Transcript Course 3

Ratios and Rates
LESSON 4-1
Course 3
Problem of the Day
How many miles are in 21,120 yd? (Hint: 1 mi = 5,280 ft)
12 mi
Lesson
Main
Lesson
4-1
Feature
Ratios and Rates
LESSON 4-1
Course 3
Check Skills You’ll Need
(For help, go to Lesson 2-3.)
1. Vocabulary Review What is the least common
denominator of two rational numbers?
Determine which rational number is greater.
2. 3 , 1
9
6
3. 15 , 4
25 5
4. 45 , 2
54 3
5. 4 , 7
7 12
Check Skills You’ll Need
Lesson
Main
Lesson
4-1
Feature
Ratios and Rates
LESSON 4-1
Course 3
Check Skills You’ll Need
Solutions
1. The least common denominator is the smallest multiple the
denominators have in common.
2. 3
3. 4
9
4. 45
5
Lesson
Main
54
Lesson
4-1
5. 7
12
Feature
Ratios and Rates
LESSON 4-1
Course 3
Additional Examples
Write the ratio 36 seconds to 12 minutes in
simplest form.
36 s
36 s
=
12 min
720 s
36
36 ÷ 36
=
720
720 ÷ 36
=
1
20
Convert minutes to seconds so that both
measures are in the same units.
Divide the common units.
Divide the numerator and denominator
by the GCF, 36.
Simplify.
1
The ratio of 36 seconds: 12 minutes is 20 .
Lesson
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Lesson
4-1
Quick Check
Feature
Ratios and Rates
LESSON 4-1
Course 3
Additional Examples
Computer time costs $4.50 for 30 min. What is the
unit rate?
cost
$4.50
=
number of minutes
30 min
Write a rate comparing cost to minutes.
Divide.
= $.15/min
The unit rate is $.15 per minute.
Quick Check
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Lesson
4-1
Feature
Ratios and Rates
LESSON 4-1
Course 3
Additional Examples
Keneesha drove her car 267 mi using 11 gal of gas. Vanessa
drove her car 210 mi using 9 gal. Give the unit rate for each. Which car
got more miles per gallon of gas?
Keneesha
miles
267 mi
=
gallons
11 gal
24.27272727 mi/gal
24.3 mi/gal
Write the rates
comparing
miles to gallons.
Divide.
Round to the
nearest tenth.
Vanessa
miles
210 mi
=
gallons
9 gal
23.33333333 mi/gal
23.3 mi/gal
Keneesha’s car got more miles per gallon.
Lesson
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Lesson
4-1
Feature
Ratios and Rates
LESSON 4-1
Course 3
Additional Examples
(continued)
Check for Reasonableness 24.3 • 11 = 267.3 and 267.3 267.
Also, 23.3 • 9 = 209.7 and 209.7 210. The answers are reasonable.
Quick Check
Lesson
Main
Lesson
4-1
Feature
Ratios and Rates
LESSON 4-1
Course 3
Lesson Quiz
Express each ratio in simplest form.
1. 27 laps : 81 minutes
1
3
2. 12 minutes : 3 hours
1
15
3. Carli walked 16 miles in 5 hours. Find the unit rate.
3.2 mi/h
4. A 21-oz bottle of shampoo costs $2.80. A 12-oz bottle costs $1.35.
Which has the better unit rate?
12-oz bottle
Lesson
Main
Lesson
4-1
Feature
Converting Units
LESSON 4-2
Course 3
Problem of the Day
Write a fraction in lowest terms, with a single digit numerator, that is about
the same as each decimal: 0.52, 0.13, 0.74, 0.88.
1 1 3 7
2 , 8 , 4, 8
Lesson
Main
Lesson
4-2
Feature
Converting Units
LESSON 4-2
Course 3
Check Skills You’ll Need
(For help, go to Lesson 2-5.)
1. Vocabulary Review What is the product of a number
and its reciprocal?
Find each product. Write the answer in simplest form.
