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.

1 Are ${\tfrac {3}{9}}$ = ${\tfrac {5}{7}}$ a proportion?

	Yes
	No

2 Are ${\tfrac {9}{3}}$ = ${\tfrac {12}{4}}$ a proportion?

	Yes
	No

Solving with x

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

Problem #1

Solve the proportion

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

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:

$3x=42$

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

$x=14$

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

← Lesson 1

Geometry/Chapter 5