 Types of Measurements The types of measurements used most frequently in the United States are listed below: 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 both the volume and weight.  Converting Units When performing mathematical operations, it is neces- sary to convert units of measure to simplify a problem. Units of measure are converted by using either multipli- cation or division: ■ To change a larger unit to a smaller unit, simply multiply the specific number of larger units by the number of smaller units that makes up one of the larger units. For example, to find the number of inches in 5 feet, simply multiply 5, the number of larger units, by 12, the number of inches in one foot: 5 feet = how many inches? 5 feet × 12 inches (the number of inches in a single foot) = 60 inches Therefore, there are 60 inches in 5 feet. Try another: Change 3.5 tons to pounds. 3.5 tons = how many pounds? 3.5 tons × 2,000 pounds (the number of pounds in a single ton) = 6,500 pounds Therefore, there are 6,500 pounds in 3.5 tons. ■ To change a smaller unit to a larger unit, simply divide the specific number of smaller units by the number of smaller units in only one of the larger units. For example, to find the number of pints in 64 ounces, simply divide 64, the smaller unit, by 16, the number of ounces in one pint. = 4 pints Therefore, 64 ounces are equal to four pints. Here is one more: Change 24 ounces to pounds. = 2 pounds Therefore, 32 ounces are equal to two pounds.  Basic Operations with Measurement It will be necessary for you to review how to add, sub- tract, 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: 1. Add like units. 2. Simplify the answer. 32 ounces ᎏᎏ 16 ounces 64 ounces ᎏᎏ 16 ounces specific number of the smaller unit ᎏᎏᎏᎏᎏ the number of smaller units in one larger unit – MEASUREMENT AND GEOMETRY– 390 Example: Add 4 pounds 5 ounces to 20 ounces. 4 lb. 5 oz. Be sure to add ounces to ounces. + 20 oz. 4 lb. 25 oz. 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 lb. + 25 oz.  1 lb. 4 lb. + 16ͤ25 ෆ −16 9 oz. 4 pounds 25 ounces = 4 pounds + 1 pound 9 ounces = 5 pounds 9 ounces Subtraction with Measurements 1. Subtract like units. 2. Regroup units when necessary. 3. Write the answer in simplest form. For example, to subtract 6 pounds 2 ounces from 9 pounds 10 ounces, 9 lb. 10 oz. Subtract ounces from ounces. − 6 lb. 2 oz. Then, subtract pounds from pounds. 3 lb. 8 oz. Sometimes, it is necessary to regroup units when subtracting. Example: Subtract 3 yards 2 feet from 5 yards 1 foot. 5 4 ΋ yd. 1 4 ΋ ft. − 3 yd. 2 ft. 1 yd. 2 ft. From 5 yards, regroup 1 yard to 3 feet. Add 3 feet to 1 foot. Then subtract feet from feet and yards from yards. Multiplication with Measurements 1. Multiply like units. 2. Simplify the answer. Example: Multiply 5 feet 7 inches by 3. 5 ft. 7 in. Multiply 7 inches by 3, then multiply 5 × 3 feet by 3. Keep the units separate. 15 ft. 21 in. Since 12 inches = 1 foot, simplify 21 inches. 15 ft. 21 in. = 15 ft. + 1 ft. + 9 inches = 16 feet 9 inches Example: Multiply 9 feet by 4 yards. First, change yards to feet by multiplying 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 by 12 feet = 108 square feet. (Note: feet × feet = square feet) Division with Measurements 1. Divide into the larger units first. 2. Convert the remainder to the smaller unit. 3. Add the converted remainder to the existing smaller unit if any. 4. Then, divide into smaller units. 5. Write the answer in simplest form. Example: Divide 5 quarts 4 ounces by 4. 1 qt. R1 First, divide 5 ounces 1. 4ͤ5 ෆ ෆ by 4, for a result of 1 −4 quart and a reminder 1 of one. 2. R1 = 32 oz. Convert the remainder to the smaller unit (ounces). 3. 32 oz. + 4 oz. = 36 oz. Add the converted remainder to the existing smaller unit. 4. 9 oz. Now divide the smaller 4ͤ36 ෆ units by 4. 5. 