2. 10 • 1
4
3. 4 • 5
6
6
4. 4 • 3
9
2
5. 6 • 8
7
3
3
Check Skills You’ll Need
Lesson
Main
Lesson
4-2
Feature
Converting Units
LESSON 4-2
Course 3
Check Skills You’ll Need
Solutions
5
1. 1
2. 10 • 1 = 10 • 1 = 5
3•4
3 • 42
6
2
3. 4 • 5 = 4 • 5 = 10 = 5
6•6
6 • 6 3 18
9
2
1
4. 4 • 3 = 4 • 3 = 2
9 • 2 3 9 • 21
3
2
5. 6 • 8 = 6 • 8 = 16 = 2 2
7•3
7 • 31
7
7
Lesson
Main
Lesson
4-2
Feature
Converting Units
LESSON 4-2
Course 3
Additional Examples
Convert 0.7 mi to ft.
Since 5,280 ft = 1 mi, use the conversion factor 5,280 ft.
1 mi
0.7 =
Multiply by a conversion
factor 5,280 ft .
0.7 mi
5,280 ft
•
1
1 mi
1 mi
(0.7)(5,280) ft
=
1
Simplify.
= 3,696 ft
Divide.
There are 3,696 feet in 0.7 miles.
Quick Check
Lesson
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Lesson
4-2
Feature
Converting Units
LESSON 4-2
Course 3
Additional Examples
Quick Check
A rowing team completed a 2000-m course at a
rate of 6.84 m/s. Convert this rate to kilometers per minute.
Estimate 6.84
7. Then, 7 • 60 ÷ 1000 = 0.42.
6.84 m
6.84 m
1 km
=
•
•
1s
1s
1000 m
60 s
1 min
(6.48)(1)(60) km
Multiply by two ratios that
each equal one.
Divide by the common units.
= (1)(1,000)(1) min
Simplify.
= 0.4104
Use a calculator.
The team rowed at a rate of 0.4104 km/min.
Check for Reasonableness The answer 0.4104 km/min is close to the
estimate 0.42. The answer is reasonable.
Lesson
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Lesson
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Feature
Converting Units
LESSON 4-2
Course 3
Additional Examples
Use compatible numbers to estimate the number
of gallons in 33 quarts.
1 gal
The conversion factor for changing gallons to quarts is 4 qt .
Round to the nearest number
33 qt 32 qt
divisible by 4.
=
32 qt
1 gal
•
1
4 qt
Multiply by the conversion factor.
Divide by the common units.
32
= 4 gallons
Simplify.
= 8 gallons
Divide.
There are about 8 gallons in 33 quarts.
Lesson
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Lesson
4-2
Quick Check
Feature
Converting Units
LESSON 4-2
Course 3
Additional Examples
Convert 650 g to ounces.
Multiply by the conversion
650 g
1 oz
•
1
28.4 g
650 g =
1 oz
factor 28.4 g .
=
(650)(1) oz
28.4
 22.9 oz
Simplify. Divide using
a calculator.
There are about 22.9 oz in 650 g.
Quick Check
Lesson
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Lesson
4-2
Feature
Converting Units
LESSON 4-2
Course 3
Lesson Quiz
1.
Convert 0.75 hours to seconds.
2,700 seconds
2.
$150 per hour is how much per minute?
$2.50 per min
3.
69.2 cm is about how many meters?
0.7 m
4.
Convert 12 qt to liters.
about 11.3L
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Lesson
4-2
Feature
Solving Proportions
LESSON 4-3
Course 3
Problem of the Day
Write each word phrase as an algebraic expression.
a. 12 times a number 12n
b. 8 less than a number n – 8
c. twice the sum of 5 and a number 2(5 + n)
Lesson
Main
Lesson
4-3
Feature
Solving Proportions
LESSON 4-3
Course 3
Check Skills You’ll Need
(For help, go to Lesson 2-2.)
1. Vocabulary Review Is the fraction a + 2 in
b+2
simplest form? Explain.
Write each fraction in simplest form.
