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= =
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
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
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
In the diagram we can use proportions to
determine the height of the tree.
5/x = 8/28 8x = 140
x = 17.5 ft