Download Ratios and Proportions(Pages 195–200)

NAME
4-1
DATE
Ratios and Proportions (Pages 195–200)
A ratio is a comparison of two numbers by division. The ratio of x to y can be
x
expressed as x to y, x: y, or y . An equation stating that two ratios are equal is
c
a
called a proportion. In , the numbers a and d are the extremes and
b
d
the numbers b and c are the means.
Means-Extremes
Property of
Proportions
In a proportion, the product of the extremes is equal to the product of the means.
If
a
b
c
,
d
then ad bc. The cross products, ad and bc, in a proportion are equal.
You can write proportions that involve a variable and then use cross products
to solve the proportion.
EXAMPLES
x2
3
B Solve the proportion .
4
x
4
3
A Do the ratios and form a proportion?
4
3
3
4
4
3
3344
9 16
Since 9 16,
Set the cross products equal to each other.
3(x) 4(x 2)
3x 4x 8
Distribute.
3x 4x 4x 8 4x
1x 8
Check cross products.
False
3
4
4
and do not form a proportion.
3
1x
1
8
1
x 8
The solution is 8.
PRACTICE
Use cross products to determine whether each pair of ratios forms a proportion.
15 5
2. , 20 7
100
5
3. , 240 12
7.5 21
4. , 10 28
8
x
5. 5
35
a
6
6. 12
18
2.2
11
7. 6
y
5
7
8. p
8.4
20
2x
9. 35
7
1
22
10. 2
c5
12
1
11. v3
4
d1
8
12. 9
18
7 28
1. , 8 34
Solve each proportion.
13. Medicine Your doctor has prescribed two teaspoons of medicine to be
taken every six hours. How much medicine will you have taken in 4 days?
(Hint: Convert 4 days into hours.)
B
3.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
14. Standardized Test Practice Two out of every seven people at a particular
high school play in the band. If the school has 742 students, how many of
them are in the band?
A 106 students
B 212 students
C 371 students
D 1484 students
Answers: 1. no 2. no 3. yes 4. yes 5. 56 6. 4 7. 30 8. 6 9. 2 10. 49 11. 45 12. 17 13. 32 teaspoons 14. B
4.
© Glencoe/McGraw-Hill
29
CA Parent and Student Study Guide, Algebra 1
NAME
4-2
DATE
Similar Triangles (Pages 201–205)
Two figures are similar () if they have the same shape, but not necessarily
the same size.
Similar
Triangles
• If the corresponding angles of two triangles have equal measures, the triangles are
similar. The sides opposite the corresponding angles are corresponding sides.
• If two triangles are similar, the measures of their corresponding sides are proportional,
and the measures of their corresponding angles are equal.
EXAMPLES
A Determine whether the
pair of triangles shown
at the right are similar.
B In the figure below, ABC ADE.
Find the value of x.
A
A
63
B
D
C
2
Write a proportion
matching corresponding
sides of each triangle.
Two triangles are similar
27
if the measures of their
corresponding angles are equal.
mC 180° (90° 63°)
E
F
27°
mF 180° (90° 27°)
63°
Since corresponding angles have equal measures,
triangle ABC is similar to triangle FED, or
ABC FED.
BC
DE
x
8
C
x
D
AC
AE
3
E
8
2
23
(2 3)(x) 8(2)
5x 16
5x
5
B
Find the cross products.
16
5
x 3.2
PRACTICE
Determine whether each pair of triangles is similar.
1.
59
2.
59
64
27
Triangle PQR is similar to triangle XYZ. For each set of
measures given, find the measures of the remaining sides.
P
q
r
Q
3. p 4, q 3.5, r 3, x 8
X
4. p 5, q 5, r 2, z 3
Y
6. x 22.5, y 18, z 15, r 10
B
C
C
A
B
5.
C
B
6.
A
7.
8.
Z
x
B
A
7. Standardized Test Practice The triangles in the figure at the
right are similar. Find the value of x.
