Geometry (A): READ: Ratios and Proportions Reading

Ratios and Proportions

Learning Objectives

Write and simplify ratios.
Formulate proportions.
Use ratios and proportions in problem solving.

Introduction

Words can have different meanings, or even shades of meanings. Often the exact meaning depends on the context in which a word is used. In this chapter you’ll use the word similar.

What does similar mean in ordinary language? Is a rose similar to a tulip? They’re certainly both flowers. Is an elephant similar to a donkey? They’re both mammals (and symbols of national political parties in the United States!). Maybe you’d rather say that a sofa is similar to a chair? In loose terms, by similar we usually mean that things are like each other in some way or ways, but maybe not the same.

Similar has a very precise meaning in geometry, as we’ll see in upcoming lessons. To understand similar we first need to review some basic skills in ratios and proportions.

Using Ratios

A ratio is a type of fraction. Usually a ratio is a fraction that compares two parts. “The ratio of $x$ to $y$ ” can be written in several ways.

$\frac {x}{y}$
$x: y$
$x$ to $y$

Example 1

Look at the data below, giving sales at Bagel Bonanza one day.

Bagel Bonanza Monday Sales

Type of bagel	Number sold
Plain	$80$
Cinnamon Raisin	$30$
Sesame	$25$
Garlic	$20$
Whole grain	$45$
Everything	$50$

a) What is the ratio of the number of cinnamon raisin bagels sold to the number of plain bagels sold?

Ratio of cinnamon raisin to plain $= \frac{30}{80}, 30: 80$ , or $30$ to $80$ .

Note: Depending on the problem, ratios are often written in simplest form. In this case the ratio can be reduced or simplified because $\frac{30}{80} = \frac{3}{8}$ .

b) What is the ratio, in simplest form, of the number of whole grain bagels sold to the number of "everything" bagels sold?

Ratio of whole grain to everything $= \frac{45}{50} = \frac{9}{10},9:10$ , or $9$ to $10$ .

c) What is the ratio, in simplest form, of everything bagels sold to the number of whole grain bagels sold?

Answer: This ratio is just the reciprocal of the ratio in b. If the ratio of whole grain to everything is, $\frac{45}{50} = \frac{9}{10}, 9: 10$ , or $9$ to $10$ , then the ratio of everything to whole grain is, $\frac{10}{9}, 10: 9$ , or $10$ to $9$ .

d. What is the ratio, in simplest form, of the number of sesame bagels sold to the total number of all bagels sold?

First find the total number of bagels sold: $80 + 30 + 25 + 20 + 45 + 50 = 250$ .

Ratio of sesame to total sold $= \frac{25}{250} = \frac{1}{10}, 1:10$ , or $1$ to $10$ .

Note that this also means that $\frac{1}{10}$ , or $10\%$ , of all the bagels sold were sesame.

In some situations you need to write a ratio of more than two numbers. For example, the ratio, in simplest form, of the number of cinnamon raisin bagels to the number of sesame bagels to the number of garlic bagels is $6:5:4$ (or $30:25:20$ before simplifying).

Example 2

A talent show features only dancers and singers.

The ratio of dancers to singers is $3:2$ .
There are $30$ performers in all.

How many singers are there?

There is a whole number $n$ so that the total number of each group can be represented as

$\text{dancers} = 3n,\ \text{singers} = 2n.$

Since there are $30$ dancers and singers in all,

$3n+2n &=30 \\ 5n &= 30\\ n&= 6$

The number of dancers is $3n = 3 \cdot 6 = 18$ . The number of singers is $2n = 2 \cdot 6 = 12$ . It’s easy to check these answers. The numbers of dancers and singers have to add up to $30$ , and they have to be in a $3-\;\mathrm{to}-2$ ratio.

Check: $18 + 12 = 30$ . The ratio of dancers to singers is $\frac{18}{12} = \frac{3}{2}$ , or $3$ to $2$ .

Proportions

A proportion is an equation. The two sides of the equation are ratios that are equal to each other. Proportions are often found in situations involving direct variation. A scale drawing would make a good example.

Example 3

Leo uses a scale drawing of his barn. He recorded actual measurements and the lengths on the scale drawing that represent those actual measurements.

Barn dimensions
	Actual length	Length on scale drawing
Door opening	$16 \;\mathrm{feet}$	$4 \;\mathrm{inches}$
Interior wall	$25\;\mathrm{feet}$	$6.25\;\mathrm{inches}$
Water line	$10\;\mathrm{feet}$	?

a) Since he is using a scale drawing, the ratio of actual length to length on the scale drawing should be the same all the time. We can write two ratios that should be equal. This is the proportion below.

$\frac{16} {4} = \frac{25} {6.25}$

Is the proportion true?

