Solving Proportions
Section 2-7
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Solving Proportions: Understanding Equal Ratios and Cross Products, Study notes of Algebra
The concept of proportions, focusing on the cross products property. It provides examples of how to solve proportions involving variables and demonstrates the use of cross products. The document also includes exercises for practice.
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Solving Proportions
Section 2-
Goals
Goal •^
To solve and applyproportions.
What’s a Proportion?
- A
proportion
is an equation stating that two
ratios are equal.
- Example:
c d
a b
0 &
0
b
d
Proportions
-^
If the ratio of a/b is equal tothe ratio c/d; then thefollowing proportion can bewritten:
-^
The values
a
and
d
are the
extremes
. The values
b
and
c
are the
means
. When the
proportion is written as
a
: b
=
c :
d , the extremes are in the first and last positions. Themeans are in the two middlepositions.
Means
Extremes
Cross Product Property
The product of the extremes equals the
product of the means.
If
then
.
ad
bc
Solving a Proportion
To solve a proportion involving a variable,
simply set the two cross products equal toeach other. Then solve!
25^ x
x
x
or
x
x
Solve each proportion.^ y
4 g
=
^23 g
A.
B.
Use cross
products. Divide both
sides by 2.
Use cross
products. Subtract 12
from bothsides. Divide both
sides by 4.
y^
y ) = –5(8)
g
4 g
+12 = 35
Your Turn:
Example: Solving Multi-Step
Proportions
Solve the proportion
8
3
5
4
b^
b
8
3
5
4
b^
b
^
^
^
4
8
5
3
b^
b
4
32
5
15
b^
b
b
b
b
b
32
15 b
b
b
The equationlooks a lot like thisexample. Can you usecross products to findthe value of b?
8
3
1
5
4
b^
b
^
^
No
, there are 2 terms on the left side of theequation.
Write a ratio comparing bones in ears to bones in
skull. Write a proportion. Let x be the number of bones
in ears. Since x is divided by 22, multiply both sides of the
equation by 22.
There are 6 bones in the ears. The ratio of the number of bones in a human
’s ears to the number
of bones in the skull is 3:11. There are 22 bones in the skull. Howmany bones are in the ears?
Example: Application of
Ratios
The ratio of red marbles to green marbles is 6:5. There are 18red marbles. How many green marbles are there?
green
red
Write a ratio comparing green to red marbles.
x
Write a proportion. Let x be the number
green marbles. Since x is divided by 18, multiply both sides
by 18. There are 15 green marbles.