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Skills Handbook
Problem Solving Strategies
You may find one or more of these strategies helpful in solving a word problem.
Strategy
When to Use it
Draw a Diagram
You need help in visualizing the problem.
Try, Check, Revise
Solving the problem directly is too complicated.
Make a Table
The problem has data that need organizing.
Look for a Pattern
The problem describes a relationship.
Solve a Simpler Problem
The problem is complex or has numbers that
are too unmanageable to use at first.
Use Logical Reasoning
You need to reach a conclusion from some
given information.
Work Backward
You undo various operations to arrive at
the answer.
Problem Solving: Draw a Diagram
EXAMPLE
Antoine is 1.91 m tall. He measures his shadow and finds
that it is 2.34 m long. He then measures the length of the
shadow of a flagpole and finds that it is 13.2 m long. How tall
is the flagpole?
x
Start by drawing a diagram showing the given information.
The diagram shows that you can solve the problem by using
a proportion.
1.91 = x
2.34
13.2
x < 10.77
Write a proportion.
Solve for x.
1.91 m
13.2 m
2.34 m
not to scale
The flagpole is about 10.8 m tall.
EXERCISES
1. Five people meet and shake hands with one another. How many handshakes
are there in all? 10 handshakes
2. Three tennis balls fit snugly in an ordinary, cylindrical tennis ball container.
Which is greater, the circumference of a ball or the height of the container?
circumference
3. Three lines that all intersect a circle can determine at most 7 regions within
the circle, as shown in the diagram. What is the greatest number of regions
that can be determined by 5 lines? 16 regions
4. A triangle has vertices (1, 3), (2, 3), and (7, 5). Find its area. 1 unit2
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1
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Problem Solving: Try, Check, Revise
Have you ever weighed yourself on a balance scale at a doctor’s office? You start by
guessing your weight, and then you see if the scale balances. If it doesn’t, you slide
the weights back and forth until the scale does balance. This is an example of the
Try, Check, Revise strategy, a strategy helpful for solving many types of problems.
EXAMPLE
Try 1:
10 ft wide by 40 ft long
10 ? 40 = 400 ft2
Make an initial try with a perimeter
of 100 ft. Find the area.
Try 2:
20 ft wide by 30 ft long
20 ? 30 = 600 ft2
Try again and find the area.
The area is larger than the initial try.
Try 3:
35 ft wide by 15 ft long
35 ? 15 = 525 ft2
Continue trying and testing areas.
This area is smaller than the last try.
Try 4:
22 ft wide by 28 ft long
22 ? 28 = 616 ft2
Notice that the areas are larger when the
width and length are closer together.
Try 5:
25 ft wide by 25 ft long
25 ? 25 = 625 ft2
Choose dimensions that are as
close together as possible.
Skills Handbook
Skills Handbook
You have 100 ft of fencing and want to build a fence in the shape of a rectangle to
enclose the largest possible area. What should be the dimensions of the rectangle?
The dimensions of your rectangle should be 25 ft by 25 ft.
EXERCISES
1. The product of three consecutive even integers is 480. Find the integers. 6, 8, 10
2. The combined ages of a father and his twin daughters are 54 years. The
father was 24 years old when the twins were born. How old is each of the
three people? Each daughter is 10 years old; the father is 34 years old.
3. What numbers can x represent in the rectangle?
any pos. number less than 4 x
4. Alexandra has a collection of dimes and quarters. The number of dimes equals
the number of quarters. She has a total of $2.80. How many of each coin does
Alexandra have? 8
4⫺x
Use the Try, Check, Revise strategy to find the value of each variable.
5. 2a + 5 = 1 –2
6. 10 - 3c = -2 4
w + 12 = -6 54
7. 23
8. 5y - 32 = 28 12
9. 12b + 11 = 14 14
10. 0.5x - 15 = -7 16
11. Ruisa bought 7 rolls of film to take 192 pictures on a field trip. Some rolls
had 36 exposures and the rest had 24 exposures. How many of each type did
Ruisa buy? 2 rolls of 36 exposures and 5 rolls of 24 exposures
12. The sum of five consecutive integers is 5. Find the integers. –1, 0, 1, 2, 3
13. Paul buys a coupon for $20 from a local theater that allows him to see movies
for half price over the course of one year. The cost of seeing a movie is
normally $7.50. What is the least number of movies Paul would have to see to
pay less than the normal price per movie? 6 movies
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Problem Solving: Make a Table and Look for a Pattern
There are two important ways that making a table can help you solve a problem.
First, a table is a handy method of organizing information. Second, once the
information is in a table, it is easier for you to find patterns.
EXAMPLE
The squares below are made of toothpicks. How many toothpicks are in the
square with 7 toothpicks on a side?
Use a table to organize the information.
Notice the pattern in the increases in the
numbers of toothpicks in the squares. For each
increase of 1 toothpick on a side, the increase
increases by 4. The number of toothpicks in the
5 th square is 40 + 20, or 60. The number in the
6 th square is 60 + 24, or 84, and the number in
the 7 th square is 84 + 28, or 112.
Toothpicks on a side
1
2
3
4
Toothpicks in the square
4
12
24
40
+ 8 + 12 + 16
EXERCISES
1. The triangles are made of toothpicks. How many toothpicks are in Figure 10?
165 toothpicks
Figure 1
Figure 2
Figure 3
2. In each figure, the vertices of the smallest square are midpoints of the sides of
1
2
the next larger square. Find the area of the ninth shaded square.
256 in.
1 in.
3. In each figure, the midpoints of the sides of the unshaded triangles are used
as vertices of the shaded triangles. Find the total number of shaded triangles
in Figure 8. 3280 triangles
Figure 1
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Figure 2
Figure 3
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Problem Solving: Solve a Simpler Problem
Looking at a simpler version of a problem can be helpful in suggesting a problem
solving approach.
EXAMPLE
You may be tempted to divide 570 by 10, getting 57 as an answer, but looking at a
simpler problem suggests that this answer isn’t right. Suppose you have just 10 or
20 m of fencing.
10 m
two fence posts
20 m
three fence posts
These easier problems suggest that there is always one more fence post than one
tenth the length. So for a 570-m fence, there are 570
10 + 1, or 58 fence posts.
Skills Handbook
Skills Handbook
A fence along the highway is 570 m long. There is a fence post every 10 m. How
many fence posts are there?
EXERCISES
1. A farmer wishes to fence in a square lot 70 yards by 70 yards. He will install a
fence post every 10 yards. How many fence posts will he need? 28 posts
2. A snail is trying to escape from a well 10 ft deep. The snail can climb 2 ft each
day, but each night it slides back 1 ft. How many days will the snail take to climb
out of the well? 9 days
Janette's house
3. Janette is planning to walk from her house to her
friend Barbara’s house. How many different
N
paths can she take to get there? Assume that she 84
W E
walks only east and south (along the grid lines). paths.
