Ratios, Proportions and Similar Figures

Ratios, proportions and scale drawings

There are many uses of ratios and proportions. We use them in map reading, making scale drawings and

models, solving problems.

The most recognizable use of ratios and proportions is drawing models and plans for

construction. Scales must be used to approximate what the actual object will be like.

A ratio is a comparison of two quantities by division. In the rectangles below, the ratio of

shaded area to unshaded area is 1:2, 2:4, 3:6, and 4:8. All the rectangles have equivalent shaded

areas. Ratios that make the same comparison are equivalent ratios.

Using ratios

The ratio of faculty members to students in one school is 1:15. There are 675 students. How

many faculty members are there?faculty 1

students 15

1 x15 675

15x = 675

x = 45 faculty

=

A ratio of one number to another number is the quotient of the first

number divided by the second. (As long as the second

number ≠ 0)

A ratio can be written in a variety of ways.

You can use ratios to compare quantities or describe rates. Proportions are used in many fields, including construction, photography, and medicine.

a:b a/b a to b

Since ratios that make the same comparison are equivalent ratios, they all reduce to the

same value.

»

2 3 1

10 15 5= =

ProportionsTwo ratios that are equal

A proportion is an equation that states that two ratios are equal, such as:

In simple proportions, all you need to do is examine the fractions. If the fractions both

reduce to the same value, the proportion is true.This is a true proportion, since both fractions

reduce to 1/3.

5 2

15 6=

In simple proportions, you can use this same approach when solving for a missing part of a

proportion. Remember that both fractions must reduce to the same value.

To determine the unknown value you must cross multiply. (3)(x) = (2)(9)

3x = 18 3 3

x = 6 Check your proportion (3)(x) = (2)(9) (3)(6) = (2)(9) 18 = 18 True!

So, ratios that are equivalent are said to be proportional. Cross Multiply makes solving or

proving proportions much easier. In this example 3x = 18, x = 6.

If you remember, this is like finding equivalent fractions when you are adding or subtracting fractions.

1) Are the following true proportions?

2 10

3 5=

2 10

3 15=

No… 3 x 10 = 30 2 x 5 = 10

Since these values aren’t equal, the ratios are not proportional

Yes!… 3 x 10 = 30 2 x 15 = 30

Since these values are equal, the ratios are proportional

2) Solve for n:

4 n

6 42= Solution:

6(n) = 4(42) 6n = 168 6n = 168 6 6 n = 28

3) Solve for x:

25 5

x 2=

Answer on last slide

Solve the following problems.

4) If 4 tickets to a show cost $9.00, find the cost of 14 tickets.

5) A house which is appraised for $10,000 pays $300 in taxes. What should the tax be on a house appraised at $15,000.

Answers on last slide

Answers» #3: x = 10 » #4: $31.50 » #5: $450

Similar FiguresThe Big and Small of it

For Polygons to be Similar

corresponding angles must be congruent,

and corresponding sides must be

proportional

(in other words the sides must have lengths that form

equivalent ratios)

Congruent figures have the same size and shape. Similar figures have the same shape but not

necessarily the same size. The two figures below are similar. They have the same shape but not the

same size.

Let’s look at the two triangles we looked at

earlier to see if they are similar.

Are the corresponding

angles in the two triangles congruent?

Are the corresponding

sides proportional? (Do they form equivalent

ratios)

Just as we solved for variables in earlier proportions, we can solve

for variables to find unknown sides in similar figures.

Set up the corresponding sides as a proportion and then solve

for x.

Ratios x/12 and 5/10

x 5

12 10

10x = 60

x = 6

Determine if the two triangles are similar.

In the diagram we can use proportions to

determine the height of the tree.

5/x = 8/28 8x = 140

x = 17.5 ft

The two windows below are similar. Find the unknown width of the

larger window.

These two buildings are similar. Find the height of

the large building.