1 qt. 9 oz. – MEASUREMENT AND GEOMETRY– 391  Metric Measurements The metric system is an international system of meas- urement also called the decimal system. Converting units in the metric system is much easier than converting units in the English system of measurement. However, making conversions between the two systems is much more difficult. Luckily, the GED test will provide you with the appropriate conversion factor when needed. The basic units of the metric system are the meter, gram, and liter. Here is a general idea of how the two sys- tems compare: METRIC S YSTEM ENGLISH SYSTEM 1 meter A meter is a little more than a yard; it is equal to about 39 inches. 1 gram A gram is a very small unit of weight; there are about 30 grams in one ounce. 1 liter A liter is a little more than a quart. Prefixes are attached to the basic metric units listed above to indicate the amount of each unit. For example, the prefix deci means one-tenth ( ᎏ 1 1 0 ᎏ ); therefore, one decigram is one-tenth of a gram, and one decimeter is one-tenth of a meter. The following six pre- fixes can be used with every metric unit: Kilo Hecto Deka Deci Centi Milli (k) (h) (dk) (d) (c) (m) 1,000 100 10 ᎏ 1 1 0 ᎏ ᎏ 1 1 00 ᎏ ᎏ 1,0 1 00 ᎏ Examples: ■ 1 hectometer = 1 hm = 100 meters ■ 1 millimeter = 1 mm = ᎏ 1,0 1 00 ᎏ meter = .001 meter ■ 1 dekagram = 1 dkg = 10 grams ■ 1 centiliter = 1 cL* = ᎏ 1 1 00 ᎏ liter = .01 liter ■ 1 kilogram = 1 kg = 1,000 grams ■ 1 deciliter = 1 dL* = ᎏ 1 1 0 ᎏ liter = .1 liter *Notice that liter is abbreviated with a capital letter—“L.” The chart shown here illustrates some common rela- tionships used in the metric system: Length Weight Volume 1 km = 1,000 m 1 kg = 1,000 g 1 kL = 1,000 L 1 m = .001 km 1 g = .001 kg 1 L = .001 kL 1 m = 100 cm 1 g = 100 cg 1 L = 100 cL 1 cm = .01 m 1 cg = .01 g 1 cL = .01 L 1 m = 1,000 mm 1 g = 1,000 mg 1 L = 1,000 mL 1mm = .001 m 1 mg = .001 g 1 mL = .001 L Conversions within the Metric System An easy way to do conversions with the metric system is to move the decimal point to either the right or the left because the conversion factor is always ten or a power of ten. As you learned previously, when you change from a large unit to a smaller unit, you multiply, and when you change from a small unit to a larger unit, you divide. Making Easy Conversions within the Metric System When you multiply by a power of ten, you move the dec- imal point to the right. When you divide by a power of ten, you move the decimal point to the left. To change from a large unit to a smaller unit, move the decimal point to the right. kilo hecto deka UNIT deci centi milli To change from a small unit to a larger unit, move the decimal point to the left. Example: Change 520 grams to kilograms. Step 1: Be aware that changing meters to kilome- ters is going from small units to larger units, and thus, you will move the decimal point three places to the left. Step 2: Beginning at the UNIT (for grams), you need to move three prefixes to the left. یی ی k h dk unit d c m – MEASUREMENT AND GEOMETRY– 392 Step 3: Move the decimal point from the end of 520 to the left three places. 520.  Place the decimal point before the 5. .520 Your answer is 520 grams = .520 kilograms. Example: You are packing your bicycle for a trip from New York City to Detroit. The rack on the back of your bike can hold 20 kilograms. If you exceed that limit, you must buy stabilizers for the rack that cost $2.80 each. Each stabilizer can hold an additional kilogram. If you want to pack 23,000 grams of supplies, how much money will you have to spend on the stabilizers? Step 1: First, change 23,000 grams to kilograms. یی ی kg hg dkg g dg cg mg Step 2: Move the decimal point three places to the left. 23,000 g = 23.000 kg = 23 kg Step 3: Subtract to find the amount over the limit. 23 kg − 20 kg = 3 kg Step 4: Because each stabilizer holds one kilogram and your supplies exceed the weight limit of the rack by three kilograms, you must purchase three stabilizers from the bike store. Step 5: Each stabilizer costs $2.80, so multiply $2.80 by 3: $2.80 × 3 = $8.40.  