S
4. A square table can seat four people. For a banquet,
a long rectangular table is formed by placing 14
such tables edge to edge in a straight line. How many
people can sit at the long table? 30 people
Barbara's
house
5. Find the sum of the whole numbers from 1 to 999. 499,500
6. How many trapezoids are in the figure below? (Hint: Solve several simpler
problems, and then look for a pattern.) 25 trapezoids
7. At a business luncheon, 424 handshakes took place. No two people shook
hands with each other more than once. What is the least number of people in
attendance at the luncheon? 30 people
8. On the occasion of his 50th birthday, the President was honored with a 21-gun
salute. The sound of each gunshot lasted 1 second, and 4 seconds elapsed
between shots. How long did the salute last? 101 seconds
9. In a tennis tournament, each athlete plays one match against each of the other
athletes. There are 14 athletes scheduled to play in the tournament. How many
matches will be played? 91 matches
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Problem Solving: Use Logical Reasoning
Some problems can be solved without the use of numbers. They can be solved by
the use of logical reasoning, given some information.
EXAMPLE
Anna, Bill, Carla, and Doug are siblings. Each lives in a different state
beginning with W. Use these clues to determine where each sibling lives:
(1) Neither sister lives in a state containing two words.
Washington
West
Virginia
Wyoming
(2) Bill lives to the west of his sisters.
Wisconsin
(3) Anna doesn’t cross the Mississippi River when she visits Doug.
Make a table to organize what you know. Use an initial for each name.
State
A
B
C
West Virginia
✗
✗
✗
Wisconsin
✗
Wyoming
✗
✗
Washington
✓ ✗
D
✗
From clue 1, you know that neither Anna
nor Carla lives in West Virginia.
Using clues 1 and 2, you know that Bill must live
in Washington if he lives to the west of his sisters.
Use logical reasoning to complete the table.
State
A
B
C
West Virginia
✗
✗
✗ ✓
D
Doug lives in West Virginia because no other sibling does.
Wisconsin
✓
✗
✗
From clue 3, you know that Anna must live in Wisconsin.
Wyoming
✗
✗ ✓ ✗
Washington
✗
✓ ✗
✗
Carla, therefore, lives in Wyoming.
✗
EXERCISES
1. Harold has a dog, a parrot, a goldfish, and a hamster. Their names are J. T.,
Izzy, Arf, and Blinky. Izzy has neither feathers nor fins. Arf can’t bark. J. T.
weighs less than the four-legged pets. Neither the goldfish nor the dog has the
longest name. Arf and Blinky don’t get along well with the parrot. What is each
pet’s name? J.T. is the parrot; Izzy is the dog; Arf is the goldfish; Blinky is the hamster.
2. At the state basketball championship tournament, 31 basketball games are
played to determine the winner of the tournament. After each game, the loser
is eliminated from the tournament. How many teams are in the tournament? 32 teams
3. The sophomore class has 124 students. Of these students, 47 are involved in
musical activities: 25 in band and 36 in choir. How many students are involved
in both band and choir? 14 students
4. Tina’s height is between Kimiko’s and Ignacio’s. Ignacio’s height is between
Jerome’s and Kimiko’s. Tina is taller than Jerome. List the people in order from
shortest to tallest. Jerome, Ignacio, Tina, Kimiko
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Problem Solving: Work Backward
In some situations it is easier to start with the end result and work backward to
find the solution. You work backward in order to solve linear equations. The
equation 2x + 3 = 11 means “double x and add 3 to get 11.” To find x, you “undo”
those steps in reverse order.
2x + 3 = 11
Divide each side by 2.
Another time it is convenient to work backward is when you want to “reverse” a
set of directions.
EXAMPLE
1
Algebra Sandy spent 10
of the money in her purse for lunch. She then spent $23.50
for a gift for her brother, then half of what she had left on a new CD. If Sandy has
$13 left in her purse, how much money did she have in it before lunch?
Skills Handbook
x=4
Subtract 3 from each side.
Skills Handbook
2x = 8
Start with the $13 that Sandy has left in her purse.
She spent half of what she had before the $13 on a new CD, so she must have had
twice $13, or $26, before she bought the CD.
She spent $23.50 on a gift for her brother, so add $23.50 to $26. Before buying the
gift for her brother, she had $49.50.
9
1
She spent 10
of the money for lunch and was left with $49.50. That means 10
of what
she had is $49.50. Set up an equation.
9
10 x
= 49.50
x = 55
Sandy had $55 in her purse before lunch.
EXERCISES
1. To go from Bedford to Worcester, take Route 4 south, then Route 128 south,
and then Route 90 west. How do you get from Worcester to Bedford?
1. Route 90 east,
Route 128 north,
Route 4 north
2. Algae are growing on a pond’s surface. The area covered doubles each day. It
takes 24 days to cover the pond completely. After how many days will the pond
be half covered with algae? 23 days
3. Don sold 15 as many raffle tickets as Carlita. Carlita sold 3 times as many as
Ranesha. Ranesha sold 7 fewer than Russell. If Russell sold 12 tickets, how
many did Don sell? 3 tickets
4. At 6% interest compounded annually, the balance in a bank account will
double about every 12 years. If such an account has a balance of $16,000 now,
how much was deposited when the account was opened 36 years ago? about $2000
5. Solve the puzzle that Yuan gave to Inez: I am thinking of a number. If I triple
the number and then halve the result, I get 12. What number am I thinking of? 8
6. Carlos paid a $14.60 taxi fare from a hotel to the airport, including a $2.00 tip.
Green Cab Co. charges $1.20 per passenger plus $0.20 for each additional 15 mile.
How many miles is the hotel from the airport? 11.4 mi
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page 746 Using a Ruler and
Protractor
1 – 2. Answers may vary
slightly.
Using a Ruler and Protractor
Knowing how to use a ruler and protractor is crucial for success in geometry.
1. 35 mm; 46 mm
EXAMPLE
2. mlA ≠ 48; mlB ≠ 97;
mlC ≠ 35
3.
Draw a triangle that has sides of length 5.2 cm and 3.0 cm and a 688 angle between
these two sides.
The angle opens to the
3.7 cm
left, so read angle
measures from
the top
scale.
10
170 1 20 3
60 1 0
50 4
14 0
0
34
4.8 cm
4.
0
43 102
5.4 cm
1
2
3
4
Centimeters
5
6
Step 1 Use a ruler to draw a segment 5.2 cm long.
5.
68
112
214 in.
