# Mathematics exam 1

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## Mathematics exam 1

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1. – MEASUREMENT AND GEOMETRY – T ypes of Measurements 5 feet = how many inches? 5 feet × 12 inches (the number of inches in a single foot) = 60 inches The types of measurements used most frequently in the Therefore, there are 60 inches in 5 feet. United States are listed below: Try another: Units of Length Change 3.5 tons to pounds. 12 inches (in.) = 1 foot (ft.) 3.5 tons = how many pounds? 3 feet = 36 inches = 1 yard (yd.) 3.5 tons × 2,000 pounds (the number of pounds in 5,280 feet = 1,760 yards = 1 mile (mi.) a single ton) = 6,500 pounds Therefore, there are 6,500 pounds in 3.5 tons. Units of Volume 8 ounces* (oz.) = 1 cup (c.) To change a smaller unit to a larger unit, simply ■ 2 cups = 16 ounces = 1 pint (pt.) divide the speciﬁc number of smaller units by the 2 pints = 4 cups = 32 ounces = 1 quart (qt.) number of smaller units in only one of the larger 4 quarts = 8 pints = 16 cups = 128 ounces = 1 gallon units. (gal.) For example, to ﬁnd the number of pints in 64 ounces, simply divide 64, the smaller unit, by 16, Units of Weight the number of ounces in one pint. 16 ounces* (oz.) = 1 pound (lb.) 2,000 pounds = 1 ton (T.) speciﬁc number of the smaller unit the number of smaller units in one larger unit Units of Time 64 ounces 60 seconds (sec.) = 1 minute (min.) 16 ounces = 4 pints 60 minutes = 1 hour (hr.) 24 hours = 1 day Therefore, 64 ounces are equal to four pints. 7 days = 1 week 52 weeks = 1 year (yr.) Here is one more: 12 months = 1 year 365 days = 1 year Change 24 ounces to pounds. 32 ounces *Notice that ounces are used to measure both the volume and 16 ounces = 2 pounds weight. Therefore, 32 ounces are equal to two pounds. C onverting Units B asic Operations with When performing mathematical operations, it is neces- Measurement sary to convert units of measure to simplify a problem. Units of measure are converted by using either multipli- It will be necessary for you to review how to add, sub- cation or division: tract, multiply, and divide with measurement. The mathematical rules needed for each of these operations To change a larger unit to a smaller unit, simply ■ with measurement follow. multiply the speciﬁc number of larger units by the number of smaller units that makes up one of Addition with Measurements the larger units. To add measurements, follow these two steps: For example, to ﬁnd the number of inches in 5 feet, simply multiply 5, the number of larger units, 1. Add like units. by 12, the number of inches in one foot: 2. Simplify the answer. 390
2. – MEASUREMENT AND GEOMETRY – Example: Add 4 pounds 5 ounces to 20 ounces. Multiplication with Measurements 1. Multiply like units. 4 lb. 5 oz. Be sure to add ounces to ounces. 2. Simplify the answer. + 20 oz. 4 lb. 25 oz. Because 25 ounces is more than 16 Example: Multiply 5 feet 7 inches by 3. ounces (1 pound), simplify by 5 ft. 7 in. Multiply 7 inches by 3, then multiply 5 dividing by 16. Then add the 1 ×3 feet by 3. Keep the units separate. pound to the 4 pounds. 15 ft. 21 in. Since 12 inches = 1 foot, simplify 21 inches. 4 lb. + 25 oz. 15 ft. 21 in. = 15 ft. + 1 ft. + 9 inches = 16 feet 9 inches 1 lb. 4 lb. + 16 25 Example: Multiply 9 feet by 4 yards. −16 First, change yards to feet by multiplying the 9 oz. number of feet in a yard (3) by the number of 4 pounds 25 ounces = yards in this problem (4). 4 pounds + 1 pound 9 ounces = 3 feet in a yard × 4 yards = 12 feet 5 pounds 9 ounces Then, multiply 9 feet by 12 feet = Subtraction with Measurements 108 square feet. 1. Subtract like units. (Note: feet × feet = square feet) 2. Regroup units when necessary. 3. Write the answer in simplest form. Division with Measurements 1. Divide into the larger units ﬁrst. For example, to subtract 6 pounds 2 ounces 2. Convert the remainder to the smaller unit. from 9 pounds 10 ounces, 3. Add the converted remainder to the existing 9 lb. 10 oz. Subtract ounces from ounces. smaller unit if any. − 6 lb. 2 oz. Then, subtract pounds from pounds. 