogram (kg) = 1000grams (g)

1 hectogram (hg) = 100 grams

1 dekagram (dag) = 10 grams

Base Unit for Mass: 1 gram = 1g

1 decigram (dg) = .10g (1g = 10dg)

1 centigram (cg) = .01cg (1g = 100cg)

1 milligram (mg) = .001mg (1g = 1000mg)

1 metric ton = 1000 kg

CONVERSIONS BETWEEN METRIC
AND AMERICAN SYSTEMS


Length (Metric to American):

Length (American to Metric):

1km = 0.6mi

1mi = 1.6km

1m = 39.37in

1yd = 0.9m

1cm = 0.4in

1ft = 30cm

1in = 2.5cm
 
Volume (Metric to American):

Volume (American to Metric):

1L = 1.06qt

1qt = .95L

1 gal = 3.8L
 
Mass (Metric to American):

Mass (American to Metric):

1kg = 2.2lb

1lb = 0.45kg

1g = 0.04oz

1oz = 28.4g
