# The residents of a city voted on whether to raise property taxes. The ratio of yes votes to no...

## Question:

The residents of a city voted on whether to raise property taxes. The ratio of yes votes to no votes was 6 to 5. If there were 4,668 yes votes, what was the total number of votes?

## Solving Proportions:

In mathematics, a proportion is created when two ratios are equivalent, meaning that if the ratio {eq}\frac{a}{b} {/eq} is equivalent to the ratio {eq}\frac{c}{d} {/eq}, then they create a proportion of the form {eq}\frac{a}{b}=\frac{c}{d} {/eq}. We can solve proportions for any missing parts using the fact that if {eq}\frac{a}{b}=\frac{c}{d} {/eq}, then ad = bc. This is known as cross multiplication.

We are given that the ratio of yes votes to no votes was 6 to 5, or {eq}\frac{6}{5} {/eq}. We are also given that there were 4,668 yes votes. Thus, if we let x be the number of no votes, then the ratio of yes votes to no votes is {eq}\frac{4,668}{x} {/eq}. These two ratios will be equivalent since they are for the same decision, so we have the following proportion.

• {eq}\frac{6}{5}=\frac{4,668}{x} {/eq}

We can solve this for x to find the number of no votes, and then add that to the number of yes votes, 4,668, to find the total number of votes. Cross multiplication of the proportion gives the following:

• {eq}6x=5\cdot 4,668 {/eq}

Simplifying the right-hand side gives the following:

• {eq}6x=23,340 {/eq}

Dividing both sides of this equation by 6 gives the following:

• {eq}x=3,890 {/eq}

We get that x = 3,890, so there were 3,890 no votes. To find the total number of votes, we add the number of yes votes, 4,668, ad the number of no votes, 3,890.

• Total number of votes = 4,668 + 3,890 = 8,558

We get that the total number of votes is 8,558.