Section A Understanding Ratios and Proportions
7-1
7-2
7-4
Ratios and Rates
Using Tables to Explore Ratios and Rates
Proportions
Section A Quiz
0
LESSON
7-1
Proportional Relationships
Ratios and Rates
Objective
To write ratios and rates and to find unit rates
Vocabulary
ratio a comparison of two quantities using division
Example 1
Writing Ratios
You can compare the different groups by using ratios. For example, you can compare the number of
violins with the number of violas. This ratio can be written in three ways.
29
12
Notice that the ratio of violins to violas,
29 to 12
29:12
29
12
, is different from the ratio of violas to violins,
. The
12
29
order of the terms is important.
Ratios can be written to compare a part to a part, a part to the whole, or the whole to a part.
Use the table above to write each ratio in three ways.
A. flutes to clarinets
5
or
5 to 4
or
5:4
4
B. trumpets to total instruments
3
or
3 to 95 or 3:95
95
C. total instruments to basses
95
or
95 to 9 or 95:9
9
part to part
part to whole
whole to part
1
equivalent ratios ratios that name the same comparison
You can find an equivalent ratio by multiplying or dividing both terms of a ratio by the same number.
Example 2
Writing Equivalent Ratios
Write three equivalent ratios to compare the number of diamonds to the number
of spades in the pattern.
number of diamonds
=
number of spades
3
6
6
12
1
2
rate a ratios that compares two quantities that have different units of measure
unit rate a rate in which the second quantity in the comparison is one unit
Example 3
Consumer Math Application
A 3-pack of paper towels costs $2.79. A 6-pack costs $5.46. Which is the better deal?
$2.79
3 rolls
Write the rate.
$5.46
6 rolls
Write the rate.
$2.79 3
3 rolls 3
Divide both terms by 3.
$5.46 6
6 rolls 6
Divide both terms by 6.
$0.93
1 roll
$0.93 for 1 roll.
$0.91
1 roll
$0.91 for 1 roll.
The 6-pack is the better deal.
Homework:
p. 288: 1-11, 12-20 even
2
Proportional Relationships
LESSON
7-2
Using Tables to Explore Equivalent Ratios and Rates
Objective
To use a table to find equivalent ratios and rates
Example 1
Making a Table to Find Equivalent Ratios
Use a table to find three equivalent ratios.
A.
8
3
Multiply the numerator and the
denominator by 2, 3, and 4.
original
ratio
8
3
8x2
↓
16
6
↑
3x2
8x3
↓
24
9
↑
3x3
8x4
↓
32
12
↑
3x4
8 16 24
32
The ratios , ,
,and
are equivalent.
5 6 9
12
B.
40 to 16
original
ratio
40
16
The ratios
C.
40 ÷ 2
↓
20
8
↑
16 ÷ 2
Divide the numerator and
denominator by 2, 4, and 8.
40 ÷ 4
↓
10
4
↑
16 ÷ 4
40 ÷ 8
↓
5
2
↑
16 ÷ 8
40 20 10
5
are equivalent.
, , , and
16 8 4
2
4:7
original
ratio
4
7
The ratios
4x2
↓
8
14
↑
7x2
4x3
↓
12
21
↑
7x3
4x4
↓
16
28
↑
7x4
4 8 12
16
are equivalent.
, , , and
7 14 21
28
3
Example 2
Entertainment Application
A. Several groups of friends are going to take a shuttle bus to the park. The table shows how much
the different groups will pay in all. Predict how much a group of 15 friends will pay.
Number in Group
6
12
18
Bus Fare ($)
12 24
36
12 < 15 < 18; therefore the group will pay between $24 and $36.
The ratio
6
3
is equivalent to , and 3 is a factor of 15.
12
6
5 × 3 = 15
Multiply the numerator and denominator by the same factor, 5.
5 × $6 = $30
A group of 15 friends would pay $30.
B. A group of 10 friends is in line to see a movie. The table shows how much different groups will
pay in all. Predict how much the group of 10 will pay.
Number in group
Amount Paid ($)
A.
1 person
$5
10 people
$50
3
15
5
25
or
B.
4
6
30
5 people
$25
12
60
10 people
$50
Homework:
p. 292: 1-9 odd, 10-22 even
LESSON
7-4
Proportional Relationships
Proportions
Objective
To write and solve proportions
Vocabulary
proportion an equation that shows two equivalent ratios
Example 1
Modeling Proportions
Write a proportion for the model.
First write the ratio of triangles to circles.
number of triangles
number of circles
4
2
Next separate the triangles and circles into two equal groups.
Now write the ratio of triangles to circles in each group.
number of triangles in each group
number of circles in each group
2
1
A proportion shown by the model is
4
2
2
.
1
5
Example 2
Using Cross Products to Complete Proportions
A. Find the missing value in the proportion
3
4
4×n
n
16
=
3
4
n
.
16
Find the cross product.
3 × 16
The cross products are equal.
4n = 48
Divide both sides by 4 to undo the multiplication.
n = 12
Find the missing value in each proportion.
B.
9n
9n
n
12
9
n
3
12 3
36
4
C.
t
5
28
20
D.
20t 5 28
20t 140
t 7
1
c
6
12
6c 12 1
6c 12
c 2
6
E.
6
7
30
b
6b 7 30
6b 210
b 35
Example 3
Measurement Application
The label from a bottle of pet vitamins shows recommended dosages.
What dosage would you give an adult dog that weighs 15 lb?
1 tsp
20 lb
=
20v
1 15
20v
15
v
15 lb
20v 15
20 20
v 0.75 tsp.
Homework:
p. 304: 1-12, 14-20 even, 22, 23
7