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Ratios and Proportions
Keystone Geometry
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Ratio
Ratio: A ratio is a comparison of two numbers such as a : b.
When writing a ratio, always express it in simplest form.
What is the ratio of AB to CB ?
** Ratios must be compared using the same units.
A ratio can be expressed:
1. As a fraction 3
7
2. As a ratio 3 : 7
3. Using the word “to” 3 to 7
Example: What is the ratio of side AB to side CB in the
triangle?
A
AB 10

CB 6
10 5

6 3
Now try to reduce
the fraction.
10
D
3.6
8
 ratio of AB to CB  5:3.
4.8
C
6
B
Example: What is the ratio of side DB to side CD in
the triangle?
DB 3.6

CD 4.8
3.6 36 3
 ratio of DB to CD  3 : 4.


4.8 48 4
3
4
Example ……….
A baseball player goes to bat 348 times and gets 107 hits.
What is the players batting average?
Solution:
Set up a ratio that compares the number of hits to the
number of times he goes to bat. Ratio: 107
348
Convert this fraction to a decimal rounded to three decimal
places.
Decimal: 107  0.307
348
The baseball player’s batting average is 0.307 which means
he is getting approximately one hit every three times at bat.
Proportion
• Definition: A proportion is an equation stating that two
ratios are equal.
• For example,
a c
=
b d
and a : b = c : d
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Terms of a Proportion
Second Term
a c

b d
First Term
a :b  c :d
First Term
Second Term
Third Term
Fourth Term
Fourth Term
Third Term
Means and Extremes
• The first and last terms of a proportion are called
extremes.
• The middle terms are called the means.
a :b = c : d
** The product of the means is
always equal to the product of
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the extremes.
ex.
2
  18  18
9 3
Properties of Proportions
a c
=
b d
Cross-multiplication
ad = bc
b d
=
a c
Reciprocals
is equal to:
a b
=
c d
Switching
the means
Add one
a + b c + d to both
=
b
d
sides
x 5
Example: If = , then…
y 2
2x
5y = _____
2 ? y
=
5 ? x
x + y ? 5 2 7
=

y
? 2 2
x ? y
=
5 ? 2
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Proportions- examples….
Example 1:
Solve the proportion using cross
multiplication.
4 • x = 12 • 3
4x = 36
x = 9
4x = 36
4
4
Some to try…
1.
x 8
=
12 3
2. x + 5
4
3.
4.
2
=
11
x+2 4
=
x+3 5
7
9
=
6x - 4 4x + 6
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Example 2: Use a proportion to solve for the missing piece
of a triangle.
Find the value of x.
x
2

1068 252
252 x  2136
x  8.5 feet
x
2 ft
356 yards
84 yards
First! Multiply by 3
to change yards into
feet. The units must
be the same.
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Examples: Find the measure of each angle.
• Two complementary angles have measures in the ratio
2 : 3.
36 and 54
• Two supplementary angles have measures in the ratio
3 : 7.
54 and 126
• The measures of the angles of a triangle are in a ratio of
2 : 2 : 5.
40, 40, and 100
• The perimeter of a triangle is 48cm and the lengths of
the sides are in a ratio of 3 : 4 : 5. Find the length of
each side.
12cm, 16cm, and 20cm