We could write the fractions with a common denominator. One common denominator is $4 \times 6.25$ .

$\frac{16} {4} =? \frac{25} {6.25} \Rightarrow \frac{16 \cdot 6.25} {4 \cdot 6.25} =? \frac{25 \cdot 4} {6.25 \cdot 4} \Rightarrow \frac{100} {25} = \frac{100} {25}.$

The proportion is true.

b) Depending on how you think, you might have written a different proportion. You could say that the ratio of the actual lengths must be the same as the ratio of the lengths on the scale drawing.

$\frac{16} {25} =? \frac{4} {6.25} \Rightarrow \frac{16 \cdot 6.25} {25 \cdot 6.25} =? \frac{4 \cdot 25} {6.25 \cdot 25} \Rightarrow \frac{100} {25 \cdot 6.25} = \frac{100} {6.25 \cdot 25}.$

This proportion is also true. One nice thing about working with proportions is that there are several proportions that correctly represent the same data.

c) What length should Leo use on the scale drawing for the water line?

Let $x$ represent the scale length. Write a proportion.

$\left [\frac{\text{actual}} {\text{scale}} \right] \Rightarrow \frac{16} {4} = \frac{10} {x} \Rightarrow \frac{16x} {4x} = \frac{10 \cdot 4} {x(4)} \Rightarrow \frac{16x} {4x} = \frac{40} {4x}$

If two fractions are equal, and they have the same denominator, then the numerators must be equal.

$16x = 40 \Rightarrow x = \frac{40} {16} = 2.5$

The scale length for the water line is $2.5\;\mathrm{inches}$ .

Note that the scale for this drawing can be expressed as $1\;\mathrm{inch}$ to $4\;\mathrm{feet}$ , or $\frac{1}{4}\;\mathrm{inch}$ to $1\;\mathrm{foot}$ .

Proportions and Cross Products

Look at example 3b above.

$\frac{16} {25} = \frac{4} {6.25} \Rightarrow \frac{16 \cdot 6.25} {25 \cdot 6.25} = \frac{4 \cdot 25} {6.25 \cdot 25}$

$\frac{16} {25} = \frac{4} {6.25}$ is true if and only if $16 \cdot 6.25 = 4 \cdot 25$ .

In the proportion, $\frac{16}{25} = \frac{4}{6.25}, 25$ and $4$ are called the means (they’re in the middle); $16$ and $6.25$ are called the extremes (they’re on the ends). You can see that for the proportion to be true, the product of the means $(25 \cdot 4)$ must equal the product of the extremes $(16 \cdot 6.25)$ . Both products equal $100$ .

It is easy to generalize this means-and-extremes rule for any true proportion.

Means and Extremes Theorem or The Cross Multiplication Theorem

Cross Multiplication Theorem: Let $a, b, c,$ and $d$ be real numbers, with $b \ne 0$ and $d \ne 0.$

If $\frac{a}{b} = \frac{c}{d}$ then $ad = bc$ .

The proof of the cross multiplication theorem is example 4. The proof of the converse is in the Lesson Exercises.

Example 4

Prove The Cross Multiplication Theorem: For real numbers $a, b, c$ , and $d$ with $b \ne 0$ and $d \ne 0,$ If $\frac{a} {b} = \frac{c} {d}$ , then $ad = bc$ .

We will start by summarizing the given information and what we want to prove. Then we will use a two-column proof.

Given: $a, b, c,$ and $d$ are real numbers, with $b \ne 0$ and $d \ne 0,$ and $\frac{a}{b} = \frac{c}{d}$
Prove: $ad = bc$

Statement	Reason
1. $a, b, c,$ and $d$ are real numbers, with $b\ne0$ and $d\ne0$	1. Given
2. $\frac{a} {b} = \frac{c} {d}$	2. Given
3. $\frac{a} {b}\cdot \frac{d} {d} = \frac{c} {d} \cdot \frac{b} {b}$	3. $\frac{d} {d} = \frac{b} {b} = 1$ , identity property of multiplication
4. $\frac{a} {b} \cdot \frac{d} {d} = \frac{b} {b} \cdot \frac{c} {d}$	4. Commutative property of multiplication
5. $a \cdot d = b \cdot c$ or $ad = bc$	5. If equal fractions have the same denominator, then the numerators must be equal $\blacklozenge$

This theorem allows you to use the method of cross multiplication with proportions.

Lesson Summary

Ratios are a useful way to compare things. Equal ratios are proportions. With the Means-and-Extremes Theorem we have a simple but powerful method for solving any proportion.

Points to Consider

Proportions are very “forgiving”—there are many different ways to write proportions that are equivalent to each other. There are hints of some of these in the Lesson Exercises. In the next lesson, we’ll prove that these proportions are equivalent.