A 24 cm
B 48 cm
C 57.6 cm
D 67.6 cm
x
12 cm
62.4 cm
13 cm
5 cm
24 cm
Answers: 1. no 2. yes 3. y 7, z 6 4. x 7.5, y 7.5 5. p 10, r 8 6. p 15, q 12 7. C
4.
y
z
5. x 20, y 18, z 16, q 9
3.
R
p
© Glencoe/McGraw-Hill
30
CA Parent and Student Study Guide, Algebra 1
NAME
4-3
DATE
Trigonometric Ratios (Pages 206–214)
In a right triangle, the side opposite the right angle is the longest side. This
side is called the hypotenuse. The other two sides are the legs.
measure of leg opposite A
measure of hypotenuse
measure of leg adjacent A
cosine of A measure of hypotenuse
measure of leg opposite A
tangent of A measure of leg adjacent A
sine of A Definition of
Trigonometric
Ratios
a
c
b
cos A c
a
tan A b
sin A B
c
A
a
C
b
EXAMPLES
A Find the sine, cosine, and tangent of angle Q.
opposite leg
hypotenuse
sin Q cos Q 9
or 0.6
15
adjacent leg
hypotenuse
P
tan Q opposite leg
adjacent leg
9
or 0.75
12
12
or 0.8
15
15
9
R
Q
12
B Find the measure of angle P, mP, to the nearest degree.
opposite leg
sin P hypotenuse
12
⇒ sin P or 0.8
15
Use a scientific calculator to find the angle measure with a sine of 0.8.
Enter: 0.8 2nd SIN –1
Result: 53.13010235
So, mP 53°.
PRACTICE
For each triangle, find sin C, cos C, and tan C to the nearest thousandth. Use
a calculator to find the value of each trigonometric ratio to the nearest ten
thousandth if necessary.
1. A
2.
5
3
B
3.
B
8
C
4
4. sin 14°
A
15
A
10
24
17
5. cos 68°
C
26
B
C
6. tan 80°
7. cos 60°
8. sin 85°
Use a calculator to find the measure of each angle to the nearest degree.
B
14. tan K 0.2675
C
15. Standardized Test Practice Which equation can be used to find
the measure of the angle under the seesaw?
48
A sin (x°) 34
48
B cos (x°) 34
34
48
34
48
C sin (x°) D tan (x°) 34 in.
x
48 in.
8
15
8
12
5
12
© Glencoe/McGraw-Hill
31
3
B
A
4
B
8.
13. sin P 0.9052
C
B
A
7.
12. cos Y 0.7071
C
A
5.
6.
11. tan W 0.2309
3
4.
10. cos M 0.7660
Answers: 1. sin C ; cos C ; tan C 2. sin C ; cos C ; tan C 3. sin C ; cos C ; tan C 5
5
4
17
17
15
13
13
5
4. 0.2419 5. 0.3746 6. 5.6713 7. 0.5 8. 0.9962 9. 55° 10. 40° 11. 13° 12. 45° 13. 65° 14. 15° 15. C
3.
9. sin B 0.8192
CA Parent and Student Study Guide, Algebra 1
NAME
4-4
DATE
Percents (Pages 215–221)
A percent is a ratio that compares a number to 100. Percent means per
hundred, or hundredths. You can express a percent with the percent symbol
(%), as a fraction, or as a decimal.
Percent Proportion
Percentage
Base
Percentage
Base
Rate or
r
100
You can also write equations to solve problems with percents. To do this,
translate percents as decimals, the word “of” as multiplication, the phrase
“what number” as x, and the word “is” as an equal sign. Then, solve for x.
EXAMPLES
3
A Write as a percent and then as a
8
decimal.
3
8
n
100
300 8n
37.5 n
So
3
8
is equal to
B 20 is what percent of 77?
Use the percent proportion and solve for r. The
amount following the word “of” is usually the base.
Use a proportion.
Percentage
Base
Find cross products
Divide each side by 8.
37.5
100
20
77
Rate
Rate
0.26 Rate
or 37.5%.
Decimal → Percent: Move decimal point right two
places and add % symbol.
Rate 0.26 or about 26%
Percent → Decimal: Move decimal point left two
places and drop % symbol.
