Geometry/Chapter 5/Lesson 2

Introduction
We will be reviewing proportions, the properties of proportions, and how to solve them.

Proportions
A proportion is a statement that states that two ratios are equal.
 * EXAMPLE: $$\tfrac{6}{8}$$ and $$\tfrac{3}{4}$$ are equal, and thus are a proportion: $$\tfrac{6}{8}$$ = $$\tfrac{3}{4}$$.

An extended proportion is similar to an extended ration from the last lesson: A statement that states that three or more rations are equal.
 * EXAMPLE: $$\tfrac{12}{16}$$ = $$\tfrac{6}{8}$$ = $$\tfrac{3}{4}$$

All proportions have 4 parts known as the extremes and means. In $$\tfrac{6}{8}$$ = $$\tfrac{3}{4}$$, the extremes are $$6$$ and $$4$$, while the means are $$8$$ and $$3$$. The Cross-Product Property states that the products of the extremes and means are equal. So:
 * $$4$$ • $$6$$ = $$24$$
 * $$8$$ • $$3$$ = $$24$$

You can use the cross-product property to check if two ratios are a proportion. {Are $$\tfrac{3}{9}$$ = $$\tfrac{5}{7}$$ a proportion? - Yes + No
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{Are $$\tfrac{9}{3}$$ = $$\tfrac{12}{4}$$ a proportion? + Yes - No
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Solving with x
How do we solve porpotions with $$x$$ or any variable?

Problem #1
$$\tfrac{x}{6}$$ = $$\tfrac{7}{3}$$
 * Solve the proportion

Solving for $$x$$, you would multiply the extremes ($$3$$ and $$x$$) and the means ($$6$$ and $$7$$):

$$3x = 42$$

And simply work out the problem from there:

$$3x = 42$$

$$\tfrac{3x}{3}$$ = $$\tfrac{42}{3}$$

$$x = 14$$

So, alas, the $$x$$ in $$\tfrac{x}{6}$$ = $$\tfrac{7}{3}$$ is $$14$$... So: $$\tfrac{14}{6}$$ = $$\tfrac{7}{3}$$