4. Then, divide into smaller units. 3 lb. 8 oz. 5. Write the answer in simplest form. Sometimes, it is necessary to regroup units when Example: subtracting. Divide 5 quarts 4 ounces by 4. Example: Subtract 3 yards 2 feet from 5 yards 1 1 qt. R1 First, divide 5 ounces foot. 1. 4 5 by 4, for a result of 1 −4 quart and a reminder 4 4 5 yd. 1 ft. 1 of one. − 3 yd. 2 ft. 2. R1 = 32 oz. Convert the remainder 1 yd. 2 ft. to the smaller unit (ounces). From 5 yards, regroup 1 yard to 3 feet. Add 3 3. 32 oz. + 4 oz. = 36 oz. Add the converted feet to 1 foot. Then subtract feet from feet and remainder to the yards from yards. existing smaller unit. 4. 9 oz. Now divide the smaller 4 36 units by 4. 5. 1 qt. 9 oz. 391
3. – MEASUREMENT AND GEOMETRY – M etric Measurements The chart shown here illustrates some common rela- tionships used in the metric system: The metric system is an international system of meas- Length Weight Volume urement also called the decimal system. Converting units in the metric system is much easier than converting 1 km = 1,000 m 1 kg = 1,000 g 1 kL = 1,000 L units in the English system of measurement. However, 1 m = .001 km 1 g = .001 kg 1 L = .001 kL making conversions between the two systems is much 1 m = 100 cm 1 g = 100 cg 1 L = 100 cL more difﬁcult. Luckily, the GED test will provide you 1 cm = .01 m 1 cg = .01 g 1 cL = .01 L with the appropriate conversion factor when needed. 1 m = 1,000 mm 1 g = 1,000 mg 1 L = 1,000 mL The basic units of the metric system are the meter, 1mm = .001 m 1 mg = .001 g 1 mL = .001 L gram, and liter. Here is a general idea of how the two sys- tems compare: Conversions within the Metric System M ETRIC S YSTEM E NGLISH S YSTEM An easy way to do conversions with the metric system is 1 meter A meter is a little more than a to move the decimal point to either the right or the left yard; it is equal to about 39 inches. because the conversion factor is always ten or a power of 1 gram A gram is a very small unit of ten. As you learned previously, when you change from a weight; there are about 30 grams large unit to a smaller unit, you multiply, and when you in one ounce. change from a small unit to a larger unit, you divide. 1 liter A liter is a little more than a quart. Making Easy Conversions within Preﬁxes are attached to the basic metric units listed the Metric System above to indicate the amount of each unit. When you multiply by a power of ten, you move the dec- For example, the preﬁx deci means one-tenth ( 110 ); imal point to the right. When you divide by a power of therefore, one decigram is one-tenth of a gram, and one ten, you move the decimal point to the left. decimeter is one-tenth of a meter. The following six pre- To change from a large unit to a smaller unit, move ﬁxes can be used with every metric unit: the decimal point to the right. kilo hecto deka UNIT deci centi milli Kilo Hecto Deka Deci Centi Milli (k) (h) (dk) (d) (c) (m) To change from a small unit to a larger unit, move the 1 1 1 1,000 100 10 10 100 1,000 decimal point to the left. Examples: Example: ■ 1 hectometer = 1 hm = 100 meters Change 520 grams to kilograms. 1 ■ 1 millimeter = 1 mm = 1,000 meter = .001 meter Step 1: Be aware that changing meters to kilome- ■ 1 dekagram = 1 dkg = 10 grams ters is going from small units to larger units, and 1 ■ 1 centiliter = 1 cL* = 100 liter = .01 liter thus, you will move the decimal point three places to the left. ■ 1 kilogram = 1 kg = 1,000 grams 1 ■ 1 deciliter = 1 dL* = 10 liter = .1 liter Step 2: Beginning at the UNIT (for grams), you *Notice that liter is abbreviated with a capital letter—“L.” need to move three preﬁxes to the left. k h dk unit dc m 392