Step 2 Place the crosshairs of a protractor at
one endpoint of the segment. Make a small
mark at the 688 position along the protractor.
214 in.
4
3
s
r
2
ete
tim
n
e
C
214 in.
80 90 100 11
80 7 0
70 100
0
60 0 110
2
0
1
5 0
3
1
112
214 in.
1
68
0
6.
48
48
2 in.
7.
312 in.
134
Step 3 Align the ruler along the small mark and
the endpoint you used in Step 2. Place the zero
point of the ruler at the endpoint. Draw a segment
3.0 cm long.
312 in.
EXERCISES
Step 4 Complete the triangle by connecting the
endpoints of the first and second segments.
1–7. See margin.
1. Measure sides AB and BC to the nearest millimeter.
B
2. Measure each angle of #ABC to the nearest degree.
3. Draw a triangle that has sides of length 4.8 cm and 3.7 cm and a
348 angle between these two sides.
C
4. Draw a triangle that has 438 and 1028 angles and a side of length
5.4 cm between these two angles.
5. Draw a rhombus that has sides of length 2 14 in., and 688 and
1128 angles.
6. Draw an isosceles trapezoid that has one pair of 488 base angles
and a base of length 2 in. between these two base angles.
7. Draw an isosceles triangle that has two congruent sides 3 12 in. long
and a 1348 vertex angle.
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Measurement Conversions
To convert from one unit of measure to another, you multiply by a conversion factor
in the form of a fraction. The numerator and denominator are in different units, but
they represent the same amount. So, you can think of this as multiplying by 1.
An example of a conversion factor is 121 ft
in. . You can create other conversion factors
using the table on page 728.
EXAMPLE
Complete each statement.
a. 88 in. = j ft
b. 5.3 m = j cm
c. 3700 mm = j cm
d. 90 in. = j yd
cm
5.3 m ? 100
1 m = 5.3(100) cm = 530 cm
88
1
88 in. ? 121 ft
in. = 12 ft = 7 3 ft
1 yd
90
1
90 in. ? 121 ft
in. ? 3 ft = 36 yd = 2 2 yd
1 cm = 370 cm
3700 mm ? 10
mm
Skills Handbook
Skills Handbook
1
Area is always in square units, and volume is always in cubic units.
3 ft
3 ft
1 yd = 3 ft
3 ft
3 ft
3 ft
3 ft
1 yd2 = 9 ft2
1 yd3 = 27 ft3
2
EXAMPLE
Complete each statement.
a. 300 in.2 = j ft 2
b. 200,000 cm3 = j m3
1 ft = 12 in., so 1 ft 2 = (12 in.)2 = 144 in.2
1 m = 100 cm, so 1 m3 = (100 cm)3 = 1,000,000 cm3
2
1 ft2
300 in.2 ? 1 ft 2 = 2 12
200,000 cm3 ?
144 in.
1 m3
= 0.2 m3
1,000,000 cm3
EXERCISES
Complete each statement.
1. 40 cm = j m 0.4
2. 1.5 kg = j g 1500
3. 60 cm = j mm 600
200 in. = j ft 16 23
5
42 fl oz = j qt 116
5. 28 yd = j in. 1008
6. 1.5 mi = j ft 7920
8. 430 mg = j g 0.43
9. 34 L = j mL 34,000
10. 1.2 m = j cm 120
11. 43 mm = j cm 4.3
12. 3600 s = j min 60
13. 15 g = j mg 15,000
14. 12 qt = j c 48
15. 0.03 kg = j mg 30,000
4.
7.
2 14
16. 14 gal = j qt 56
17. 4500 lb = j t
19. 12 mL = j L 0.012
20. 2 pt = j fl oz 32
21. 20 m/s = j km/h 72
22. 3 ft 2 = j in.2 432
23. 108 m 2 = j cm 2 1,080,000
24. 2100 mm 2 = j cm 2 21
25. 1.4 yd2 = j ft2 12.6
26. 0.45 km 2 = j m 2
27. 1300 ft 2 = j yd 2 14449
11
28. 1030 in.2 = j ft2 772
29. 20,000,000 ft 2 = j mi 2 3125
4356
18. 234 min = j h 3.9
450,000
30. 1000 cm 3 = j m 3 0.001
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Measurement, Rounding Error, and Reasonableness
There is no such thing as an exact measurement. Measurements are always
approximate. No matter how precise it is, a measurement actually represents a
range of values.
1
EXAMPLE
Chris’s height, to the nearest inch, is 5 ft 8 in. Find the range of values this
measurement represents.
The height is given to the nearest inch, so the error is 21 in. Chris’s height, then, is
between 5 ft 712 in. and 5 ft 821 in., or 5 ft 8 in. 4 12 in. Within this range are all
measures which, when rounded to the nearest inch, equal 5 ft 8 in.
As you calculate with measurements, errors can accumulate.
2
EXAMPLE
Jean drives 18 km to work each day. The distance is given to the nearest kilometer.
a. Find the range of values this measurement represents.
The driving distance is between 17.5 and 18.5 km, or 18 4 0.5 km.
b. Find the error in the round-trip distance.
Double the lower limit, 17.5, and the upper limit, 18.5. Thus, the round trip can
be anywhere between 35 and 37 km, or 36 4 1 km. The error for the round trip
is double the error of a single leg of the trip.
So that your answers will be reasonable, keep precision and error in mind as you
calculate. For example, in finding AB, the length of the hypotenuse of #ABC, it
would be inappropriate to give the answer as 9.6566 if the sides are given to the
nearest tenth. Round your answer to 9.7.
A
?
5.1
C
B
8.2
EXERCISES
Each measurement is followed by its unit of greatest precision. Find the range of
values that each measurement represents. 2. 123 1 cm to 124 1 cm
2
1. 24 ft (ft) 23 21 ft to 24 12 ft 2. 124 cm (cm)
2
3. 340 mL (mL) 339 12 mL to 340 12 mL
4. 5 12 mi. Q 12 mi R
5. 73.2 mm (0.1 mm)
6. 34 yd 2 A yd 2 B 33 12 yd2 to 34 12 yd2
3
1
73.15
mm
to
73.25
mm
5 4 mi to 5 4 mi
7. 5.4 mi (0.1 mi)
8. 6 ft 5 in. (0.5 in.)
9. 15 12 yd Q 12 yd R 1541 yd to 1534 yd
6 ft 4.75 in. to 6 ft 5.25 in.
