1-4 Solving Proportions - Harmony Public Schools

1-4
Solving Proportions
TEKS FOCUS
VOCABULARY
TEKS (5)(A) Solve linear equations in one
variable, including those for which the
application of the distributive property
is necessary and for which variables are
included on both sides.
TEKS (1)(F) Analyze mathematical
relationships to connect and communicate
mathematical ideas.
Additional TEKS (1)(A)
ĚCross products – In the
a c
proportion b
= d , the products
ad and bc are called cross
products.
ĚProportion – A proportion is an
equation that states that two
ratios are equal. For example,
a c
= , where b ≠ 0 and d ≠ 0 is
b d
a proportion.
ĚRate – A rate is a ratio of a
to b where a and b represent
quantities measured in
different units.
ĚRatio – A ratio is the comparison
of two quantities by division.
ĚAnalyze – examine closely
objects, ideas, or relationships to
learn more about their nature
ESSENTIAL UNDERSTANDING
If two ratios are equal and a quantity in one of the ratios is unknown, you can write
and solve a proportion to find the unknown quantity.
Property Cross Products Property of a Proportion
Words The cross products of a proportion are equal.
Algebra If ab = dc , where b ≠ 0 and d ≠ 0, then ad = bc.
9
Example 34 = 12
, so 3(12) = 4(9), or 36 = 36.
You can use the Multiplication Property of Equality to prove the Cross
Products Property.
#
bd #
bd
a c
=
b d
a
= bd
b
a
= bd
b
#
#
Assume this equation is true.
c
d
c
d
Multiplication Property of Equality
Divide the common factors.
da = bc
Simplify.
ad = bc
Commutative Property of Multiplication
For this proportion, a and d are called the extremes of the proportion and b and c are
called the means. Notice that in the Cross Products Property the product of the means
equals the product of the extremes.
22
Lesson 1-4
Solving Proportions
Problem 1
How is this problem
related to problems
you’ve solved before?
It is similar to solving a
one-step equation using
multiplication. You can
simply multiply by 12 to
isolate m.
Solving a Proportion Using the Multiplication Property
m
What is the solution of the proportion 78 = 12
?
7 m
8 = 12
7
12 8 = 12
84
8 =m
#
# 12m
10.5 = m
Multiply each side by 12.
Simplify.
Divide.
Problem 2
Which property
should you use?
The Cross Products
Property can be easier to
use when the variable is
in the denominator.
TEKS Process Standard (1)(F)
Solving a Proportion Using the Cross Products Property
What is the solution of the proportion 43 = 8x ?
4 8
3=x
4x = 3(8)
Cross Products Property
4x = 24
Multiply.
x=6
Divide each side by 4 and simplify.
Problem 3
How is this proportion
different from others
you’ve seen?
This proportion looks
more complex, but the
Cross Products Property
is true for any proportion.
Treat each numerator as
a single variable when
you cross-multiply.
Solving a Multi-Step Proportion
8
b+3
What is the solution of the proportion b −
5 = 4 ?
b-8 b+3
5 = 4
4(b - 8) = 5(b + 3)
Cross Products Property
4b - 32 = 5b + 15
Distributive Property
4b - 32 - 4b = 5b + 15 - 4b
Subtract 4b from each side.
-32 = b + 15
Simplify.
-47 = b
Subtract 15 from each side and simplify.
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23
Problem 4
TEKS Process Standard (1)(A)
Using a Proportion to Solve a Problem
Music A portable media player has 2 gigabytes of storage and can hold about
500 songs. A similar but larger media player has 80 gigabytes of storage. About
how many songs can the larger media player hold?
The number of songs
the larger player
can hold
Ě6PDOOHUPHGLDSOD\HUKDV
2 gigabytes and can hold
500 songs.
Ě/DUJHUPHGLDSOD\HUKDV
80 gigabytes.
Is there only one
way to write a
proportion?
No. You can write other
proportions to solve the
problem. For example,
2 gigabytes
80 gigabytes
also works.
2 gigabytes 80 gigabytes
500 songs = s songs
500 songs
= s songs
Write a proportion to model the
situation. You can set up the
proportion so that the numerators
have the same units and the
denominators have the same
units. Then solve the proportion.
Write a proportion.
2s = 500(80)
Cross Products Property
2s = 40,000
Multiply.
s = 20,000
Divide each side by 2 and simplify.
NLINE
HO
ME
RK
O
The larger media player can hold 20,000 songs.
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Solve each proportion using the Multiplication Property of Equality.
q
For additional support when
completing your homework,
go to PearsonTEXAS.com.
-3
1. 8 = 45
3
x
2. 4 = 26
x
5. 16 = 12
9
k
6. 2 = 25
x
3
3. 4 = 5
x
1
7. 120 = 24
m
3
4. 7 = 5
h
2
8. 15
= 125
9. Use a Problem-Solving Model (1)(B) A gardener is transplanting flowers into a
flower bed. She has been working for an hour and has transplanted 14 flowers. She
has 35 more flowers to transplant. If she works at the same rate, how many more
hours will it take her?
