Chapter 6 - Ratio and Proportion

Chapter 6 - Ratio and Proportion The general proportion formula is: Another version using the verbal keywords is: Solving Proportions Proportion Pitfall

 

The general proportion formula is:

 

 

Another version using the verbal keywords is:

 

 

These type of problems are given as verbal expressions and you need to convert them into algebraic components.  Substitue the numbers into the formula and solve for the unknown.  The percent is usually easy to spot. 

The tricky part is understanding if you are looking for and setting up the equation properly.  Once done, you solve for the missing quantity using the rules you learned to solve linear equations.

 

There are three types of problems:

1)  Find the percentage.   Example: What percent of 40 is 10?

The example sets up like this:

 

 

 

Note: Don't always assume that the smaller number goes on the top and the larger number on the bottom

 

2)  Find the "portion".  Example:  What is 17.5% of 240?

The example sets up like this:

 

 

3)  Find the "total".  Example:  What number is 25% of 150?

 The example sets up like this: 

 

 

Note: Another way to solve this is to recognize that "of" is signifies multiplication.

So, you can convert the percent to a decimal and multiply it by 150:

(0.25)(150)

 

Solving Proportions

You can two proportions to extrapolate information.  In class, I found that there were 74.8 million dogs in the United States (source: US Humane Society) and 281.4 million people in the United States (source: Wikipedia).  I proposed the question of how to estimate how many dogs there are in Connecticut.  I found that there are 35.1 million people in Connecticut (source: Wikipedia).  I can use proportions to find the number of dogs:

 

note that all numbers in millions

 

 

I cross multiply and get a linear equation to solve.

 

241.8x = (74.8)(3.5)

 

241.8x = 261.8

 

divide both sides by 241.8 and x = 1.08

 

So, there are 1.08 million dogs in Connecticut.

 

Proportion Pitfall

When you set up the proportion make sure that the units of the numberators are the same and the units of the denominators are the same too.  In the above example, notice that dogs are both numerators (one number and one variable).  Both denominators are numbers of people.  By doing this, it does not matter HOW you set up the proportions.  The linear equation will still resolve to the same value (see example below):

 

equation one

 

 

10x = (5)(20)

10x = 100

x = 10

 

equation two

 

 

10x = (5)(20)

10x = 100

x = 10

 

both equations solve to x = 10