5.35 mi to 5.45 mi
10. The lengths of the sides of TJCM are given to the nearest tenth of a centimeter.
2.7 cm
T
Find the range of values for the figure’s perimeter. 8.7 cm to 9.1 cm
1.4 cm
11. To the nearest degree, two angles of a triangle are 498 and 738. What is the
range of values for the measure of the third angle? 57 to 59
M
2.8 cm
12. The lengths of the legs of a right triangle are measured as 131 m and 162 m.
You use a calculator to find the length of the hypotenuse. The calculator display
reads 2 0 8 . 3 3 8 6 7 . What should your answer be? 208 m
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J
2.0 cm
C
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The Effect of Measurement Errors on Calculations
Measurements are always approximate, and calculations with these measurements
produce error. Percent error is a measure of accuracy of a measurement or
calculation. It is the ratio of the greatest possible error to the measurement.
percent error =
greatest possible error
measurement
The dimensions of a box are measured as 18 in., 12 in., and 9 in. Find the
percent error in calculating its volume.
The measurements are to the nearest inch, so the greatest possible length error is
one half of one inch, or 0.5 in.
as measured
maximum value
minimum value
V=/?w?h
V=/?w?h
V=/?w?h
= 18 ? 12 ? 9
= 18.5 ? 12.5 ? 9.5
= 17.5 ? 11.5 ? 8.5
= 1944, or 1944 in.3
< 2196.9, or 2196.9 in.3
< 1710.6, or 1710.6 in.3
Skills Handbook
Skills Handbook
EXAMPLE
Possible Error:
maximum - measured
measured - minimum
2196.9 - 1944 = 252.9
1944 - 1710.6 = 233.4
greatest possible error
percent error =
measurement
252.9
= 1944
< 0.1300926
< 13%
The percent error is about 13%.
EXERCISES
Find the percent error in calculating the volume of each box given its dimensions.
Round to the nearest percent.
1. 10 cm by 5 cm by 20 cm 18%
2. 12 in. by 6 in. by 2 in.
3. 1.2 mm by 5.7 mm by 2.0 mm 8%
4. 7.5 m by 6.4 m by 2.7 m 3%
5. 22.5 cm by 16.4 cm by 26.4 cm 1%
6. 1.24 cm by 4.45 cm by 5.58 cm 1%
7.
814
in. by
1712
41%
8. 734 in. by 2218 in. by 614 in. 2%
in. by 5 in. 5%
Find the percent error in calculating the perimeter of each figure.
N 7%
9.
N 2%
10.
N 2%
11.
27 cm
2.5 m
8 in.
2.5 m
24 cm
22 cm
6 in.
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page 750 Mean, Median, and
Mode
5. Median; the mode does
not represent most of
the data; the mean is
significantly affected by
the single highest salary.
Mean, Median, and Mode
Measures of central tendency, such as mean, median, and mode, are numbers that
describe a set of data.
The mean, sometimes called the average, is the sum of the data items divided by
the number of data items.
The median is the middle number when data items are placed in order and there
are an odd number of data items. For an even number of data items, the median is
the mean of the middle two numbers.
The mode is the data item that appears most frequently.
A set of data may have more than one mode or no modes.
EXAMPLE
Eighteen students were asked to measure the angle formed by the three
objects in the diagram. Their answers, in order from least to greatest, are
as follows:
Oak tree
Flagpole
65, 66, 66, 66, 66, 66, 66, 67, 67, 67, 67, 67, 68, 68, 69, 70, 74, 113
Find the mean, median, and mode of the data.
18 measures = 1258 = 69 8
Mean: sum of the18
9
18
Median:
This data list is already ordered. The two middle numbers—the ninth
and the tenth numbers on the list—are both 67. So the median is 67.
Statue
Mode:
There are more 66’s than any other number, so the mode is 66.
EXERCISES
Find the mean, median, and mode of each set of data.
1. Numbers of students per school in Newtown: 234, 341, 253, 313, 273, 301, 760 353.6; 301; no mode
2. Lunch expenses: $4.50, $3.26, $5.02, $3.58, $1.25, $3.05, $4.24, $3.56, $3.31 $3.53; $3.56; no mode
3. Salaries at D. B. Widget & Co.: $15,000; 18,000; $18,000; $21,700; $26,500;
$27,000; $29,300; $31,100; $43,000; $47,800; $69,000; $140,000 $40,533; $28,150; $18,000
4. Population of towns in Brower County: 567, 632, 781, 902, 1034, 1100, 1598,
2164, 2193, 3062, 3074, 3108, 3800, 3721, 4104 2123; 2164; no mode
5. In Exercise 3, which measure or measures of central tendency do you think
best represent the data? Explain. See margin.
6. Find the mean, median, and mode of the exam scores at the right.
7. In the example, the student who reported the angle measure as 113
most likely made an error. If this measure is dropped from the list,
what are the mean, median, and mode of the remaining 17 scores?
8. In the example, if the measurement 65 were instead 51, would the mean
decrease? Would the median? Would the mode? yes; no; no
Final Exam Scores
34, 47, 53, 56, 57, 62, 62, 64, 67,
70, 74, 74, 74, 78, 82, 85, 85, 85,
85, 86, 88, 92, 93, 93, 94, 95, 97
6. 75.3; 78; 85
6
7. 67 17
, 67, 66
9. In the example, if the two students who measured the angle at 68 both reduced
their measurements to 67, would the mode be affected? How? Yes; the mode would be 67.
750
750
Skills Handbook
GEOM_3e_SKILHB_740-762 10/26/05 6:56 PM Page 751
Bar Graphs and Line Graphs
Data displayed in a table can be very useful, but a table is not always as easy to
interpret as a graph. Bar graphs and line graphs can show the same data, but
sometimes one type of graph has advantages over the other.
EXAMPLE
$63,444
2002
$79,855
1998 1999 2000 2001 2002
Year
page 751 Bar Graphs and
Line Graphs
Bar graphs are useful when you wish to compare amounts. In the example above,
the tallest bar is clearly twice the height of the shortest bar. At a glance, it is
evident that in four years the revenue approximately doubled.
1. Create a bar graph and a line graph to display the data in the table below.
See margin.
Sales of Rock Music (in millions of dollars)
$3527
$3675
$3552
SOURCE: Recording Industry Association of America.
Go to www.PHSchool.com for a data update.
Temperatures in
Grand Island, Nebraska,
on February 2
Web Code: afg-2041
.
A.
M
5. Can you tell from the graph what the actual maximum and minimum
temperatures were between 6 A.M. and 6 P.M.? Explain. See above right.
60
50
40
30
20
10
0
6
3. During which time periods did the temperature appear to increase?
8 A.M.– 2 P.M.
4. Estimate the temperature at 11 A.M. and at 5 P.M. 288F; 428F
Temperature (⬚F)
For Exercises 2 – 6, refer to the line graph at the right.
