In this year’s student election for president, there were two candidates. The winner received more votes than the loser. If there were 584 votes cast for president, how many votes did each of the two candidates receive?
When introducing fractions into an equation, the same rules for solving any equation apply. You need to keep the equations in balance by adding, subtracting, multiplying, or dividing on both sides of the equals sign in order to isolate the variable. The goal still remains to get your variable alone on one side of the equals sign with your constant terms on the other in order to solve for this variable.
With fractions, there is sometimes an added step of multiplying and dividing the equation by the numerator and denominator in order to solve for the variable. Or, if there are multiple fractions that do not have the same denominator, you must first find the least common denominator (LCD) before combining like terms.
Concept Problem Revisited
In this year’s student election for president, there were two candidates. The winner received more votes. If there were 584 votes cast for president, how many votes did each of the two candidates receive?
Let votes for candidate 1 (the winner)
Let votes for candidate 2
You must have only one variable in the equation in order to solve it. Let’s look at another relationship from the problem.
Now substitute into the original problem.
So candidate 2 received 250 votes. Candidate 1 would then receive votes.
- A fraction is a part of a whole consisting of a numerator divided by a denominator. For example, if a pizza is cut into eight slices and you ate 3 slices, you would have eaten of the pizza. is a fraction with 3 being the numerator and 8 being the denominator.
- Least Common Denominator
- The least common denominator or lowest common denominator is the smallest number that all of the denominators (or the bottom numbers) can be divided into evenly. For example with the fractions and , the smallest number that both 2 and 3 will divide into evenly is 6. Therefore the least common denominator is 6.
1. Solve for x: .
2. Solve for x: .
3. Solve for x: .
Since all the denominators are the same (12), we can simplify further:
Solve for the variable in each of the following equations.