2. 30
99
3. 42
4. 132
12
602
5. 70
25
Check Skills You’ll Need
Lesson
Main
Lesson
4-3
Feature
Solving Proportions
LESSON 4-3
Course 3
Check Skills You’ll Need
Solutions
1. Yes; there is no common factor between the
numerator and denominator.
1
30 1 3 • 10
10
2.
=
=
99 1 3 • 33
33
3. 42 = 6 • 7 = 7 = 3 1
12 1 6 • 2
2
2
132 1 2 • 66
66
4.
=
=
602 1 2 • 301 301
70
5 • 14
14
4
5. 25 = 5 • 5 = 5 = 2 5
1
Lesson
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1
Lesson
4-3
Feature
Solving Proportions
LESSON 4-3
Course 3
Additional Examples
8
Do 4 and
form a proportion? Explain.
18
9
4
9
8 gallons
18
Write as a proportion.
gallons
Use number sense to find a
common multiplier.
Since 4 = 8 ,they form a proportion.
9
18
Quick Check
Lesson
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Lesson
4-3
Feature
Solving Proportions
LESSON 4-3
Course 3
Additional Examples
The fixed rate of conversion is 1 euro = 0.7876 Irish pounds.
How many euros would you receive for 125 Irish pounds?
Let p = the number of euros.
0.7876
125
= p
1
Write the Irish pounds
euros .
proportion
0.7876 • p = 1 • 125
Write the cross products.
0.7876 • p
125
=
0.7876
0.7876
Divide each side by 0.7876.
125
Use a calculator.
0.7876
You would receive 158.71 euros.
Quick Check
Lesson
Main
Lesson
4-3
Feature
Solving Proportions
LESSON 4-3
Course 3
Lesson Quiz
1. Is 5 proportional to 10 ? Explain.
8
24
No; the fractions are not equal.
Solve each proportion.
w
2. 12 = 3
4
9
3.
4
20
=
5
r
25
4. Suppose the exchange rate for dollars to Indian rupees is 0.02. How
many rupees should you receive for $100?
5,000 rupees
Lesson
Main
Lesson
4-3
Feature
Similar Figures and Proportions
LESSON 4-4
Course 3
Problem of the Day
A football team scored 38 points in a game. They scored 3 points for a field
goal and 7 points for each touchdown with an extra point. How many field
goals did they make? How many touchdowns?
1 field goal and 5 touchdowns or 2 touchdowns and 8 field goals
Lesson
Main
Lesson
4-4
Feature
Similar Figures and Proportions
LESSON 4-4
Course 3
Check Skills You’ll Need
(For help, go to Lesson 4-3.)
1.
Vocabulary Review What are the cross products for
10
2
= ?
15
3
Solve each proportion.
2.
21
7
=
t
13
k
22
3. 50 =
10
16
324
4. 25 =
m
Check Skills You’ll Need
Lesson
Main
Lesson
4-4
Feature
Similar Figures and Proportions
LESSON 4-4
Course 3
Check Skills You’ll Need
Solutions
7t = 273
10k = 1,100
7t
273
=
7
7
10k
1,100
=
10
10
t = 39
k = 110
4. 16m = 8,100; m = 506 1
4
Lesson
Main
Lesson
4-4
Feature
Similar Figures and Proportions
LESSON 4-4
Course 3
Additional Examples
Is rectangle ABCD similar to rectangle RSTU? Explain why or
why not.
First, check to see if corresponding angles are congruent.
A
R
B
S
All right angles are 90°.
C
T
Lesson
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D
U
Lesson
4-4
Feature
Similar Figures and Proportions
LESSON 4-4
Course 3
Additional Examples
(continued)
Next, check to see if corresponding sides are in proportion.
AB
RS
6
48
6 • 24
DA
UR
3
24
AB corresponds to RS. DA corresponds to UR.
48 • 3
Write the cross products.
144 = 144
Substitute.
Simplify.
The corresponding sides are in proportion, so rectangle ABCD
is similar to rectangle RSTU.