2. What was the lowest temperature recorded between 6 A.M. and 6 P.M.? 108F
Sales of Rock Music
Sales (in millions of dollars)
$3977
6. The same data could be presented in a bar graph. Which presentation
is better for these data, a line graph or a bar graph? Explain why. Answers may vary.
Sample: A line graph; the line graph shows a change in temperature over time
more clearly.
.
$4086
0
Year
P.
M
$4127
.
Sales
5. No; you cannot tell
how the temperature
changed between the
measurements.
6
2000
P.
M
1999
.
1998
M
1997
2
1996
A.
1995
10
Year
2000
00
EXERCISES
4000
20
Did revenue increase from 1998 to 1999? It is difficult to tell by looking at either
graph; for that information, you should look back at the table.
Sales of Rock Music
99
Line graphs allow you to see how a set of data changes over time. In the example,
the slope of the line shows that revenue has increased at a steady rate since 1999.
98
Sales (in millions of dollars)
1.
19
0
2001
97
10,000
$51,772
19
20,000
2000
96
30,000
$40,019
19
40,000
1999
19
50,000
$39,780
80,000
70,000
60,000
50,000
40,000
30,000
20,000
10,000
0
1998 1999 2000 2001 2002
Year
95
60,000
Revenue
1998
19
Revenue (dollars)
Revenue (dollars)
70,000
Year
Revenue of HJL Co.
4000
2000
0
19
96
19
97
19
98
19
99
20
00
80,000
Line Graph
19
95
Revenue of HJL Co.
Skills Handbook
Bar Graph
Skills Handbook
Revenue of HJL Co.
Make a bar graph and a line graph showing the data in the table at the right.
Year
Time of Day
Skills Handbook
751
751
GEOM_3e_SKILHB_740-762 10/26/05 6:56 PM Page 752
Box-and-Whisker Plots
A box-and-whisker plot is a way to display data on a number line. It provides a
picture of how tightly the data cluster around the median and how wide a range
the data have. The diagram below shows the various points associated with a
box-and-whisker plot.
Lower quartile:
median of lower
half of data
Least
data point
Upper quartile:
median of upper
half of data
Greatest
data point
Median
The box represents the
middle half of the data.
The “whiskers” represent the
upper and lower fourths of the data.
EXAMPLE
The heights, in inches, of 23 geometry students are as follows.
58, 61, 63, 63, 63, 64, 64, 65, 65, 65, 67, 68, 68, 68, 69, 70, 70, 70, 72, 72, 72, 74, 75
Draw a box-and-whisker plot.
The heights range from 58 in. to 75 in. Show 58 and 75 as endpoints on a line
segment. The median is 68, so locate 68 in relation to 58 and 75. The lower quartile
(the median of the lower eleven heights) is 64. The upper quartile (the median of
the upper eleven heights) is 70. Locate 64 and 70 and draw a box enclosing them.
Draw a vertical segment inside the box through the median.
58
64
68
70
75
EXERCISES
1. All of the physical education students at Martin Luther King, Jr., High School
were timed sprinting the 100-meter dash. The box-and-whisker plot below
summarizes the data. Use it to find the following.
a. median 16.6
b. lower quartile 15.1
c. upper quartile 19.4
12.4
15.1
16.6
19.4
23.7
2. Make a box-and-whisker plot for the following data set, which lists the
weights, in pounds, of the students trying out for the wrestling team at
Benjamin Banneker High School.
2.
Wrestlers' Weights
104
130
144
155 171
104, 121, 122, 130, 130, 131, 140, 144, 147, 147, 148, 155, 160, 163, 171
3. Make a box-and-whisker plot for the following set of data, which lists the 3.
numbers of pages in a set of books. (Hint: Order the data from smallest
to largest.)
185
205, 198, 312, 254, 185, 268, 297, 242, 356, 262
205
752
752
Skills Handbook
Pages in Book
356
258 297
GEOM_3e_SKILHB_740-762 10/26/05 6:56 PM Page 753
Squaring Numbers and Finding Square Roots
The square of a number is found by multiplying the number by itself. An exponent
of 2 is used to indicate that a number is being squared.
1
EXAMPLE
Simplify.
(72)2 = 72 ? 27
4
= 49
(-3.5)2 = (-3.5) ? (-3.5)
= 25
= 12.25
The square root of a number is itself a number that, when squared, results in the
original number. A radical symbol (" ) is used to represent the positive square
root of a number.
2
EXAMPLE
Skills Handbook
c. (27)2
b. (-3.5)2
52 = 5 ? 5
Skills Handbook
a. 52
Simplify. Round to the nearest tenth if necessary.
a. !36
!36 = 6 since
b. !174
62
= 36
!174 < 13.2 since 13.22 < 174
You can solve equations that include squared numbers.
3
EXAMPLE
Algebra Solve.
a. x2 = 144
b. a2 + 32 = 52
x = 12 or -12
a2 + 9 = 25
a2 = 16
a = 4 or -4
EXERCISES
Simplify.
1. 112
121
2. 162 256
3. (-14)2
5. 5.12
26.01
6. (37)2
7. (85)2
9
49
196
64
25
4. (-21)2 441
8. -62
–36
x 2 Simplify. Round to the nearest tenth if necessary.
9. !100 10
13. !400 20
10. !169 13
11. !74 8.6
12. !50 7.1
14. !289 17
15. Å 49 2
3
16.
49 7
Å 81 9
Algebra Solve. Round to the nearest tenth if necessary.
17. x2 = 49 w7
18. a2 = 9 w3
19. y2 + 7 = 8 w1
20. 5 + x2 = 11 w2.4
21. 82 + b2 = 102 w6
22. 52 + 42 = c2 w6.4
23. p2 + 122 = 132 w5
24. 202 = 152 + a2 w13.2
Skills Handbook
753
753
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Evaluating and Simplifying Expressions
You evaluate an expression with variables by substituting a number for each
variable. Then simplify the expression using the order of operations. Be
especially careful with exponents and negative signs. For example, the expression
-x 2 always yields a negative or zero value, and (-x)2 is always positive or zero.
1
Order of Operations
1. Perform any operation(s)
inside grouping symbols.
2. Simplify any term with
exponents.
EXAMPLE
3. Multiply and divide in
order from left to right.
Algebra Evaluate each expression for r = 4.
a.
4. Add and subtract in
order from left to right.
-r 2
-r 2 = - A 42 B = -16
b. -3r 2
-3r 2 = -3 A 42 B = -3(16) = -48
c. (-3r)2
(-3r)2 = (-3 ? 4)2 = (-12)2 = 144
To simplify an expression, you eliminate any parentheses and combine like terms.
2
EXAMPLE
Algebra Simplify each expression.
a. 5r - 2r + 1
Combine like terms.
