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Ratios, Proportions and Similar FiguresRatios, 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= =
ProportionsProportions
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 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 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=
2) Solve for x:
4 x
6 42=
3) Solve for x:
25 5
x 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 FiguresSimilar Figures
The Big and Small of it
For Polygons to be Similar
corresponding angles must be congruent,
andcorresponding 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/12and5/10
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 = 8/288x = 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.