Measurement for Praxis I: Pre-Professional Skills Test Study Guide (page 4)
This section reviews the basics of measurement systems used in the United States (sometimes called customary measurement) and other countries, methods of performing mathematical operations with units of measurement, and the process of converting between different units.
The use of measurement enables a connection to be made between mathematics and the real world. To measure any object, assign a number and a unit of measure. For instance, when a fish is caught, it is often weighed in ounces and its length measured in inches. The following lesson will help you become more familiar with the types, conversions, and units of measurement.
Types of Measurements
The types of measurements used most frequently in the United States are listed here:
- Units of Length
- 12 inches (in.) = 1 foot (ft.)
- 3 feet = 36 inches = 1 yard (yd.)
- 5,280 feet = 1,760 yards = 1 mile (mi.)
- Units of Volume
- 8 ounces* (oz.) = 1 cup (c.)
- 2 cups = 16 ounces = 1 pint (pt.)
- 2 pints = 4 cups = 32 ounces = 1 quart (qt.)
- 4 quarts = 8 pints = 16 cups = 128 ounces = 1 gallon (gal.)
- Units of Weight
- 16 ounces* (oz.) = 1 pound (lb.)
- 2,000 pounds = 1 ton (T.)
- Units of Time
- 60 seconds (sec.) = 1 minute (min.)
- 60 minutes = 1 hour (hr.)
- 24 hours = 1 day
- 7 days = 1 week
- 52 weeks = 1 year (yr.)
- 12 months = 1 year
- 365 days = 1 year
- *Notice that ounces are used to measure the dimensions of both volume and weight.
When performing mathematical operations, it may be necessary to convert units of measure to simplify a problem. Units of measure are converted by using either multiplication or division.
- To convert from a larger unit into a smaller unit, multiply the given number of larger units by the number of smaller units in only one of the larger units:
- To change a smaller unit to a larger unit, divide the given number of smaller units by the number of smaller units in only one of the larger units:
(given number of the larger units) × (the number of smaller units per larger unit) = answer in smaller units
For example, to find the number of inches in five feet, multiply 5, the number of larger units, by 12, the number of inches in one foot:
5 feet = ? inches
5 feet × 12 (the number of inches in a single foot) = 60 inches: 5 ft. × = 60 in.
Therefore, there are 60 inches in five feet.
Change 3.5 tons to pounds.
3.5 tons = ? pounds
3.5 tons × = 7,000 pounds
Therefore, there are 7,000 pounds in 3.5 tons.
For example, to find the number of pints in 64 ounces, divide 64, the number of smaller units, by 16, the number of ounces in one pint.
64 ounces = ? pints
Therefore, 64 ounces equals four pints.
Change 32 ounces to pounds.
32 ounces = ? pounds
Therefore, 32 ounces equals two pounds.
Basic Operations with Measurement
You may need to add, subtract, multiply, and divide with measurement. The mathematical rules needed for each of these operations with measurement follow.
Addition with Measurements
To add measurements, follow these two steps:
- Add like units.
- Simplify the answer by converting smaller units into larger units when possible.
Add 4 pounds 5 ounces to 20 ounces.
Be sure to add ounces to ounces.
Because 25 ounces is more than 16 ounces (1 pound), simplify by dividing by 16:
Then add the 1 pound to the 4 pounds:
4 pounds 25 ounces = 4 pounds + 1 pound 9 ounces = 5 pounds 9 ounces
Subtraction with Measurements
- Subtract like units if possible.
- If not, regroup units to allow for subtraction.
- Write the answer in simplest form.
For example, 6 pounds 2 ounces subtracted from 9 pounds 10 ounces.
Subtract ounces from ounces.
Then subtract pounds from pounds.
Sometimes, it is necessary to regroup units when subtracting.
Subtract 3 yards 2 feet from 5 yards 1 foot.
Because 2 feet cannot be taken from 1 foot, regroup 1 yard from the 5 yards and convert the 1 yard to 3 feet. Add 3 feet to 1 foot. Then subtract feet from feet and yards from yards:
Therefore, 5 yards 1 foot – 3 yards 2 feet = 1 yard 2 feet.
Multiplication with Measurements
- Multiply like units if units are involved.
- Simplify the answer.
Multiply 5 feet 7 inches by 3.
Multiply 7 inches by 3, then multiply 5 feet by 3. Keep the units separate.
Because 12 inches = 1 foot, simplify 21 inches.
15 ft. 21 in. = 15 ft. + 1 ft. 9 in. = 16 ft. 9 in.