5r - 2r + 1 = 3r + 1
b. p(3r - 1)
c. (r + p)(r - p)
Use the distributive property.
p(3r - 1) = 3pr - p
Multiply polynomials.
(r + p)(r - p) = r 2 - p2
EXERCISES
x Algebra Evaluate each expression for x ≠ 5 and y ≠ –3.
2
1. -2x 2 –50
2. -y + x 8
3. -xy 15
4. (x + 5y) 4 x –2
5. x + 5y 4 x 2
6. (-2y)2 36
7. (2y)2 36
8. (x - y)2 64
1 –2
9. x 1
y
13. x ? y - x –20
10. y - (x - y) –11
11. -yx 243
14. x - y ? x 20
y3 2 x
15. x 2 y –4
17. Which expression gives the area of the shaded figure at the right? B
B. p A r 2 - s2 B
A. p(r - s)2
2
2
C. p A s - r B
D. pr2 - 2ps
2(1 2 x)
12. y 2 x
1
16. -y(x - 3)2 12
s
r
x 2 Algebra Simplify.
2x ± 3
18. 6x - 4x + 8 - 5
19. 2(/ + w) 2< ± 2w
20. -(4x + 7) –4x – 7
21. -4x(x - 2) 4x 2 ± 8x
22. 3x - (5 + 2x) x – 5
23. 2t 2 + 4t - 5t 2–3t 2 ± 4t 24. (r - 1)2 r 2 – 2r ± 1
25. (1 - r)2 1 – 2r ± r 2
26. (y + 1)(y - 3)
27. 4h + 3h - 4 + 3 7h – 1 28. pr - (1 + pr) –1
2 – 2y – 3
y
31. 3y 2 - A y 2 + 3y B
32. -(x + 4)2
30. 2ph(1 - r)2
2πhr2 – 4πhr ± 2πh
2y 2 – 3y
–x 2 – 8x – 16
29. (x + 4)(2x - 1)
2x 2 ± 7x – 4
754
754
Skills Handbook
GEOM_3e_SKILHB_740-762 10/26/05 6:56 PM Page 755
Simplifying Radicals
A radical expression is in its simplest form when all three of the following
statements are true.
1. The expression under the radical sign contains no perfect square factors
(other than 1).
2. The expression under the radical sign does not contain a fraction.
EXAMPLE
Simplify.
a.
4
b. !12
Å9
4 5 !4 5 2
3
!9
!12 5 !4 ? !3 5 2 !3
Å9
2
EXAMPLE
Find the length of the diagonal of rectangle HJKL.
c2
=
c2
+
= 50
72
12
Use the Pythagorean Theorem.
Simplify the right side.
c = !50
3
7
H
1
L
Skills Handbook
1
Skills Handbook
3. The denominator does not contain a radical expression.
J
1
K
c
7
Find the square root of each side.
= !25 ? 2
Find a perfect square factor of 50.
= 5 !2
Simplify the radical.
EXAMPLE
Simplify 1 .
!3
1 ? !3 5 !3
3
!3 !3
Multiply by
!3
, or 1, to eliminate the radical in the denominator.
!3
EXERCISES
Simplify each radical expression.
1. !27 3!3
2. !24 2 !6
6. !228 !57
2
7.
!16
2
Å5
3. !150 5!6
!10
8.
5
27
Å 75
4.
9. 3
3
5
72 2!2
9
6
10. 6 !18 3 !
2
!48
1 1
3
5.
Å9
!8
3 !2
4
Å
x 2 Algebra Find the value of x. Leave your answer in simplest radical form.
11.
13
9
7
x
15.
x
12.
13. 2 Á 39
4 Á 10
14
2 Á 51
10
4
x
18
3!5
x
17.
20
x
10
x
4!11
16.
22
6
3
16
15
5!10
14.
x
2 Á 29
x
18.
18
15
3 Á 61
Skills Handbook
755
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GEOM_3e_SKILHB_740-762 10/26/05 6:56 PM Page 756
Simplifying Ratios
The ratio of the length of the shorter leg to the length of the longer leg
for this right triangle is 4 to 6. This ratio can be written in several ways.
4
6
4 to 6
4
4;6
6
EXAMPLE
Algebra Simplify each ratio.
b. 3ab ; 27ab
a. 4 to 6
3ab
3ab ; 27ab = 27ab
4 to 6 = 46
= 22 ?? 23
= 9 3ab
? 3ab
Find and remove the common factor.
= 91
= 23
4b
c. 4aa 1
1b
4a 1 4b = 4(a 1 b)
a1b
a1b
=4
Factor the numerator. The denominator cannot be
factored. Remove the common factor (a ± b).
EXERCISES
2
x Algebra Simplify each ratio.
1. 25 to 15 35
2. 6 ; 9 23
36
3. 54 23
4. 0.8 to 2.4 13
7 1
5. 14x 2x
12c
6. 14c 67
7. 22x2 to 35x 22x
35
1
8. 0.5ab ; 8ab 16
9. 0.25x 16y
4xy
2 1 x x 1 1
11. x 2x
2
x23
14. (2x - 6) ; (6x - 4) 3x
22
π
17. 2 pr
r±π
3
10. 121x to 5x 10
1
13. 0.72t ; 7.2t2 10t
16. (9x - 9y) to (x - y) 9
r
1 pr
1
r
12. 4r2 to 6r 24
15. 12xy : 8xy 32
8ab ab
18. 32xy
4xy
Express each ratio in simplest form.
5
19. shorter leg ; longer leg 12
20. hypotenuse to shorter leg 13
5
shorter leg 5
21. hypotenuse 13
13
22. hypotenuse ; longer leg 12
23. longer leg to shorter leg
x2
12
5
15
longer leg
24. hypotenuse 12
13
36
area of shaded figure
Algebra Write an expression in simplest form for area of blue figure .
25.
26.
27.
5
14
2
π
1
9
4y
2
兹苵
756
Skills Handbook
3x
x
2
兹苵
756
39
5y
2y
GEOM_3e_SKILHB_740-762 10/26/05 6:56 PM Page 757
Absolute Value
Absolute value is used to represent the distance of a number from 0 on a number
line. Since distance is always referred to as a nonnegative number, the absolute
value of an expression is nonnegative.
On the number line at the right, both 4 and -4 are four
units from zero. Therefore, Δ4« and Δ-4« are both equal
to four.
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
4 units
0
1
2
3
4
5
EXAMPLE
Simplify each expression.
a. Δ-7«
b. Δ15«
Δ-7« = 7
c. Δ4« + Δ-19«
Δ15« = 15
Δ4« + Δ-19« = 4 + 19 = 23
When working with more complicated expressions, always remember to simplify
within absolute value symbols first.