37.5% 0.375
PRACTICE
Write each ratio as a percent and then as a decimal.
5
1. 50
9
2. 24
35
3. 40
12
5. 10
3
4. 10
Use a proportion or an equation to answer each question.
B
11. What percent of 40 is 100?
12. 30 is 75% of what number?
13. 24 is 120% of what number?
C
B
8.
10. What is 50% of 48?
C
B
A
7.
9. $5.60 is what percent of $8.00?
C
A
5.
6.
8. What percent of 75 is 50?
B
A
14. Standardized Test Practice How much will you save on a stereo that is
marked down 30% if it is normally priced at $299.99?
A $210.00
B $90.00
C $9.00
D $8.99
2
4.
7. Ten is what percent of 5?
Answers: 1. 10%, 0.1 2. 37.5%, 0.375 3. 87.5%, 0.875 4. 30%, 0.3 5. 120%, 1.2 6. 40% 7. 200% 8. 66 % 9. 70%
3
10. 24 11. 250% 12. 40 13. 20 14. B
3.
6. Twelve is what percent of 30?
© Glencoe/McGraw-Hill
32
CA Parent and Student Study Guide, Algebra 1
NAME
4-5
Finding
Percent
of Change
DATE
Percent of Change (Pages 222–227)
percent of change amount of change
original amount
amount of change original amount new amount
percent of decrease ⇒ new amount is less than original amount
percent of increase ⇒ new amount is more than original amount
EXAMPLES
A Find the percent of change if the original
price of an item is $56 and the new price
$32. Is this change a percent of increase
or decrease?
B A book with an original price of $15 is on
sale at a discount of 25%. If the sales tax
is 10%, what is the final price of the
book?
amount of change: 56 32 or 24
amount of change
original amount
24
56
Discount 25% of original price
0.25 15 or $3.75
Sale price $15 $3.75 or $11.25
Tax 10% of sale price
0.10 $11.25 or $1.13
Final $11.25 $1.13
or about 0.43
The percent of change is 43%.
Since the new amount is less than the original
amount, 32 56, this is a percent of decrease.
$12.38
Try This Together
1. original: 500 tons
new: 640 tons
Is this change a percent of increase or decrease? Find the percent of change.
HINT: Subtract to find the amount of change.
PRACTICE
State whether each percent of change is a percent of increase or a percent
of decrease. Then find the percent of increase or decrease. Round to the
nearest whole percent.
2. original: 12 cm
new: 30 cm
3. original: 40 mph
new: 70 mph
4. original: $14.99
new: $8.99
5. original: 100 lb
new: 120 lb
6. original: 50¢
new: 69¢
7. original: 16 oz
new: 20 oz
Find the final price of each item.
4.
C
C
A
B
5.
C
B
6.
A
7.
8.
B
A
8. printer: $101.98
discount: 15%
11.
9. notebook: $1.49
sales tax: 7.5%
10. gum: $0.45
sales tax: 8%
Standardized Test Practice All shirts at a store are reduced by 40%. If
Answers: 1. increase; 28% 2. increase; 150% 3. increase; 75% 4. decrease; 40% 5. increase; 20% 6. increase; 38%
7. increase; 25% 8. $86.68 9. $1.60 10. $0.49 11. C
B
3.
© Glencoe/McGraw-Hill
33
CA Parent and Student Study Guide, Algebra 1
NAME
4-6
DATE
sales tax is 8.5%, find the final price of a
shirt that normally costs $18.
A $7.20
B $10.80
C $11.72
You can calculate the chance, or probability, that a particular event will
happen by finding the ratio of the number of ways the event can occur to the
number of possible outcomes. The probability of an event may be written as a
fraction, decimal, or percent. When outcomes have an equal chance of
occurring, they are equally likely. When an outcome is chosen without any
preference, the outcome occurs at random.
Definition of
Probability
probability of an event or P(event) Definition
of Odds
number of ways the number of ways the
odds of an event event can occur : event cannot occur
successes : failures
number of favorable outcomes
total number of possible outcomes
EXAMPLES
A Find the probability of randomly
choosing the letter p in the word “apple.”