You know about figures that are congruent. But many figures that are alike are not congruent. They may have the same shape, even though they are not the same size. These are similar figures; ratios and proportions are integral to defining and understanding similar figures.

Review Questions

The votes for president in a club election were:

$\text{Suarez}, 24 && \text{Milhone}, 32 && \text{Cho}, 20$

Write each of the following ratios in simplest form.
1. votes for Milhone to votes for Suarez
2. votes for Cho to votes for Milhone
3. votes for Suarez to votes for Milhone to votes for Cho
4. votes for Suarez or Cho to total votes
Use the diagram below for exercise 2.
Write each of the following ratios in simplest form.
1. $MN: MQ$
2. $MN: NP$
3. $NP: MN$
4. $MN: MP$
5. $\;\mathrm{area\ of}\ MNRQ: \;\mathrm{area\ of}\ NPSR$
6. $\;\mathrm{area\ of}\ NPSR: \;\mathrm{area\ of}\ MNRQ$
7. $\;\mathrm{area\ of}\ MNRQ: \;\mathrm{area\ of}\ MPSQ$
The measures of the angles of a triangle are in the ratio $3: 3: 4$ . What are the measures of the angles?
The length and width of a rectangle are in a $3: 5$ ratio. The area of the rectangle is $540\;\mathrm{square\ inches}$ . What are the length and width?
Prove the converse of Theorem 7-1: For real numbers $a, b, c,$ and $d,$ with, $b \ne 0$ and $d \ne 0, ad = bc \Rightarrow a/b = c/d$ .

Given: $a, b, c,$ and $d$ are real numbers, with $b \ne 0$ and $d \ne 0$ and $ad = bc$

Prove: $\frac{a} {b} = \frac{c} {d}$

Which of the following statements are true for all real numbers and and ?
1. If $\frac{a} {b} = \frac{c} {d}$ then $\frac{a} {d} = \frac{c} {b}$ .
2. If $\frac{a} {b} = \frac{c} {d}$ then $\frac{a} {c} = \frac{b} {d}$ .
3. If $\frac{a} {b} = \frac{c} {d}$ then $\frac{b} {a} = \frac{d} {c}$ .
4. If $\frac{a} {b} = \frac{c} {d}$ then $\frac{b} {c} = \frac{a} {d}$ .
Solve each proportion for .
1. $\frac{6} {w} = \frac{4} {5}$
2. $\frac{w} {3} = \frac{12} {w}$
3. $\frac{3} {4} = \frac{w} {w+2}$
Shawna drove $245\;\mathrm{miles}$ and used $8.2\;\mathrm{gallons}$ of gas. At that rate, she would use $x\;\mathrm{gallons}$ of gas to drive $416\;\mathrm{miles}$ . Write a proportion that could be used to find the value of $x$ .
Solve the proportion you wrote in exercise 8. How much gas would Shawna expect to use to drive $416\;\mathrm{miles}?$
Rashid, Leon, and Maria are partners in a company. They divide the profits in a $3: 2: 4$ ratio, with Rashid getting the largest share and Leon getting the smallest share. In $2006$ the company had a total profit of $\$1,800,000$ . How much profit did each person receive?

Review Answers

1. $4: 3$
2. $5: 8$
3. $6: 8: 5$
4. $11: 19$
1. $1:1$
2. $2:1$
3. $1:2$
4. $2:3$
5. $2:1$
6. $1:2$
7. $2:3$
$54^\circ, 54^\circ, 72^\circ$
$30\;\mathrm{inches}$ and $18\;\mathrm{inches}$

Statement	Reason
A. $a, b, c,$ and $d,$ with $b \ne 0$ and $d \ne 0$ , and $ad = bc$ .	A. Given
B. $ad \times \frac{1} {bd} = bc \times \frac{1} {bd}$	B. Multiplication Property of Equality
C. $\frac{ad} {bd} = \frac{bc} {bd}$	C. Arithmetic
D. $\frac{a} {b} \times \frac{d} {d} = \frac{b} {b} \times \frac{c} {d}$	D. Arithmetic
E. $\frac{a} {b} = \frac{c} {d}$	E. $\frac{d} {d} = \frac{b} {b} = 1$ , identity property of equality

1. No
2. Yes
3. Yes
4. No
1. $w = 7.5$
2. $w = 6$ or $w = -6$
3. $w = 6$
$\frac{245} {8.2} = \frac{416} {x}$ or equivalent
$x \approx 13.9$ . At that rate she would use about $13.9\;\mathrm{gallons}$ of gas.
Rashid gets $\$ 800,000$ , Leon gets $\$ 400,000$ , and Maria gets $\$ 600,000$

Last modified: Thursday, May 13, 2010, 9:05 AM