10. Apply Mathematics (1)(A) A florist uses 2 dozen roses to make 5 centerpieces.
How many dozens of roses will he need to make 20 centerpieces?
11. Analyze Mathematical Relationships (1)(F) If 5 lb of pasta salad serves 14 people,
how much pasta salad should you take to a picnic with 49 people?
12. Apply Mathematics (1)(A) Approximately 3 people out of every 30 are left-handed.
About how many left-handed people would you expect in a group of 140 people?
24
Lesson 1-4
Solving Proportions
13. Maya runs 100 m in 13.4 s. Amy can run 100 m in 14.1 s. If Amy were to finish a
100-m race at the same time as Maya, how much of a head start, in meters, would
Amy need?
Solve each proportion using the Cross Products Property.
-9
5
14. b = 6
8
-3
3
15. p = 10
m
6
17. -25 = t
16. 4 = 22
Solve each proportion using any method.
18.
b+4 7
5 =4
2c
19. 11 =
c-3
4
20.
q + 2 2q - 11
7
5 =
c+1
21. c - 2 = 47
22. Analyze Mathematical Relationships (1)(F) The electric bill
for Ferguson’s Furniture is shown at the right. The cost of
electricity per kilowatt-hour and the total charges for one
month are given. To the nearest tenth, how many
kilowatt-hours of electricity did Ferguson’s Furniture
use in that month?
Centerville Electric
Account Name: Ferguson’s Furniture
Account Number: 34-14567-89
Cost per kilowatt-hour
Total charges
Previous balance
Total Amount Due
23. Apply Mathematics (1)(A) A particular computer takes
3 min to download a 45-min TV show. How long will it take
the computer to download a 2-hour movie?
$.07
$143.32
$.00
$143.32
Solve each proportion. Tell whether you used the Multiplication Property of
Equality or the Cross Products Property for your first step. Explain your choice.
p
7
m
1.5
24. 4 = 8
25. 4.5 = 25
27.
28. b - 4 = 7
b + 13 - 5b
2 = 3
3b
2.5
26. y = 7
x+2
3
3
29. 2x - 6 = 8
30. Explain Mathematical Ideas (1)(G) Describe and correct
the error in solving the proportion at the right.
31. Apply Mathematics (1)(A) A bakery sells packages of 10
bagels for $3.69. If the bakery starts selling the bagels in
packages of 12, how much would you expect a package of
12 to cost?
A. $3.08
C. $4.43
B. $4.32
D. $4.69
8
x+3
= 2
3
16 = 3x + 3
13 = 3x
13
= x
3
32. Display Mathematical Ideas (1)(G) Write a proportion that contains a variable.
Name the extremes, the means, and the cross products. Solve the proportion. Tell
whether you used the Multiplication Property of Equality or the Cross Products
Property to solve the proportion. Explain your choice.
STEM
33. Apply Mathematics (1)(A) Many trees have concentric rings that can be counted
to determine the tree’s age. Each ring represents one year’s growth. A maple tree
with a diameter of 12 in. has 32 rings. If the tree continues to grow at about the
same rate, how many rings will the tree have when its diameter is 20 in.?
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25
34. Apply Mathematics (1)(A) Allie wants to meet Grace at a fountain in a park 4 mi
away from Allie’s house. Allie will bike to the park at an average rate of 10 mi/h.
Grace lives 1.2 mi away from the park and walks at an average rate of 3 mi/h. How
many minutes ahead of Allie should Grace start out so that they meet at the
fountain at the same time?
Allie’s
House
Grace’s
House
Fountain
4 miles
1.2 miles
Solve each proportion.
35.
4y - 3 4
=
y2 + 1 y
w2 + 3
w
36. 2w + 2 = 2
37.
5x
5
=
x3 + 5 x2 - 7
38. Use a Problem-Solving Model (1)(B) A group of high school students is making a
parade float by stuffing pieces of tissue paper into a wire frame. They use 150 pieces
of tissue paper to fill an area 3 ft long and 2 ft wide. The total area they want to fill is
8 ft long and 7 ft wide. How many pieces of tissue paper they will need?
39. Apply Mathematics (1)(A) It takes an insect 15 s to crawl 1 ft. How many hours
would it take the insect to crawl 1 mi if the insect crawls at the same rate?
TEXAS End-of-Course PRACTICE
40. A high school soccer team is making trail mix to sell at a fundraiser. The recipe calls
for 3 lb of raisins and 2 lb of peanuts. If the team purchases 54 lb of peanuts, how
many pounds of raisins will they need?
A. 27
B. 36
C. 81
D. 162
41. One day during flu season, 13 of the students in a class were out sick, and only
24 students were left. How many students are in the class?
F. 16
G. 30
H. 36
42. An art gallery owner is framing a rectangular painting, as
shown. The owner wants the width of the framed painting
to be 38 12 in. How wide should each of the vertical sections
of the frame be?
26
Lesson 1-4
A. 4 18 in.
C. 4 12 in.
B. 4 14 in.
D. 8 12 in.
Solving Proportions
J. 72
x in.
x in.
30 in.