Mathematics exam 1 ppt

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

 Ty p e s o f M e a s u r e m e n t s

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.

 C o n v e r t i n g U n i t s

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,

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

 B a s i c O p e r a t i o n s w i t h

M e a s u r e m e n t

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.

32 ounces

16 ounces

64 ounces

16 ounces

specific number of the smaller unit

 the number of smaller units in one larger unit

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 + 1625

−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

54

 yd 1 ft.4

− 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 45 by 4, for a result of 1

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

 M e t r i c M e a s u r e m e n t s

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:

M ETRIC S YSTEM E NGLISH S YSTEM

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 (110);

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:

1,000 100 10 110 1100 1,0100

Examples:

■ 1 hectometer = 1 hm = 100 meters

■ 1 millimeter = 1 mm = 1,0100meter =

.001 meter

■ 1 dekagram = 1 dkg = 10 grams

■ 1 centiliter = 1 cL* = 1100liter = 01 liter

■ 1 kilogram = 1 kg = 1,000 grams

■ 1 deciliter = 1 dL* = 110liter = 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:

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

  

k h dk unit d c m

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.

  

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

 G e o m e t r y

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

 A n g l e s

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

Acute Angle

1 2

D B

Endpoint (or Vertex)

ray ray

C OMPLEMENTARY A NGLES

Two angles are complementary if the sum of their

meas-ures is equal to 90 degrees

S UPPLEMENTARY A NGLES

Two angles are supplementary if the sum of their

meas-ures is equal to 180 degrees

A DJACENT A NGLES

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°

A

C

S S

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°

Complementary Angles