B Find the odds of randomly selecting the
letter p in the word “Mississippi.”
There are 2 p’s and 5 letters in all.
P(choosing a p) There are 11 letters in the word. Two letters are p’s
and 11 2 or 9 letters are not p’s.
Odds of selecting a p
number of p’s : number not p’s
2:9 2:9 is read “2 to 9.”
2
5
2
The probability is , 0.4, or 40%.
5
Try These Together
1. What is the probability of rolling a 1
or a 2 using a 6-sided number cube?
2. From a group of 125 boys and 150 girls, what
are the odds of randomly selecting a girl?
HINT: The number of favorable outcomes is 2.
HINT: Remember to simplify your ratio.
PRACTICE
Determine the probability of each event.
3. You toss a coin and get heads.
4. A person was born on a weekday.
Find the probability of each outcome if a computer randomly chooses a letter
in the word “mathematical.”
5. the letter t
6. the letter a or c
7. the letter d
8. not an m
Find the odds of each outcome if a computer randomly chooses a letter in
the word “Alabama.”
9. the letter a
B
C
C
B
C
13. Standardized Test Practice What are the odds of randomly selecting a dime
from a dish containing 11 pennies, 6 nickels, 5 dimes, and 3 quarters?
C 1:4
9. 4:3 10. 1:6 11. 3:4 12. 7:0 13. C
34
5
7. 0 8. 6
© Glencoe/McGraw-Hill
D 4:1
1
B 1:5
6. 3
A 5:1
1
B
A
5. 6
8.
5
B
A
7.
4. 7
A
5.
6.
12. not a g
1
2. 6:5 3. 2
4.
11. a consonant
1
Answers: 1. 3
3.
10. the letter b
CA Parent and Student Study Guide, Algebra 1
NAME
4-7
DATE
15.0
S
T
A
N
D
A
R
D
S
Weighted Averages (Pages 233–238)
Sometimes the numbers that go into an average do not all have the same
weight or importance. In such cases, you may want to use a weighted
average. Two applications of weighted averages are mixture problems and
problems involving uniform motion, or motion at a constant rate or speed.
The formula distance rate time, or d rt is used to solve uniform motion
problems.
EXAMPLE
How much pure juice and 20% juice should you mix to make 4 quarts
of 50% juice?
Let p the amount of pure juice to be
added. Then, make a table of the information.
Quarts
Next, write an equation with the expression
for each amount of juice.
Pure juice (100%)
20% juice
4p
p
pure juice 20% juice 50% juice
p 0.2(4 p) 2
p 0.8 0.2p 2
(1 0.2)p 0.8 2
0.8p 0.8 2
0.8p 1.2
p 1.5
50% juice
4
Amount of Juice
100% of p 1 p or p
20% of 4 p 0.2(4 p)
50% of 4 0.5 4 or 2
You should mix 1.5 quarts of pure juice with 4 1.5 or 2.5 quarts of
20% juice to obtain a 4 quart mixture that is 50% juice.
PRACTICE
1. Entertainment Symphony tickets cost
$16 for adults and $8 for students. A total
of 634 tickets worth $8432 were sold. Use
the table to find how many adult and
student tickets were sold.
Number
Sold
Adult Tickets
x
Student Tickets 634 x
2. Transportation A truck and a jeep leave Melbourne,
the truck heading east and the jeep heading west.
The jeep is traveling 5 mph slower than the truck.
In 3 hours, the vehicles are 465 miles apart. Draw a
diagram of the situation and then use the table to find
the speed of each vehicle. (Hint: eastbound distance westbound distance total distance apart.)
B
C
C
A
B
5.
C
B
6.
A
7.
8.
Rate
Time Distance
(mph) (hours) (miles)
Truck
x
3
Jeep
3
B
A
3. Standardized Test Practice A group of twenty people bought popcorn at a
movie. A regular popcorn cost $2 and a large popcorn cost $3. If the total
bill for popcorn was $49, how many bags of each size did they buy?