Multiply 9 feet by 4 yards.
First, decide on a common unit: either change the 9 feet to yards, or change the 4 yards to feet. Both options are explained below:
To change yards to feet, multiply the number of feet in a yard (3) by the number of yards in this problem (4).
- 3 feet in a yard · 4 yards = 12 feet
Then multiply: 9 feet × 12 feet = 108 square feet.
(Note: feet · feet = square feet = ft.2)
To change feet to yards, divide the number of feet given (9), by the number of feet in a yard (3).
- 9 feet ÷ 3 feet in a yard = 3 yards
Then multiply 3 yards by 4 yards = 12 square yards.
(Note: yards × yards = square yards = yd.2)
Division with Measurements
- Divide into the larger units first.
- Convert the remainder to the smaller unit.
- Add the converted remainder to the existing smaller unit, if any.
- Divide into smaller units.
- Write the answer in simplest form.
Divide 5 quarts 4 ounces by 4.
- Divide into the larger unit:
- Convert the remainder:
- Add remainder to original smaller unit:
- Divide into smaller units:
- Write the answer in simplest form:
1 qt. = 32 oz.
32 oz. + 4 oz. = 36 oz.
36 oz. ÷ 4 = 9 oz.
1 qt. 9 oz.
The metric system is an international system of measurement also called the decimal system. Converting units in the metric system is much easier than converting units in the customary system of measurement. However, making conversions between the two systems is much more difficult. The basic units of the metric system are the meter, gram, and liter. Here is a general idea of how the two systems compare:
Prefixes are attached to the basic metric units to indicate the amount of each unit. For example, the prefix deci means one-tenth therefore, one decigram is one-tenth of a gram, and one decimeter is one-tenth of a meter. The following six prefixes can be used with every metric unit:
- 1 hectometer = 1 hm = 100 meters
- 1 millimeter = 1 mm = meter = .001 meter
- 1 dekagram = 1 dkg = 10 grams
- centiliter = 1 cL* liter = .01 liter
- 1 kilogram = 1 kg = 1,000 grams
- 1 deciliter = 1 dL* = liter = .1 liter
*Notice that liter is abbreviated with a capital letter—L.
- The following chart illustrates some common relationships used in the metric system:
Conversions within the Metric System
An easy way to do conversions with the metric system is to move the decimal point either to the right or to the left, because the conversion factor is always ten or a power of ten. Remember, when changing from a large unit to a smaller unit, multiply. When changing from a small unit to a larger unit, divide.
Making Easy Conversions within the Metric System
When multiplying by a power of ten, move the decimal point to the right, because the number becomes larger.
When dividing by a power of ten, move the decimal point to the left, because the number becomes smaller.
- To change from a larger unit to a smaller unit, move the decimal point to the right.
- kilo hecto deka deci centi milli
An easy way to remember the metric prefixes is to remember the mnemonic: "King Henry Died Of Drinking Chocolate Milk." The first letter of each word represents a corresponding metric heading from Kilo down to Milli: "King"—Kilo, "Henry"—Hecto, "Died"—Deka, "Of"—Original Unit, "Drinking"—Deci, "Chocolate"—Centi, and "Milk"—Milli.
- To change from a smaller unit to a larger unit, move the decimal point to the left.
- Change 520 grams to kilograms.
- Be aware that changing meters to kilometers is going from small units to larger units and, thus, requires that the decimal point move to the left
- Beginning at the unit (for grams), note that the kilo heading is three places away. Therefore, the decimal point will move three places to the left.
- Move the decimal point from the end of 520 to the left three places.
Place the decimal point before the 5: .520
The answer is 520 grams = .520 kilograms.
- Ron's supply truck can hold a total of 20,000 kilograms. If he exceeds that limit, he must buy stabilizers for the truck that cost $12.80 each. Each stabilizer can hold 100 additional kilograms. If he wants to pack 22,300,000 grams of supplies, how much money will he have to spend on the stabilizers?
- First, change 22,300,000 grams to kilograms.
- Move the decimal point three places to the left: 22,300,000 g = 22,300.000 kg = 22,300 kg.
- Subtract to find the amount over the limit: 22,300 kg – 20,000 kg = 2,300 kg.
- Because each stabilizer holds 100 kilograms and the supplies exceed the weight limit of the truck by 2,300 kilograms, Ron must purchase 23 stabilizers: 2,300 kg ÷ 100 kg per stabilizer = 23 stabilizers.
- Each stabilizer costs $12.80, so multiply $12.80 by 23: $12.80 × 23 = $294.40.
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