Geometry As previously defined, geometry is the study of shapes and the relationships among them. Basic concepts in geometry will be detailed and applied in this section. The study of geometry always begins with a look at basic vocabulary and concepts. Therefore, here is a list of def- initions of important terms: area—the space inside a two-dimensional figure bisect—cut in two equal parts circumference—the distance around a circle diameter—a line segment that goes directly through the center of a circle—the longest line you can draw in a circle equidistant—exactly in the middle of hypotenuse—the longest leg of a right triangle, always opposite the right angle line—an infinite collection of points in a straight path point—a location in space parallel—lines in the same plane that will never intersect perimeter—the distance around a figure perpendicular—two lines that intersect to form 90- degree angles quadrilateral—any four-sided closed figure radius—a line from the center of a circle to a point on the circle (half of the diameter) volume—the space inside a three-dimensional figure – MEASUREMENT AND GEOMETRY– 393  Angles An angle is formed by an endpoint, or vertex, and two rays. Naming Angles There are three ways to name an angle. 1. An angle can be named by the vertex when no other angles share the same vertex: ∠A. 2. An angle can be represented by a number written across from the vertex: ∠1. 3. When more than one angle has the same vertex, three letters are used, with the vertex always being the middle letter: –1 can be written as ∠BAD or as ∠DAB; –2 can be written as ∠DAC or as ∠CAD. Classifying Angles Angles can be classified into the following categories: acute, right, obtuse, and straight. ■ An acute angle is an angle that measures less than 90 degrees. ■ A right angle is an angle that measures exactly 90 degrees. A right angle is represented by a square at the vertex. ■ An obtuse angle is an angle that measures more than 90 degrees, but less than 180 degrees. ■ A straight angle is an angle that measures 180 degrees. Thus, its sides form a straight line. Straight Angle 180° Obtuse Angle Right Angle A cute Angle 1 2 A C D B Endpoint (or Vertex) ray ray – MEASUREMENT AND GEOMETRY– 394 COMPLEMENTARY ANGLES Two angles are complementary if the sum of their meas- ures is equal to 90 degrees. SUPPLEMENTARY ANGLES Two angles are supplementary if the sum of their meas- ures is equal to 180 degrees. ADJACENT ANGLES Adjacent angles have the same vertex, share a side, and do not overlap. The sum of the measures of all adjacent angles around the same vertex is equal to 360 degrees. Angles of Intersecting Lines When two lines intersect, two sets of nonadjacent angles called vertical angles are formed. Vertical angles have equal measures and are supplementary to adjacent angles. ■ m∠1 = m∠3 and m∠2 = m∠4 ■ m∠1 + m∠2 = 180 and m∠2 + m∠3 = 180 ■ m∠3 + m∠4 = 180 and m∠1 + m∠4 = 180 Bisecting Angles and Line Segments Both angles and lines are said to be bisected when divided into two parts with equal measures. Example Line segment AB is bisected at point C. According to the figure, ∠A is bisected by ray AC. 35° 35° 1 2 3 4 1 2 3 4 ∠1 + ∠2 + ∠3 + ∠4 = 360° 1 2 ∠ 1 and ∠2 are adjacent. Adjacent Angles 1 2 ∠1 + ∠2 = 180° Supplementary Angles 1 2 ∠1 + ∠2 = 90° Complementar y Angles – MEASUREMENT AND GEOMETRY– 395 . Volume 1 km = 1, 000 m 1 kg = 1, 000 g 1 kL = 1, 000 L 1 m = .0 01 km 1 g = .0 01 kg 1 L = .0 01 kL 1 m = 10 0 cm 1 g = 10 0 cg 1 L = 10 0 cL 1 cm = . 01 m 1 cg = . 01 g 1 cL = . 01 L 1 m = 1, 000 mm 1 g = 1, 000. (m) 1, 000 10 0 10 ᎏ 1 1 0 ᎏ ᎏ 1 1 00 ᎏ ᎏ 1, 0 1 00 ᎏ Examples: ■ 1 hectometer = 1 hm = 10 0 meters ■ 1 millimeter = 1 mm = ᎏ 1, 0 1 00 ᎏ meter = .0 01 meter ■ 1 dekagram = 1 dkg = 10 grams ■ 1 centiliter. dekagram = 1 dkg = 10 grams ■ 1 centiliter = 1 cL* = ᎏ 1 1 00 ᎏ liter = . 01 liter ■ 1 kilogram = 1 kg = 1, 000 grams ■ 1 deciliter = 1 dL* = ᎏ 1 1 0 ᎏ liter = .1 liter *Notice that liter is abbreviated