refresher

def
A ratio is a comparison of two (or more) quantities.
The ratio of one number to another number is the quotient of the first number divided by the second number, where the second number is not zero. (A ratio is a fraction.)

While a ratio is most commonly written as a fraction, it may also appear in other forms:
ratio1

Since a ratio can be written as a fraction, it can also be written in any form that is equivalent to that fraction. All of the following statements are equivalent:

ratio2

Equivalent ratios are ratios that can be reduced to the same value: ratio4

A continued ratio refers to the comparison of more than two quantities: a : b : c.

When working with ratios in an algebraic setting, remember that 3 : 4 : 7
may need to be expressed as 3x : 4x : 7x (an equivalent form).

example
The sides of a pentagon are in the ratio of 2 : 3 : 5 : 1 : 4. If the perimeter of the pentagon is 90 units, find the lengths of the five sides.

Solution: Represent the sides of the pentagon as 2x, 3x, 5x, x, and 4x, an equivalent form.
Since 2 + 3 + 5 + 1 + 4 does not equal 90, we know that the side lengths will be an equivalent form of this continued ratio.
2x + 3x + 5x + x + 4x = 90
15x = 90
x = 6
The sides of the pentagon are 12, 18, 30, 6 and 24 units.

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def
A proportion is an equation stating that two ratios are equivalent (equal), written in the form ratio5.

Proportions always have an equal sign!

A proportion can be written in two forms: ratio6

For example, ratio7
where both are read "6 is to 9 as 2 is to 3".

In each proportion, the first and last terms (6 and 3) are called the extremes.
The second and third terms (9 and 2) are called the means.

mean       or       mean2

bullet RULE: In a proportion, the product of the means
equals the product of the extremes.

(You may see this rule referred to as "cross multiply" or "cross product".)
proprules

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Properties of Proportions:

proportionrules
Notice that all of these proportions "cross multiply" to yield the same result.
a • d = c • b


Some additional properties:

proportion2

 

reminder Keep in mind that there are many different ways to express
equivalent proportions.

 

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example
Solve for x:   ratio8

Solution: Apply the rule that "in a proportion, the product of the means equals the product of the extremes."
ratio9 gives (5)•(12) = 8 • x;      60 = 8x;      x = 7.5 ANSWER

 

example
In Geometry, we also use this rule when working with similar triangles.

pfp1 geoprod


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