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Computing Ratios
8.1 Ratio and Proportion
Objectives

* Find and simplify the ratio of two
numbers.
* Use proportions to solve real-life
problems, such as computing the width
of a painting.
If a and b are two quantities that are measured
in the same units, then the ratio of “a” to “b”
is a/b. The ratio of a to b can also be written
as a:b. Because a ratio is a quotient (meaning
division), its denominator cannot be zero
(can’t divide by zero). Ratios are usually
expressed in simplified form. For instance,
the ratio of 6:8 is usually simplified to 3:4.
(You divided by 2)
8.1 Ratio & Proportion
8.1 Ratio & Proportion
Ex. 1: Simplifying Ratios
Ex. 1: Simplifying Ratios
Simplify the ratios:
a. 12 cm
b. 6 ft
4 cm
18 ft
OR
a. 12:4
b. 6:18


c. 9 in.
18 in.
c. 9:18
The fraction and ratio are both ways to express
ratios
a.
Simplify the ratios:
12 cm
b. 6 ft
4m
18 in
Solution: To simplify the ratios with unlike
units, convert to like units so that the units
divide out. Then simplify the fraction, if
possible.
8.1 Ratio & Proportion
Ex. 1: Simplifying Ratios

a.
Simplify the ratios:
12 cm
4m
12 cm
12 cm
4m
4·100cm
8.1 Ratio & Proportion
Ex. 1: Simplifying Ratios

12
400
3
100
.
Simplify the ratios:
b. 6 ft
18 in
6 ft
6·12 in
18 in
18 in.
72 in.
18 in.
4
1
4
3:100
8.1 Ratio & Proportion
8.1 Ratio & Proportion
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Ex. 2: Using Ratios

The perimeter of
rectangle ABCD is 60
centimeters. The ratio
of AB: BC is 3:2
3:2. Find
the length and the
width of the rectangle
Ex. 2: Using Ratios

B
C
w
A
l
D
SOLUTION: Because
the ratio of AB:BC is
3:2, you can represent
the length of AB as 3x
and the width of BC as
2x.
B
w
A
8.1 Ratio & Proportion
Solution:
Statement
2l + 2w = P
2(3x) + 2(2x) = 60
6x + 4x = 60
10x = 60
x=6
l
D
8.1 Ratio & Proportion
Ex. 3: Using Extended Ratios
Reason
Formula for perimeter of a rectangle
Substitute l, w and P
Multiply
Combine like terms
Divide each side by 10


So, ABCD has a length of 18 centimeters and a width of 12 cm.
The measures of the angles
in ∆JKL are in the
extended ratio 1:2:3.
Find the measures of the
angles.
l
Begin by sketching a
triangle. Then use the
extended ratio of 1:2:3 to
label the measures of
the angles as x°, 2x°, and
3x°.
K
2x°
Solution:
3x°
x°
J
8.1 Ratio & Proportion
Statement
x°+ 2x°+ 3x° = 180°
6x = 180
x = 30
C
L
8.1 Ratio & Proportion
Ex. 4: Using Ratios
Reason
Triangle Sum Theorem
Combine like terms
Divide each side by 6

The ratios of the side
lengths of ∆DEF to
the corresponding
side lengths of ∆ABC
are 2:1. Find the
unknown lengths.
F
C
3 in.
A
D
8 in.
B
E
So, the angle measures are 30°, 2(30°) = 60°, and 3(30°) = 90°.
8.1 Ratio & Proportion
8.1 Ratio & Proportion
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Ex. 4: Using Ratios
Using Proportions
SOLUTION:






F
DE is twice AB and DE =
8, so AB = ½(8) = 4
Use the Pythagorean
Theorem to determine
6 in
what side BC is.
DF is twice AC and AC =
3, so DF = 2(3) = 6
EF is twice BC and BC =
D
5, so EF = 2(5) or 10
C
5 in
3 in.
A
4 in
B
a2 + b2 = c2
8 in.
E
32 + 42 = c2
9 + 16 = c2
An equation that
equates two ratios is
called a proportion.
For instance
instance, if the
ratio of a/b is equal to
the ratio c/d; then the
following proportion
can be written:
Means
25 = c2
5=c
CROSS PRODUCT PROPERTY. The
product of the extremes equals the product of
the means.
1.
=
The numbers a and d are the
extremes of the proportions.
The numbers b and c are the
means of the proportion.
8.1 Ratio & Proportion
8.1 Ratio & Proportion
Properties of proportions
Extremes
Properties of proportions
2.
RECIPROCAL PROPERTY. If two ratios
are equal, then their reciprocals are also
equal.
If
 = , then ad = bc
If
 = , then
b
a =
To solve the proportion, you find the
value of the variable.
8.1 Ratio & Proportion
Ex. 5: Solving Proportions
4
x
x
4 4
x
=
5
7
=
7
5
=
28
5
Write the original
proportion.
Reciprocal prop.
4
Multiply each side by 4
Simplify.
8.1 Ratio & Proportion
8.1 Ratio & Proportion
Ex. 5: Solving Proportions
3
2
=
y+2 y
3y = 2(y+2)
3y = 2y+4
y
=
4
Write the original
proportion.
Cross Product prop.
Distributive Property
Subtract 2y from each
side.
8.1 Ratio & Proportion
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