Similar Polygons: Ratio and Proportion

Objectives

Find and simplify the ratio of two numbers

Use proportions to solve real-life problems

Def: A ratio is a comparison of two numerical quantities measured in the same units.

8 cm to 12 cm

8 cm : 12 cm

cm

cm

12

8

Since a ratio is a quotient, what do we know about the denominator?

Simplifying Ratios:

cm

cm

12

8

a.) 12x to 16x

This can also be written as:

2 to 3

2 : 3

3

2=

Simplify the ratios.

b.) 6 ft.18 in.

The ratio of two numbers is 4 : 5. If the sum of the two numbers is 36, what are the two numbers?

Lets think about this:

4 and 5 ?

8 and 10

12 and 15

16 and 20 YES!

Let’s look at an easier way to solve this problem:

5

4

5

4

x

x

4

369

3654

x

x

xx

Example:

A poster is 8ft long and 52 in wide. Find the ratio of width to length.

52 in : 8 ft

52 in : 96 in

13 : 24 in

Comparing 3 or more numbers a,b,c

a : b : c

Example: The measure of the three angles of a triangle are in a ratio of 2:2:5. Find the measure of each angle.

Solution: 40,40,100

Def: A proportion is a statement that two ratios are equal to each other.

Examples:

3

2

4

5

45

27

5

3

xxd

nc

b

a

Properties of Proportions

d

c

b

a

meansextremes

Def: In the proportion, a and d are called extremes while b and c are called the means.

“The product of the means is equal to the product of the extremes” (Cross-Product).

Note: Two proportions are equivalent if their cross products are equivalent.

Therefore: If

d

c

b

a then, ad = bc

Reciprocal Property

If two ratios are equal, then their reciprocals are also equal.

d

c

b

aIf , then

c

d

a

b

Def: To solve a proportion means to find all values of the variable that satisfy the proportion (that makes the proportion true).

Examples: Solve each of the following proportions

5

4

3

1

x

x1.

7

518

3

53

xx2.

x

x

x

x 4

5

3.

4.

23

10

2

1

xx

x

Practical Examples

Practical examples:If a punch recipe calls for 1 part of 7-up and 2

parts of orange juice, then you need to use the same ratio (no matter how much of the punch you want) in order to keep the taste consistent.  

If you are mixing paint to paint your house, you need to keep the ratio (of color pigments to white paint) constant to ensure that the color will remain exactly the same.

Proportion Word Problems

If you can buy one can of pineapple chunks for $2 then how many can you buy with $10?First set up a proportion then solve for your

variable.Remember proportions are two equivalent

ratios set equal to each other.1 can = x

$2 $10

1 can = x $2 $10

And one more…

Shawna reduced the size of a rectangle to a height of 2 in. What is the new width if it was originally 24 in wide and 1 foot tall?