Quick Check
Lesson
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Lesson
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Feature
Similar Figures and Proportions
LESSON 4-4
Course 3
Additional Examples
A stonemason’s sketch of a carving to be made
on a building includes the letter “E” shown below. If the
width of the actual letter in the arrangement is 22 in.,
what is the height?
22 in.
2.75 in.
=
x
5 in.
2.75 • x = 5 • 22
Set up a proportion.
Write the cross products.
2.75 x = 110
Simplify.
2.75x
110
=
2.75
2.75
Divide each side by 2.75.
x = 40
Simplify.
The height of the letter is 40 inches.
Quick Check
Lesson
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Lesson
4-4
Feature
Similar Figures and Proportions
LESSON 4-4
Course 3
Additional Examples
RST ~
14
12
=
d
21
12 • d = 21 •14
12d = 294
12d 294
=
12
12
d = 24.5
PSU. Find the value of d.
Write a proportion.
Write the cross products.
Simplify.
Divide each side by 12.
Simplify.
The value of d is 24.5.
Quick Check
Lesson
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Lesson
4-4
Feature
Similar Figures and Proportions
LESSON 4-4
Course 3
Lesson Quiz
1. Are the triangles similar? Explain.
No; their sides are not proportional.
2. A model of a building is 18 in. tall and 24 in. wide. The building is 30 ft
tall. How wide is the building?
40 ft
Lesson
Main
Lesson
4-4
Feature
Similar Figures and Proportions
LESSON 4-4
Course 3
Lesson Quiz
3. In the figure at the right,
 MNO ~  LNP.
Find the value of a.
18
4. If all the lengths in Exercise 3 are doubled, are the
triangles still similar? Explain why or why not.
Yes; corresponding values are multiplied by the same factor.
Lesson
Main
Lesson
4-4
Feature
Similarity Transformations
LESSON 4-5
Course 3
Problem of the Day
There are three different 1-digit numbers greater than zero and all odd. Their
sum is 15. What are the numbers?
3, 5, 7 or 1, 5, 9
Lesson
Main
Lesson
4-5
Feature
Similarity Transformations
LESSON 4-5
Course 3
Check Skills You’ll Need
(For help, go to Lesson 3-4.)
1. Vocabulary Review The first coordinate in an ordered
pair is the ? -coordinate.
Graph each point on a coordinate plane.
2. A(3, 6)
3. B(–2, 7)
4. C(5, –1)
5. D(–3, 0)
Check Skills You’ll Need
Lesson
Main
Lesson
4-5
Feature
Similarity Transformations
LESSON 4-5
Course 3
Check Skills You’ll Need
Solutions
1. x
Lesson
Main
2-5.
Lesson
4-5
Feature
Similarity Transformations
LESSON 4-5
Course 3
Additional Examples
Quick Check
Find the image of
scale factor of 3.
ABC after a dilation with center A and a
A C is 3 times AC.
Since A is the
center of dilation
A=A.
A=A
A B C is the image of ABC after
a dilation with a scale factor of 3.
ABC ~ A B C
Lesson
Main
Lesson
4-5
A B is 3 times AB.
Feature
Similarity Transformations
LESSON 4-5
Course 3
Additional Examples
Quick Check
Find the coordinates of the image of quadrilateral KLMN after
1
a dilation with a scale factor of . Quadrilateral KLMN has vertices
2
K (–2, –1), L (0, 2), M (4, 2), and N (4, –1).
Step 1 Multiply the x- and
y-coordinates of each point by 1 .
Step 2 Graph the image.
2
K (–2, –1)
L (0, 2)
M (4, 2)
N (4, –1)
Lesson
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K (–1, – 1 )
2
L (0, 1)
M (2, 1)
N (2, – 1 )
2
Lesson
4-5
Feature
Similarity Transformations
LESSON 4-5
Course 3
Additional Examples
The figure below PQR shows the outline of a
playing field. A city planner dilates the design to show
the area available for community youth to play sports.
Find the scale factor. Is it an enlargement or a reduction?
