The document defines ratios and proportions, and provides examples of how to set up and solve proportions to find missing values. It explains that a ratio compares two numbers using division, and a proportion states that two ratios are equal. Examples are given of setting up proportions to solve problems involving student-faculty ratios, triangle side lengths, and more. Step-by-step workings are shown for solving various proportions.

1. Obj. 34 Ratio and Proportion
The student is able to (I can):
• Write and simplify ratios
• Use proportions to solve problems

2. ratio
A comparison of two numbers by division.
The ratio 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
1
be written 1 to 2, 1 : 2, or
.
2
Note: To compare more than two numbers,
use “dot” notation. Ex. 3 : 7 : 9

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

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?
students 15
x
: =
faculty
1 500
x = (15)(500)
= 7500 students

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=8
This means that the sides measure
2(8) = 16 ft.
3(8) = 24 ft.
5(8) = 40 ft.

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

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=4
11(4) = 44 m
4(4) = 16 m
The longer part is 28 m longer than
the shorter part.

10. Examples
(PAP only)
6. Find the ratio of x to y if
a) 2x = 3y
2x 3y
=
2y 2y
x 3
=
y 2
b) 6 ( y + 3 ) = 2 ( x + 9 )
6y + 18 = 2x + 18
6y = 2x
x 6
= =3
y 2