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
15
x
= 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
10
=
3
15
=
1
5
Proportions
Two 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
15
=
2
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
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
3
=
10
5
2
3
=
10
15
2) Solve for x:
4
6
=
x
42
3) Solve for x:
25
x
=
5
2
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.
Similar Figures
The 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.
x
5
12
10
10x = 60
x = 6
Ratios
x/12
and
5/10
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.