Ratios and proportions can be used to solve problems involving comparisons of quantities. A ratio compares two numbers using division, while a proportion states that two ratios are equal. To solve proportions, the cross product property is used, setting up equivalent fractions and solving with cross multiplication. Similar polygons have corresponding angles that are congruent and side lengths that are proportional, allowing their similarities to be described through statements listing corresponding vertices.

1. 3.8.2 Ratio & Proportion
The student is able to (I can):
• Write and simplify ratios
• Use proportions to solve problems• Use proportions to solve problems
• Identify similar polygons

2. ratio A comparison of two numbers by division.
The ratioratioratioratio of two numbers a and b, where b
does not equal 0 (b ≠ 0) can be written as
a to b
a : b
a
b
Example: The ratio comparing 1 and 2 can
be written 1 to 2, 1 : 2, or .
Note: To compare more than two numbers,
use “dot” notation. Ex. 3 : 7 : 9
b
1
2

3. proportion An equation stating that two ratios are
equal. Two sets of numbers are
proportionalproportionalproportionalproportional if they use the same ratio.
Example: or a : b = c : d
Cross Products Property
a c
b d
=
Cross Products Property
In a proportion, if , and b and d ≠ 0,
then
ad = bc
a c
b d
=

4. Solving Problems with Ratios
If a problem contains a ratio of numbers,
set up a proportion and cross-multiply.
Example: The student-faculty ratio at a
college is 15: 1. If there are 500 faculty,
how many students are there?
=:
student 15s x
x = (15)(500)
= 7500 students
=
facul
:
student
ty
15
1
s x
500

5. If a problem contains a ratio comparing
more than two numbers, let x be the
common factor and set up an equation to
solve for x. Once we know x, we can find the
original quantities.

6. Example The ratio of the side lengths of a triangle is
2 : 3 : 5, and its perimeter is 80 ft. What
are the lengths of each side?
Let the side lengths be 2x, 3x, and 5x.
2x + 3x + 5x = 80
10x = 80
x = 8x = 8
This means that the sides measure
2(8) = 16 ft.
3(8) = 24 ft.
5(8) = 40 ft.

7. Examples Solve each proportion:
1.
8x = 96 x = 12
2.
2x = 20 x = 10
3 x
8 32
=
4 2
x 5
=
2x = 20 x = 10
3.
3x = 6(x — 2)
3x = 6x — 12
—3x = —12 x = 4
x x 2
6 3
−
=

8. Examples 4. The ratio of the angles of a triangle is
2: 2: 5. What is the measure of each
angle?
2x + 2x + 5x = 180˚
9x = 180˚
x = 20
2(20) = 40˚
2(20) = 40˚
5(20) = 100˚

9. Examples 5. A 60 meter steel pole is cut into two
parts in the ratio of 11 to 4. How much
longer is the longer part than the
shorter?
11x + 4x = 60
15x = 60
x = 4x = 4
11(4) = 44 m
4(4) = 16 m
The longer part is 28 m longer than
the shorter part. (44 — 16)

10. similar
polygons
Two polygons are similar if and only if their
corresponding angles are congruent and
their corresponding side lengths are
proportional.
Example:
N O
X M
6
54
12
8
∠N ≅ ∠X
∠L ≅ ∠S
∠E ≅ ∠A
∠O ≅ ∠M
EL
AS
5
3
4
10
6
8
3 4 5 6
6 8 10 12
= = =
NOEL ~ XMAS

11. Note: A similarity statementsimilarity statementsimilarity statementsimilarity statement describes
two similar polygons by listing their
corresponding vertices.
Example: NOEL ~ XMAS
Note: To check whether two ratios are
equal, cross-multiply them–theequal, cross-multiply them–the
products should be equal.
Example: Is ?=
3 4
6 8
=24 24

12. Example Determine whether the rectangles are
similar. If so, write the similarity ratio and
a similarity statement.
Q U
AD
R E
15
6
25
10
All of the angles are right angles, so all the
angles are congruent.
QUAD ~ RECT
sim. ratio:
150 = 150
CT
10
6 15
?
10 25
= 3
5