In a mayoral election, the number of votes for the incumbent...

In a mayoral election, the number of votes for the incumbent was 25$\%$ more than the number for the opponent. Altogether, the two candidates received 5175 votes. How many votes did the incumbent mayor receive?

Key Concepts

Ratio and Proportion

This concept involves understanding the relative sizes of two or more quantities by means of a ratio. It is essential for comparing parts to a whole or one part to another, especially when the problem states that one quantity is a certain percent more or less than another.

Percentage Increase

Percentage increase is used to describe how much one quantity is larger than another by a specific percent. It involves converting a percentage into its decimal multiplier, which then allows for the proportional adjustment of values in mathematical expressions.

Algebraic Representation

This concept entails assigning variables to unknown quantities and expressing the relationships between them with equations. It is critical in translating verbal descriptions from word problems into mathematical formulations that can be manipulated to find numerical answers.

Linear Equations

Linear equations are used to model relationships where the unknown variable appears in a first-degree equation. They provide a structured method for solving for unknown values when given a total sum or balance, allowing one to isolate and compute the values of individual components.

Transcript

00:01 Okay, so to answer this question, we're going to have to use some algebra and do some substitution to figure it out.

00:05 So the first thing we need to do is figure out what our two equations are going to be.

00:09 So the first one is pretty simple.

00:11 It says that in a mayoral election, the number of votes for the incumbent was 25 % more than the number for their opponent, but altogether the two candidates received 5 ,175 votes.

00:23 So let's just kind of use x and y here.

00:25 We can use x to be the incumbent.

00:29 So we'll just put x as the incumbent and y will be the opponent.

00:37 Okay, and so it says that the two candidates together received 5 ,175 votes.

00:44 So our first equation is just going to be x plus y equals 5 ,175.

00:56 So now we got to figure out how we're going to write our second equation.

01:00 So that's the number of votes for the incumbent was 25 % more than the opponent.

01:11 And so basically, what is 25 % more? so we know that the incumbent got 100 % of y's votes, whatever amount they got, they got 100 % of those plus an extra 25 % higher than what they got, plus 25 % more than what y received.

01:31 So a way that we can write that is that x equals 125 % y, which would be 125 % of what y received.

01:42 Like i said, they received 100%, which is the total number of y plus an extra 25 % of whatever y's numbers were.

01:50 So that's 125%...

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