PQ
image
original
PQ
=
6
4
=
3
2
= 1.5
The scale factor is 1.5.
The dilation is an enlargement.
Quick Check
Lesson
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Lesson
4-5
Feature
Similarity Transformations
LESSON 4-5
Course 3
Lesson Quiz
ABC has coordinates A(0, 0), B(10, 0), and C(5, 5). Find the
coordinates of the image of ABC after a dilation with each scale factor.
1. 1
A (0, 0), B (2, 0), C (1, 1)
2. 4
A (0, 0), B (40, 0), C (20, 20)
5
3. Figure ABCD shows the outline of a porch. The figure
A′B′C′D′ is the outline of a table formed by dilating
ABCD. Find the scale factor. Is it an enlargement
or a reduction?
1
, reduction
3
Lesson
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Lesson
4-5
Feature
Scale Models and Maps
LESSON 4-6
Course 3
Problem of the Day
Mirror primes are pairs of prime numbers in which the digits are reversed,
such as 13 and 31. Find all the mirror primes less than 100.
13 and 31, 17 and 71, 37 and 73, and 79 and 97; 11 is its own mirror image.
Lesson
Main
Lesson
4-6
Feature
Scale Models and Maps
LESSON 4-6
Course 3
Check Skills You’ll Need
(For help, go to the Skills Handbook page 632.)
1. Vocabulary Review A product is the result of which
operation?
Multiply.
2. 4  3.2
3. 7.6  5.9
4. 1.8  22
5. 13  6.5
Check Skills You’ll Need
Lesson
Main
Lesson
4-6
Feature
Scale Models and Maps
LESSON 4-6
Course 3
Check Skills You’ll Need
Solutions
1. multiplication
2. 12.8
3. 44.84
4. 39.6
5. 84.5
Lesson
Main
Lesson
4-6
Feature
Scale Models and Maps
LESSON 4-6
Course 3
Additional Examples
On a blueprint, the cellar is 4 in. by 3 in. The scale is
1 in. = 8 ft. What are the length and width of the actual cellar?
2
First, find the actual length of the cellar.
Let
= the actual length of the cellar.
blueprint measure (in.)
actual measure (ft)
Lesson
Main
1
2
4
=
8
blueprint length (in.)
actual length (ft)
Lesson
4-6
Feature
Scale Models and Maps
LESSON 4-6
Course 3
Additional Examples
(continued)
1
•
2
1
2
1
2
1
2
=8•4
Write the cross
products.
= 32
Simplify.
=
32
1
2
= 64
Lesson
Main
Divide each side
1
by .
2
Simplify.
Lesson
4-6
Feature
Scale Models and Maps
LESSON 4-6
Course 3
Additional Examples
(continued)
The length of the actual room is 64 ft.
Next, find the actual width of the cellar.
Let w = the actual width of the cellar.
blueprint measure (in.)
actual measure (ft)
Lesson
Main
1
2
3
=
8
w
blueprint length (in.)
actual length (ft)
Lesson
4-6
Feature
Scale Models and Maps
LESSON 4-6
Course 3
Additional Examples
(continued)
1
• w=8•3
2
Write the cross
products.
1
w = 24
2
Simplify.
1w
24
2
=
1
1
2
2
Divide each side
1
by .
w = 48
2
Simplify.
The width of the actual room is 48 ft.
Lesson
Main
Lesson
4-6
Quick Check
Feature
Scale Models and Maps
LESSON 4-6
Course 3
Additional Examples
The map distance from El Paso, Texas, to Chihuahua,
Mexico, measures about 7.5 cm. The scale is 1 cm = 50 km. What is
the actual distance?
Let d be the actual distance from El Paso, Texas to Chihuahua,
Mexico.
map (cm)
actual (km)
1
7.5
= d
50
map (cm)
actual (km)
1 • d = 50 • 7.5
Set up a proportion.
Write the cross products.
d = 375
Simplify.
The actual distance from El Paso, Texas to Chihuahua, Mexico
is 375 kilometers.
