Calculating with Units
Calculations involving units of measurement follow simple rules and procedures.
Rule #1:When adding or subtracting, the units must be identical and the answer is in the same units.
inches + inches = inches
Rule #2:When multiplying, units are also multiplied.
(inches)(inches) = square inches (in2)
Rule #3:When dividing, some units cancel out.
= (no units)This is a ratio, so it is only a number.
Practice
Copy the problem onto another sheet of paper. Show all of your work.
Pay attention to the UNITS!
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1) 12 in + 6 in =
2)
3) 20 cups 5 cups =
4)
5) (5 ft2)(3 ft) =
6)
7)
8)
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Using Unit Fractions to Do Unit Conversions
Unit Fractions make use of the fact that any number can be multiplied by 1 without changing the value of the number.
Unit fractions may be made from any two terms that describe the same or equivalent "amounts" of what we are interested in. For example, we know that:
1 foot = 12 inches
We can make two unit fractions from this information. Both fractions have a value of one.
Practice
Copy the problem onto another sheet of paper.
Write two unit fractions for the following pairs of units. Use the Units Conversion Charts in the manual to select equivalencies.
Revised 7/20/13 sw/pe
9) pints and quarts
10) yards and miles
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To convert between units, set up each problem by writing down the known measurement. Decide what unit fraction you need, based on the units you start with and want to end up with. Place the units of the known data in the denominator of the unit fraction. This allows you to cancel the units you start with. Multiply by as many unit fractions as necessary to end up with the desired units.
Examples
a. How many inches are in 6 feet? Since we are starting with feet, we choose the unit fraction that has feet in the denominator so the feet units will cancel.Then we multiply, using the remaining units in the answer.
b. Express 6.5 quarts in gallons.
Conversion factor: 4 quarts = 1 gallon
Possible unit fractions: or
Since we are starting with quarts, we choose the unit fraction that has quarts in the denominator so that the quarts units will cancel, and then multiply using the remaining unitsin the answer:
= ______gallon
You can also string more than two unit factors together.
Example
e. How many seconds are in 2 years?
Practice Problem
Solve this problem on another sheet of paper. First, list the possible UNIT FRACTIONS.Circle the unit fractions you choose to use in solving the problem. Show all of your work.
11) There are 85 households and each one burns 5 pounds of trash per day. How many tons of trash are burned in one year?Unit Conversions with Metric Units
Scientists generally work in metric units and often use the following prefixes.
Prefix / Abbreviation / Meaning / Examplemega- / M / 106 / 1 megameter (Mm) = 1 x 106 m
kilo- / k / 103 / 1 kilogram (kg) = 1 x 103 g
centi- / c / 10-2 / 1 centimeter (cm) = 1 x 10-2 m
milli- / m / 10-3 / 1 milligram (mg) = 1 x 10-3 g
micro- / µ / 10-6 / 1 micrometer (µm) = 1 x 10-6 m
nano- / n / 10-9 / 1 nanogram (ng) = 1 x 10-9 g
Larger to Smaller Units
Since the metric system of measurement is based on powers of 10, to convert to smaller units of the same linear measurement we can multiply by powers of 10 by moving the decimal point to the right.
Examples
12 meters = 1200 centimeters (multiplying by 100, the decimal point is moved 2 places to the right, adding 2 zeros)
5 kilograms = 5000 grams (multiplying by 1000 the decimal point is moved 3 places to the right, adding 3 zeros)
Smaller to Larger Units
To convert to larger units, we can divide by powers of 10 by moving the decimal point to the left.
Examples:
250 mL = 0.250 liters (dividing by 1000, the decimal point is moved three places to the left)
4320 m = 4.320 km (dividing by 1000, the decimal point is moved three places to the left)
To convert between different units of measurement, sometimes moving the decimal point is not sufficient. For example, converting from Standard (or English) units to Metric units requires the use of equivalencies not based on powers of 10.
Unit fractions may be made from any two terms that describe the same or equivalent "amounts" of what we are interested in. For example, we know that:
1 inch = 2.54 centimeters
We can make two unit fractions from this information. Both fractions have a value of one.
Practice
Copy the problem onto another sheet of paper.
Write two unit fractions for the following pairs of units. Use the Units Conversion Charts in the manual to select equivalencies.
Revised 7/20/13 sw/pe
12) grams and kilograms
13) liters and milliliters
Revised 7/20/13 sw/pe
To convert between units, set up each problem by writing down the known measurement. Decide what unit fraction you need, based on the units you start with and want to end up with. Place the units of the known data in the denominator of the unit fraction. This allows you to cancel the units you start with. Multiply by as many unit fractions as necessary to end up with the desired units.
Examples
a. How many centimeters are in 6.00 inches? Since we are starting with inches, we choose the unit fraction that has inches in the denominator so the inches units will cancel.Then we multiply, using the remaining units in the answer.
b. Express 24.0 cm in inches.
c. Express 6.5 quarts in liters.
Conversion factor: 0.946 liters = 1 quart
Possible unit fractions: or
Since we are starting with quarts, we choose the unit fraction that has quarts in the denominator so that the quarts units will cancel. Then we multiply, using the remaining unitsin the answer:
= ______liters
d. Express 3.25 yards in millimeters. This problem may be done with two unit fractions, converting first to meters, and then millimeters.
Conversion factors: 1 yard = 0.9144 m; 1 meter = 1000 mm
Possible unit fractions: or
or
Because we are starting with yards, we choose the unit fraction that has yards in the denominator. Then we multiply by the unit fraction that will allow us to cancel the metersand leave us with millimeters.
______mm
Practice
Copy the problem onto another sheet of paper. Show all of your work.
List the UNIT FRACTIONS.
Convert the following:
14) 0.73 km to meters
15) 8 oz to grams
16) 25 in to cm
f. Convert 50.0 milliliters to liters (milliliters to liters is a common conversion.)
g. The density of mercury is 13.6 g/cm3. Express in units of kg/m3.
We also can use unit fractions for solving problems.
h. How many g of PM per cubic meter of air is the equivalent of 0.000003 grams of PM per liter of air?
To do this calculation, we need two unit fractions , derived from the following equivalencies:
1 m3 = 1000 liters or 1 liter = 0.001 m3
1 g = 1,000,000 g or 1 g = 1.0 x 10-6 g
For each equivalency, we have a choice in which unit fraction to use. Writing the unit fractions so that the 1 is in the denominator will allow us to use multiplication of the numerators to calculate the result.
Practice
Copy the problem onto another sheet of paper. Show all of your work.
Pay attention to the UNITS! List UNIT FRACTIONS where needed and circle the unit fractions you choose to use.
17) / 12 ml – 5 ml =18) / (9 mm2)(3 mm) =
19) /
20) / How many millimeters are present in 20.0 inches?
21) / A sample of 7060 liters of air contains 0.0002 grams of particulate matter. This needs to be reported in micrograms per cubic meter. Make this conversion.
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