A 5 regular, 15 large
B 12 regular, 8 large
C 11 regular, 9 large
D 7 regular, 13 large
2. See Answer Key for diagram; truck: 80 mph, jeep: 75 mph 3. C
4.
Total
Price
© Glencoe/McGraw-Hill
35
Answers: 1. 420 adult; 214 student
3.
Price Per
Ticket
CA Parent and Student Study Guide, Algebra 1
NAME
4-8
Direct
Variation
DATE
Direct and Inverse Variation (Pages 239–244)
A direct variation is described by an equation of the form y kx, where k 0. In this
equation, k is called the constant of variation. In a direct variation, as x increases in value,
y increases in value.
Direct proportion:
Inverse
Variation
x1
x2
y1
y2
An inverse variation is described by an equation of the form xy k, where k 0. In an
indirect variation, as x increases in value, y decreases in value.
Inverse proportion:
x1
x2
y2
y1
EXAMPLES
18
A Does c represent an inverse or a
d
B If y 4 when x 6, and y varies directly
as x, find y when x 9.
direct variation? What is the constant of
variation in this equation?
As d increases, the value of c will decrease, therefore
the equation represents an inverse variation. The
constant of variation is 18.
y
x1
x2
1
y
direct proportion
6
9
4
y
x1 6, y1 4, and x2 9
2
2
6y2 36
y2 6
Find the cross products.
Divide each side by 6.
PRACTICE
Determine which equations represent inverse variations and which represent
direct variations. Then find the constant of variation.
8
1. a b
y
1
3. x 7y
2. 9 x
4. d 65t
Solve. Assume that y varies directly as x.
5. If y 8 when x 5,
find x when y 64.
6. If y 14 when x 84,
find x when y 2.
7. If y 15 when x 27,
find y when x 9.
8. If y 3 when x 4,
find y when x 52.
Solve. Assume y varies inversely as x.
B
C
B
C
8.
B
A
13. Standardized Test Practice The amount an employee earns varies directly
as the number of hours she works. If she gets paid $58.80 for 8 hours of
work, how much will she get paid for 15 hours of work?
A $110.25
B $112.50
C $117.60
4. direct; 65 5. 40 6. 12 7. 5 8. 39 9. 6 10. 4 11. 27 12. 3
B
A
7.
12. If y 21 when x 4,
find y when x 28.
C
A
5.
6.
11. If y 9 when x 6,
find y when x 2.
© Glencoe/McGraw-Hill
36
D $120.00
1
4.
10. If y 8 when x 5,
find x when y 10.
Answers: 1. inverse; 8 2. direct; 9 3. inverse; 7
13. A
3.
9. If y 3 when x 14,
find x when y 7.
CA Parent and Student Study Guide, Algebra 1
NAME
4
DATE
Chapter 4 Review
Age Ratios and Proportions
You may have read in a newspaper that the population of the United States
is getting older. This means that the ratio of older people to younger people is
increasing. How about in your neighborhood? What is the ratio of adults to
children in your neighborhood or community?
1. Pick three public places in your community. Parks, shopping malls, or
street corners are good examples. Make sure you pick a place that is not
going to have more than the usual number of children or adults. With a
parent, stay at each of the three locations for about 30 minutes and count
the number of children and adults you see. Consider children to be anyone
under the age of 18 and adults to be anyone age 18 or older. Record the
information in the table below. Then find the ratio of children to the total
number of people and the ratio of adults to the total number of people
observed at each location.
Location 1
Location 2
Location 3
Children
Adults
Total
Ratio of children to total
Ratio of adults to total
2. Suppose you go back to the three locations on another day and observe
50 people at each location. Use the ratios found in Exercise 1 to estimate
how many of these people you would expect to be adults and how many
you would expect to be children? Use proportions and round answers to
the nearest whole number.
Location 1
Location 2
Location 3
Number of children out of 50
Number of adults out of 50
3. Based on the information you gathered from your three locations, what
percent of the people living in your community are children? What percent
of the people living in your community are adults?
Answers are located on page 107.
© Glencoe/McGraw-Hill
37
CA Parent and Student Study Guide, Algebra 1