Lesson
Main
Lesson
4-6
Quick Check
Feature
Scale Models and Maps
LESSON 4-6
Course 3
Lesson Quiz
1. A 6-ft man is designing a new chair that would make him feel like a 2.5-ft
child. The seat of a normal chair is 1.5 ft high. How high should he make
the seat in his new chair?
3.6 ft
2. A map scale shows 4 cm to represent 6 km. Two intersections measure
1 cm apart on the map. What is the actual distance?
1.5 km
Lesson
Main
Lesson
4-6
Feature
Scale Models and Maps
LESSON 4-6
Course 3
Lesson Quiz
For Exercises 3–4, use the diagram.
3. A tennis court is 36 ft wide. A drawing of the court is
2 1 in. long and 1 in. wide. Find the scale used.
4
1 in. = 36 ft
4. Find the actual length of the court.
81 ft
Lesson
Main
Lesson
4-6
Feature
Similarity and Indirect Measurement
LESSON 4-7
Course 3
Problem of the Day
A rectangular field is 120 yd long and 53 yd 1 ft wide. How much longer is the
field than it is wide?
66 yd 2 ft
Lesson
Main
Lesson
4-7
Feature
Similarity and Indirect Measurement
LESSON 4-7
Course 3
Check Skills You’ll Need
(For help, go to Lesson 4-4.)
1. Vocabulary Review Similar figures have the
same ? but not necessarily the same size.
2. If ABC ~ XYZ, which angle is congruent to B?
Check Skills You’ll Need
Lesson
Main
Lesson
4-7
Feature
Similarity and Indirect Measurement
LESSON 4-7
Course 3
Check Skills You’ll Need
Solutions
1. shape
2. Y
Lesson
Main
Lesson
4-7
Feature
Similarity and Indirect Measurement
LESSON 4-7
Course 3
Additional Examples
When a 6-ft student casts a 17-ft shadow, a flagpole casts a
shadow that is 51 ft long. Find the height of the flagpole.
Set up a proportion for the similar triangles.
Words
flagpole’s height
length of flagpole’s shadow
=
student’s height
length of student’s shadow
Let h = the flagpole’s height.
Proportion
17h = 6 • 51
17h
6 • 51
=
17
17
h = 18
h
6
51
17
=
Write the cross products.
Divide each side by 17.
Simplify.
Quick Check
The height of the flagpole is 18 ft.
Lesson
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Lesson
4-7
Feature
Similarity and Indirect Measurement
LESSON 4-7
Course 3
Additional Examples
In the figure below,
ABC ~
EDC. Find d.
Use similar triangles to set up a proportion involving the lengths
of corresponding sides.
ED
CD
=
AB
CB
ED corresponds to AB.
CD corresponds to CB.
d
141
=
416
312
Substitute.
312 • d = 416 • 141
Lesson
Main
Write the cross products.
Lesson
4-7
Feature
Similarity and Indirect Measurement
LESSON 4-7
Course 3
Additional Examples
(continued)
312d = 58,656
Simplify.
312d
58,656
=
312
312
Divide each side by 312.
58,656
312
Use a calculator.
188
The length d is 188 m.
Quick Check
Lesson
Main
Lesson
4-7
Feature
Similarity and Indirect Measurement
LESSON 4-7
Course 3
Lesson Quiz
1. A 5-ft tall student casts a 12-ft shadow. A tree casts a
27-ft shadow. How tall is the tree?
11.25 ft tall
2. A 6-ft man casts a 9-ft shadow. A sculpture casts a 45-ft shadow. How
tall is the sculpture?
30 ft
Lesson
Main
Lesson
4-7
Feature
Similarity and Indirect Measurement
LESSON 4-7
Course 3
Lesson Quiz
Use the diagram for Exercise 3. EFG ~ JHG
3. The diagram shows an outline of a village green EFG
next to a small park JHG. The length of JH is 47.4 m,
FG is 31 m, and HG is 15.8 m. Find the length of EF.
93 m
Lesson
Main
Lesson
4-7
Feature