2
Skills Handbook
Skills Handbook
1
4 units
EXAMPLE
Simplify each expression.
a. Δ4 - 8«
b. -3Δ-7 - 4«
Δ4 - 8« = Δ-4«
-3Δ-7 - 4« = -3Δ-11«
=4
= -3 ? 11
= -33
To solve an equation involving absolute value, remember that absolute value
symbols cause both negative and positive values to become positive.
3
EXAMPLE
Algebra Solve.
a. Δx« = 7
b. Δx« - 3 = 22
x = 7 or -7
Δx« - 3 = 22
Δx« = 25
x = -25 or 25
EXERCISES
Simplify each expression.
1. Δ-8«
2. Δ11«
8
5. Δ-7« + Δ15«
22
9. 10 - Δ-20«
–10
3. Δ16«
11
6. Δ-12« - Δ-12«
10. Δ-9 - 11«
20
x 2 Algebra Solve.
13. Δx« = 16 –16 or 16
14. 2 = Δx« –2 or 2
0
16
7. Δ5« - Δ10« –5
11. 2Δ-21 + 16« 10
4. Δ-23«
23
8. Δ4« + Δ2«
6
12. -8Δ-9 + 4« –40
–24 or 24
15. Δx« + 7 = 27 –20 or 20 16. Δx« - 9 = 15
Skills Handbook
757
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Solving and Writing Linear Equations
To solve a linear equation, use the properties of equality and properties of
real numbers to find the value of the variable that satisfies the equation.
1
EXAMPLE
Algebra Solve each equation.
a. 5x - 3 = 2
b. 1 - 2(x + 1) = x
5x - 3 = 2
5x = 5
x=1
1 - 2(x + 1) = x
1 - 2x - 2 = x
Add 3 to each side.
-1 - 2x = x
Divide each side by 5.
-1 = 3x
-13 = x
Use the Distributive Property.
Simplify the left side.
Add 2x to each side.
Divide each side by 3.
You will sometimes need to translate word problems into equations. Look for
words that suggest a relationship or some type of mathematical operation.
2
EXAMPLE
Algebra A student has grades of 80, 65, 78, and 92 on four tests. What is the
minimum grade she must earn on her next test to ensure an average of 80?
Relate average of 80, 65, 78, 92, and next test, is 80
Pull out the key words and numbers.
Define Let x = the grade on the next test.
Let a variable represent what you are looking for.
Write
80 1 65 1 78 1 92 1 x = 80
5
315 1 x = 80
5
315 + x = 400
x = 85
Write an equation.
Combine like terms.
Multiply each side by 5.
Subtract 315 from each side.
The student must earn 85 on the next test for an average of 80.
EXERCISES
2
x Algebra Solve each equation.
1. 3n + 2 = 17 5
2. 5a - 2 = -12 –2
4. 3(n - 4) = 15 9
7. 6 - (3t + 4) = -17
10. 75 p - 10 = 30 56
3. 2x + 4 = 10 3
5. 4 - 2y = 8 –2
19
3
8. 7 = -2(4n - 4.5)
m - 3 = 1 –8
11. 22
6. -6z + 1 = 13 –2
1
4
9. (w + 5) - (2w + 5) = 5 –5
12. 5k + 2(k + 1) = 23 3
x 2 Algebra Write an equation and solve the problem.
13. Twice a number subtracted from 35 is 9. What is the number? 35 – 2x ≠ 9; 13
14. A new tenant pays the landlord the amount of the first month’s rent, the same
amount for the last month’s rent, and half a month’s rent for a security deposit.
The total is $2437.50. How much is the monthly rent? 2.5r ≠ $2437.50; $975
15. The Johnsons pay $9.95 a month plus $0.035 per minute for local phone service.
Last month, they paid $12.75. How many minutes of local calls did they make? $9.95 ± $0.035m ≠ $12.75;
80 min
758
758
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Solving Literal Equations
An equation with two or more variables is called a literal equation. It is often
necessary to solve a literal equation for a particular variable.
1
EXAMPLE
P = 2(O + w)
P = 2O + 2w
P - 2w = 2O
P 2 2w = O
2
2
Use the Distributive Property.
Subtract 2w from each side.
Divide each side by 2.
EXAMPLE
Algebra The formula A = 12(b1 + b2)h gives the area A of a trapezoid with bases
b1 and b2 and height h. Solve for h.
A = 12(b1 + b2)h
2A = h(b1 + b2)
2A
b1 1 b2 = h
3
Skills Handbook
Skills Handbook
Algebra The formula P = 2(O + w) gives the perimeter P of a rectangle with
length O and width w. Solve the equation for O.
Multiply each side by 2.
Divide each side by (b1 ± b2).
EXAMPLE
Algebra The formula for converting from degrees Celsius C to degrees
Fahrenheit F is F = 95C + 32. Solve for C.
F = 95 C + 32
F - 32 = 95 C
5(F - 32) = C
9
Subtract 32 from each side.
Multiply each side by 59 .
EXERCISES
2
S
15. < ≠ S 2πrπr ≠ πr
–r
16. b1 ≠
2A 2 hb2
h
≠ 2A
h – b2
x 2 Algebra Solve each equation for the variable in red.
1. Perimeter of rectangle: P = 2w + 2O w ≠ P 22 2<
Á πS
3. Surface area of sphere: S = 4pr 2 r ≠ 12 Î S
π ≠ 2π
5. Area of kite or rhombus: A = 12 d1d2 d2 ≠ 2A
d1
7. Area of regular polygon: A = 12ap a ≠ 2A
p
9. Area of triangle: A = 12 bh h ≠ 2A
b
11. Euler’s Formula: F + V = E + 2 V ≠ E ± 2 – F
V
2. Volume of prism: V = Owh w ≠ <h
A
4. Lateral area of cylinder: A = 2prh r ≠ 2πh
Á πA
6. Area of circle: A = pr 2 r ≠ Î A
π ≠ π
8. Volume of cylinder: V = pr 2h h ≠ V2
y
y
πr
10. Tangent of &A: tan A = x x ≠ tan A
C
12. Circumference of circle: C = 2pr r ≠ 2π
3V ≠ Á 3πhV
13. Cosine of &A: cos A = bc b ≠ c cos A
14. Volume of cone: V = 13 pr 2h r ≠ Î πh
πh
15. Surface area of right cone: S = pr 2 + prO See above. 16. Area of trapezoid: A = 12(b1 + b2)h See above.
17. Volume of pyramid: V = 13 Bh B ≠ 3V
h
19. Surface area of regular pyramid: S = B + 12 pO
2B
< ≠ 2S 2
p
18. Pythagorean Theorem: a 2 + b 2 = c 2 b ≠ Í c 2 2 a 2
20. Surface area of right cylinder: S = 2pr 2 + 2prh
2πr 2
S – r
h ≠ S 22πr
≠ 2πr
Skills Handbook
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Systems of Linear Equations
Normally, there are many ordered pairs that satisfy a given equation. For example,
(3, 4), (4, 5), (5, 6), and infinitely many other pairs all satisfy the equation
y = x + 1. In solving a system of two linear equations, however, you need to find
ordered pairs that satisfy both equations at once. Ordinarily, there is just one such
ordered pair; it is the point where the graphs of the two lines intersect.
