### 29_ Ratios and Proportions

```Bell Ringer
Ratios and Proportions
• A ratio is a comparison of a number “a” and a
nonzero number “b” using division
Example 1
Simplify Ratios
Simplify the ratio.
a. 60 cm : 200 cm
3
ft
b. 18 in.
SOLUTION
a.
60 cm
60 cm : 200 cm can be written as the fraction
.
200 cm
60 cm
60 ÷ 20
200 cm = 200 ÷ 20
3
=
10
Divide numerator and denominator
by their greatest common factor, 20.
Simplify.3 is read as “3 to 10.”
10
Example 1
b.
Simplify Ratios
3 ft
= 3 · 12 in.
18 in.
18 in.
=
36 in.
18 in.
=
36 ÷ 18
18 ÷ 18
=
2
1
Substitute 12 in. for 1 ft.
Multiply.
Divide numerator and denominator
by their greatest common factor, 18.
Simplify. 12 is read as “2 to 1.”
Example 2
Use Ratios
In the diagram, AB : BC is 4 : 1 and AC = 30. Find
AB and BC.
SOLUTION
Let x = BC. Because the ratio of AB to BC is 4 to 1, you
know that AB = 4x.
AB + BC = AC
4x + x = 30
5x = 30
x=6
Substitute 4x for AB, x for BC, and 30 for A
Divide each side by 5.
Example 2
Use Ratios
To find AB and BC, substitute 6 for x.
AB = 4x = 4 · 6 = 24
BC = x = 6
ANSWER So, AB = 24 and BC = 6.
Example 3
Use Ratios
The perimeter of a rectangle is 80 feet. The ratio of the length
to the width is 7 : 3. Find the length and the width of the
rectangle.
SOLUTION
The ratio of length to width is 7 to 3. You can let the length l = 7x
and the width w = 3x.
2l + 2w = P
Formula for the perimeter of a rectangle
2(7x) + 2(3x) = 80 Substitute 7x for l, 3x for w, and 80 for
14x + 6x = 80 Multiply.
20x = 80
x=4
Divide each side by 20.
Example 3
Use Ratios
To find the length and width of the rectangle,
substitute 4 for x.
l = 7x = 7 · 4 = 28
w = 3x = 3 · 4 = 12
The length is 28 feet, and the width
is 12 feet.
Now You Try 
1.
In the diagram, EF : FG is 2 : 1 and EG = 24.
Find EF and FG.
EF = 16; FG = 8
2. The perimeter of a rectangle is 84 feet. The ratio of
the length to the width is 4 : 3. Find the length and
the width of the rectangle.
length, 24 ft; width, 18 ft
An equation that states that two ratios are
equal is called a proportion
Example 4
Solve a Proportion
y+2
5
=
Solve the proportion
.
3
6
SOLUTION
y+2
5
=
3
6
5 · 6 = 3(y + 2)
30 = 3y + 6
30 – 6 = 3y + 6 – 6
24 = 3y
24 3y
=
3
3
8=y
Write original proportion.
Cross product property
Multiply and use distributive property
Subtract 6 from each side.
Simplify.
Divide each side by 3.
Simplify.
Now You Try 
Solve the proportion.
3
6
=
3. x 8
15
=
4. 5
y
3