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Do NowFind the unit rate.
1. 18 miles in 3 hours
2. 6 apples for $3.30
3. 3 cans for $0.87
4. 5 CD’s for $43
Course 2
5-4 Identifying and Writing Proportions
6 mi/h
$0.55 per apple
$0.29 per can
$8.60 per CD
Hwk: p 45 & 46
EQ: How do I find equivalent ratios and to identify proportions and solve proportions by using cross products?
Course 2
5-4 Identifying and Writing Proportions
M7N1.b Compare and order rational numbers, including repeating decimals; M7N1.d Solve problems using rationalnumbers;
M7A2.a Given a problem, define a variable, write an equation, solve the equation, and interpret the solution;M7A2.b Use the addition and multiplication properties of equality to solve one- and two-step linear equations
Course 2
5-4 Identifying and Writing Proportions
An equation stating that two ratios are equivalent is called a proportion. The equation, or proportion, below states that the ratios and are equivalent. 10
62515
106
= 2515
Read the proportion by saying “ten is to six
as twenty-five is to fifteen.”
Reading Math
106
= 2515
Course 2
5-4 Identifying and Writing Proportions
If two ratios are equivalent, they are said to be proportional to each other, or in proportion.
Determine whether the ratios are proportional.
Additional Example 1A: Comparing Ratios in Simplest Forms
Course 2
5-4 Identifying and Writing Proportions
,2451
72128
72 ÷ 8128 ÷ 8
= 916
24 ÷ 351 ÷ 3
= 817
Simplify .2451
Simplify . 72128
817
Since = , the ratios are not proportional.9
16
Determine whether the ratios are proportional.
Additional Example 1B: Comparing Ratios in Simplest Forms
Course 2
5-4 Identifying and Writing Proportions
,150 105
9063
90 ÷ 9 63 ÷ 9
= 107
150 ÷ 15105 ÷ 15
= 10 7
107
Since = , the ratios are proportional.107
Determine whether the ratios are proportional.
Check It Out: Example 1A
Course 2
5-4 Identifying and Writing Proportions
,5463
72144
72 ÷ 72144 ÷ 72
= 1 2
54 ÷ 963 ÷ 9
= 67
67
Since = , the ratios are not proportional.12
Determine whether the ratios are proportional.
Check It Out: Example 1B
Course 2
5-4 Identifying and Writing Proportions
, 135 75
94
9 4
135 ÷ 15 75 ÷ 15
= 9 5
95
Since = , the ratios are not proportional.94
Directions for making 12 servings of rice call for 3 cups of rice and 6 cups of water. For 40 servings, the directions call for 10 cups of rice and 19 cups of water. Determine whether the ratios of rice to water are proportional for both servings of rice.
Additional Example 2: Comparing Ratios Using a Common Denominator
Course 2
5-4 Identifying and Writing Proportions
Write the ratios of rice to water for 12 servings and for 40 servings.
Ratio of rice to water, 12 servings:36
Ratio of rice to water, 40 servings: 1019
36
= 3 · 196 · 19
= 57114
1019
= 10 · 619 · 6
= 60114
57114
60114Since = , the two ratios are not proportional.
Servings of Rice
Cups of Rice
Cups of Water
12 3 6
40 10 19
Use the data in the table to determine whether the ratios of beans to water are proportional for both servings of beans.
Check It Out: Example 2
Course 2
5-4 Identifying and Writing Proportions
Write the ratios of beans to water for 8 servings and for 35 servings.
Ratio of beans to water, 8 servings:43
Ratio of beans to water, 35 servings: 13 9
43
= 4 · 93 · 9
= 3627
13 9
= 13 · 3 9 · 3
= 3927
3627
3927
Servings of Beans Cups of Beans Cups of Water
8 4 3
35 13 9
Since = , the two ratios are not proportional.
Course 2
5-4 Identifying and Writing Proportions
You can find an equivalent ratio by multiplying or dividing the numerator and the denominator of a ratio by the same number.
Additional Example 3: Finding Equivalent Ratios and Writing Proportions
Find a ratio equivalent to each ratio. Then use the ratios to find a proportion. Possible Answers:
Insert Lesson Title Here
Course 2
5-4 Identifying and Writing Proportions
A. 3535
= 3 · 25 · 2
610
=
35
= 610
B. 28162816
= 28 ÷ 416 ÷ 4
2816
= 74
= 74
Check It Out: Example 3
Find a ratio equivalent to each ratio. Then use the ratios to find a proportion. Possible Answers:
Insert Lesson Title Here
Course 2
5-4 Identifying and Writing Proportions
A. 2323
= 2 · 33 · 3
69
=
23
= 69
B. 16121612
= 16 ÷ 412 ÷ 4
1612
= 43
= 43
Course 2
5-5 Solving Proportions
For two ratios, the product of the numerator in one ratio and the denominator in the other is a cross product. If the cross products of the ratios are equal, then the ratios form a proportion.
5 · 6 = 30
2 · 15 = 30=2
56
15
Course 2
5-5 Solving Proportions
You can use the cross product rule to solve proportions with variables.
CROSS PRODUCT RULE
In the proportion = , the cross products,
a · d and b · c are equal.
ab
cd
Use cross products to solve the proportion.
Additional Example 1: Solving Proportions Using Cross Products
Course 2
5-5 Solving Proportions
15 · m = 9 · 5
15m = 45
15m15
= 4515
m = 3
= m5
915
Check It Out: Example 1
Insert Lesson Title Here
Course 2
5-5 Solving Proportions
Use cross products to solve the proportion.
7 · m = 6 · 14
7m = 84
7m7
= 847
m = 12
67
= m14
Course 2
5-5 Solving Proportions
When setting up a proportion to solve a problem, use a variable to represent the number you want to find. In proportions that include different units of measurement, either the units in the numerators must be the same and the units in the denominators must be the same or the units within each ratio must be the same.
16 mi4 hr
= 8 mix hr
16 mi8 mi
= 4 hrx hr
Additional Example 2: Problem Solving Application
Course 2
5-5 Solving Proportions
If 3 volumes of Jennifer’s encyclopedia takes up 4 inches of space on her shelf, how much space will she need for all 26 volumes?
Course 2
5-5 Solving Proportions
34
= 26x
3 · x = 4 · 26
3x = 104
3x3
= 1043
x = 34 23
She needs 34 23
inches for all 26 volumes.
Additional Example 2 Continued
Check It Out: Example 2
Course 2
5-5 Solving Proportions
John filled his new radiator with 6 pints of coolant, which is the 10 inch mark. How many pints of coolant would be needed to fill the radiator to the 25 inch level?
Quiz
Insert Lesson Title Here
Course 2
5-4 Identifying and Writing Proportions
Determine whether the ratios are proportional.
1. 930 ,
1240
2. 1221 ,
1015
Find a ratio equivalent to each ratio. Then use the ratios to write a proportion.
3. 38
4. 37
Quiz
Insert Lesson Title Here
Course 2
5-4 Identifying and Writing Proportions
5. In pre-school, there are 5 children for everyone teacher. In another preschool there are 20 children for every 4 teachers. Determine whether the ratios of children to teachers are proportional in both preschools.
20 4
5 1
=
Lesson Quiz: Part I
Insert Lesson Title Here
Course 2
5-5 Solving Proportions
Use cross products to solve the proportion.
6. 2520
= 45t
7. x9
= 1957
8. 23
= r36
9. n10
=288
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