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

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 : ba

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

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=

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

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.

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.

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

−=

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˚

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)

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 M6

54

12

8

∠N ≅ ∠X

∠L ≅ ∠S

∠E ≅ ∠A

∠O ≅ ∠M

ELAS

5

3

410

6

8

3 4 5 6

6 8 10 12= = =

NOEL ~ XMAS

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–the equal, cross-multiply them–the products should be equal.

Example: Is ?=3 4

6 8

=24 24 �

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