4. – MEASUREMENT AND GEOMETRY – Step 3: Move the decimal point from the G eometr y end of 520 to the left three places. 520. As previously deﬁned, geometry is the study of shapes and the relationships among them. Basic concepts in Place the decimal point before the 5. .520 geometry will be detailed and applied in this section. The Your answer is 520 grams = .520 kilograms. study of geometry always begins with a look at basic vocabulary and concepts. Therefore, here is a list of def- Example: initions of important terms: You are packing your bicycle for a trip from New York City to Detroit. The rack on the back area—the space inside a two-dimensional ﬁgure of your bike can hold 20 kilograms. If you bisect—cut in two equal parts exceed that limit, you must buy stabilizers for circumference—the distance around a circle the rack that cost $2.80 each. Each stabilizer can diameter—a line segment that goes directly through hold an additional kilogram. If you want to pack the center of a circle—the longest line you can 23,000 grams of supplies, how much money will draw in a circle you have to spend on the stabilizers? equidistant—exactly in the middle of hypotenuse—the longest leg of a right triangle, Step 1: First, change 23,000 grams to kilograms. always opposite the right angle line—an inﬁnite collection of points in a straight kg hg dkg g dg cg mg path Step 2: Move the decimal point three places to the point—a location in space left. parallel—lines in the same plane that will never 23,000 g = 23.000 kg = 23 kg intersect perimeter—the distance around a ﬁgure Step 3: Subtract to ﬁnd the amount over the limit. 23 kg − 20 kg = 3 kg perpendicular—two lines that intersect to form 90- degree angles Step 4: Because each stabilizer holds one kilogram quadrilateral—any four-sided closed ﬁgure and your supplies exceed the weight limit of the radius—a line from the center of a circle to a point rack by three kilograms, you must purchase three on the circle (half of the diameter) stabilizers from the bike store. volume—the space inside a three-dimensional ﬁgure Step 5: Each stabilizer costs$2.80, so multiply $2.80 by 3:$2.80 × 3 = \$8.40. 393
5. – MEASUREMENT AND GEOMETRY – A ngles An acute angle is an angle that measures less than ■ 90 degrees. An angle is formed by an endpoint, or vertex, and two rays. Acute Angle y ra A right angle is an angle that measures exactly 90 ■ ray degrees. A right angle is represented by a square at the vertex. Endpoint (or Vertex) Naming Angles There are three ways to name an angle. Right Angle B An obtuse angle is an angle that measures more ■ D than 90 degrees, but less than 180 degrees. 1 2 A C Obtuse 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, A straight angle is an angle that measures 180 ■ three letters are used, with the vertex always degrees. Thus, its sides form a straight line. being the middle letter: –1 can be written as ∠BAD or as ∠DAB; –2 can be written as ∠DAC Straight Angle or as ∠CAD. 180° Classifying Angles Angles can be classiﬁed into the following categories: acute, right, obtuse, and straight. 394
6. – MEASUREMENT AND GEOMETRY – C OMPLEMENTARY A NGLES Angles of Intersecting Lines Two angles are complementary if the sum of their meas- When two lines intersect, two sets of nonadjacent angles ures is equal to 90 degrees. called vertical angles are formed. Vertical angles have equal measures and are supplementary to adjacent angles. Complementary 1 2 Angles 2 1 3 ∠1 + ∠2 = 90° 4 S UPPLEMENTARY A NGLES Two angles are supplementary if the sum of their meas- m∠1 = m∠3 and m∠2 = m∠4 ■ ures is equal to 180 degrees. m∠1 + m∠2 = 180 and m∠2 + m∠3 = 180 ■ m∠3 + m∠4 = 180 and m∠1 + m∠4 = 180 ■ Supplementary Angles Bisecting Angles and Line 1 Segments 2 Both angles and lines are said to be bisected when ∠1 + ∠2 = 180° divided into two parts with equal measures. A DJACENT A NGLES Example Adjacent angles have the same vertex, share a side, and do not overlap. S S C A B 1 Adjacent Angles 2 ∠1 and ∠2 are adjacent. Line segment AB is bisected at point C. The sum of the measures of all adjacent angles around the same vertex is equal to 360 degrees. 2 C 35° ∠1 + ∠2 + ∠3 + ∠4 = 360° 3 1 35° 4 A According to the ﬁgure, ∠A is bisected by ray AC. 395