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: $\frac{6}{8}$ and $\frac{3}{4}$ are equal, and thus are a proportion: $\frac{6}{8}$ = $\frac{3}{4}$.

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: $\frac{12}{16}$ = $\frac{6}{8}$ = $\frac{3}{4}$

All proportions have 4 parts known as the extremes and means. In $\frac{6}{8}$ = $\frac{3}{4}$, 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. <quiz display=simple> {Are $\frac{3}{9}$ = $\frac{5}{7}$ a proportion? |type="()"} - Yes + No

{Are $\frac{9}{3}$ = $\frac{12}{4}$ a proportion? |type="()"} + Yes - No </quiz>

Solving with x

How do we solve porpotions with $x$ or any variable?

Problem #1

Solve the proportion

$\frac{x}{6}$ = $\frac{7}{3}$

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:

$3x=42$

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

$x=14$

So, alas, the $x$ in $\frac{x}{6}$ = $\frac{7}{3}$ is $14$... So: $\frac{14}{6}$ = $\frac{7}{3}$ Template:Subpage navbar