One method you can always use to solve a system of linear equations is the
substitution method.
EXAMPLE
Algebra Solve the system. 2x - y = -10
-3x - 2y = 1
Solve one of the equations for a variable. Looking at the two equations,
it seems easiest to solve the first equation for y.
2x - y = -10
-y = -2x - 10
Subtract 2x from each side.
y = 2x + 10
Multiply each side by –1.
Now substitute 2x + 10 for y in the other equation.
-3x - 2y = 1
-3x - 2(2x + 10) = 1
-3x - 4x - 20 = 1
-7x = 21
x = -3
Write the other equation.
Substitute (2x ± 10) for y.
Use the Distributive Property.
Simplify and add 20 to each side.
Divide each side by –7.
So x = -3. To find y, substitute -3 for x in either equation.
2x - y = -10
2(-3) - y = -10
-6 - y = -10
-y = -4
y=4
Write one of the equations.
y
Substitute –3 for x.
(⫺3, 4)
Simplify.
2
Add 6 to each side.
Multiply each side by –1.
So the solution is x = -3 and y = 4, or (-3, 4). If you graph 2x - y = -10
and -3x - 2y = 1, you’ll find that the lines intersect at (-3, 4).
⫺4 ⫺2 O
x
⫺2
EXERCISES
x Algebra Solve each system. 5. inf. many solutions: all points on the line y ≠ 14 x – 34
2
1. x + y = 3 (4, –1)
x-y=5
2. y - x = 4 no sol.
x+3=y
3. y = 1 (4, 1)
5x - 2y = 18
4. 3x - y = 5 (–2, –11)
x = -2
5. 4y - x = -3 See above. 6. 8x - 1 = 4y (43 , 54 )
2x - 6 = 8y
3x = y + 1
7. 2x + 2y = -4 (–3, 1)
-x + 3y = 6
8. 12y - 3x = 11 (–31 , 56 )
x - 2y = -2
9. 5x + 7y = 1 (3, –2)
4x - 2y = 16
10. Give an example of a system of linear equations with no solution. What do you
know about the slopes of the lines of such a system? Answers may vary. Sample: y ≠ 2x – 3, 4x – 2y ≠ 5.
They are equal or they are undefined.
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Percents
A percent is a ratio in which a number is compared to 100. For example,
the expression 60 percent means “60 out of 100.” The symbol %
stands for “percent.”
EXAMPLE
Convert each percent to a decimal.
a. 42%
b. 157%
42% = 0.42
157% = 1.57
c. 12.4%
d. 4%
12.4% = 0.124
4% = 0.04
To calculate a percent of a number, write the percent as a decimal and multiply.
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A percent can be written in decimal form by first writing it in ratio form, and then
25
writing the ratio as a decimal. For example, 25% is equal to the ratio 100
or 14.
1
As a decimal, 4 is equal to 0.25. Note that 25% can also be written
directly as a decimal by moving the decimal point two places to the left.
EXAMPLE
Simplify. Where necessary, round to the nearest tenth.
a. 30% of 242
b. 7% of 38
30% of 242 = 0.3 ? 242
7% of 38 = 0.07 ? 38
= 72.6
= 2.66 ≈ 2.7
For a percent problem, it is a good idea to check that your answer is reasonable by
estimating it.
3
EXAMPLE
Estimate 23% of 96.
23% < 25% and 96 < 100. 25% Q or 14 R of 100 = 25. A reasonable estimate is 25.
EXERCISES
Convert each percent to a decimal.
1. 50% 0.5
2. 75% 0.75
3. 27% 0.27
4. 6% 0.06
5. 32.5% 0.325
6. 84.6% 0.846
7. 9% 0.09
8. 2.5% 0.025
Simplify. Where necessary, round to the nearest tenth.
9. 21% of 40 8.4
13. 80.4% of 52 41.8
10. 45% of 200 90
11. 6% of 120 7.2
12. 2% of 54 1.1
14. 23.8% of 176 41.9
15. 7.5% of 32 2.4
16. 9.25% of 89 8.2
19. 73% of 64 45
20. 77% of 42 32
Estimate. Answers may vary. Samples are given.
17. 12% of 70 7
18. 48% of 87 43
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Probability
Probability is a measure of the likelihood of an event occurring. All probabilities
range from 0 to 1 where 0 is the probability of an event that cannot happen and
1 is the probability of an event that is certain to happen. An event with probability
0.5 or 50% has an equal chance of happening or not happening.
of favorable outcomes
The formula P(E) = number
number of possible outcomes is used to calculate the
probability of event E.
1
EXAMPLE
The numbers 2 through 21 are written on pieces of paper and placed in a hat.
One piece of paper is drawn at random. Determine the probability of selecting a
perfect square.
The total number of outcomes, 2, 3, 4, c, 21, for this event is 20.
There are 3 favorable outcomes: 4, 9, 16.
3
P(selecting a perfect square) = 20
2
EXAMPLE
Determine the probability of getting exactly two heads when two coins are tossed.
The total number of outcomes, (H, H), (H, T), (T, H), (T, T), for this event is 4.
There is 1 favorable outcome, (H, H).
P(two heads) = 14
EXERCISES
A jar contains 3 white balls, 7 red balls, and 4 green balls. A ball is selected at
random from the jar. Determine the probability of selecting a ball with the
given color.
1. red 12
3
2. white 14
3. green 72
4. green or white 12
5. A red ball is removed from the jar. Determine the probability that the next ball
4
selected will be green. 13
6. Two green balls are removed from the jar. Determine the probability that the
next ball selected will be green. 61
You roll a 12-sided polyhedron with the numbers 1–12 on its congruent faces.
Determine the probability of each outcome.
1
7. rolling a 2 12
10. rolling an odd number 12
8. rolling a 4 or a 5 16
5
11. rolling a prime number 12
9. rolling an even number 12
12. rolling a factor of 8 31
A coin is flipped three times. Determine the probability of each outcome.
13. exactly two tails 38
14. two heads and one tail 38
15. no more than two tails 87
16. no more than one head 12
17. at least one tail 87
18. all tails or all heads 14
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