Problem 34 - Prealgebra

Question

Select the appropriate operation. Justify your selection. Then solve. VOTING In the class election, Murray received \(\frac{1}{3}\) of the votes and Sara received \(\frac{2}{5}\) of the votes. Makayla received the rest. What fraction of the votes did Makayla receive?

Step-by-Step Solution

Answer

Makayla received \(\frac{4}{15}\) of the votes.

Step 1: Analyze the Problem

We need to determine the fraction of votes Makayla received in the class election. Murray received \(\frac{1}{3}\) of the votes, and Sara received \(\frac{2}{5}\). Together, they received a total of \(\frac{1}{3} + \frac{2}{5}\) of the votes. We will need to subtract this sum from the whole, which is \(1\) (since 100% or the whole fraction of votes were distributed among all candidates).

Step 2: Find a Common Denominator

To add the fractions \(\frac{1}{3}\) and \(\frac{2}{5}\), we need a common denominator. The denominators are 3 and 5, so the least common multiple is 15. We convert \(\frac{1}{3}\) to \(\frac{5}{15}\) and \(\frac{2}{5}\) to \(\frac{6}{15}\).

Step 3: Add the Fractions

Add \(\frac{5}{15}\) and \(\frac{6}{15}\) which results in \(\frac{11}{15}\). This represents the portion of the votes received by Murray and Sara together.

Step 4: Calculate Makayla's Portion

Since the total votes are represented by 1, we subtract the portion received by Murray and Sara from the whole: \(1 - \frac{11}{15} = \frac{15}{15} - \frac{11}{15} = \frac{4}{15}\).

Step 5: Verify the Solution

Check that the sum of all portions equals the whole: \(\frac{5}{15} + \frac{6}{15} + \frac{4}{15} = \frac{15}{15} = 1\). This confirms our calculations are correct.

Key Concepts

Addition of Fractions

When you add fractions, the first step is to ensure that they have a common denominator. Imagine you have a pie cut into equal slices. You can't easily combine slices from differently sized pies unless they have the same size slices.
For the class election problem, we needed to add Murray's and Sara's votes: \(\frac{1}{3}\) and \(\frac{2}{5}\). To do this, first, we find the least common multiple (LCM) of the denominators 3 and 5 which is 15. By converting these fractions so they have the same denominator, we turn \(\frac{1}{3}\) into \(\frac{5}{15}\) and \(\frac{2}{5}\) into \(\frac{6}{15}\). Now it's simple to add them:

• \(\frac{5}{15} + \frac{6}{15} = \frac{11}{15}\)

Now you know how much of the vote they got together.

Subtraction of Fractions

To find Makayla's portion of the votes, you'll need to subtract the total amount of votes received by Murray and Sara from 1, which represents the whole. Think of it as starting with a full pie and taking out the pieces for Murray and Sara to find out what's left for Makayla.
Here's how you do it step by step. After finding that Murray and Sara together got \(\frac{11}{15}\) of the votes, subtract this from the whole pie:

• \(1 - \frac{11}{15}\)

Convert the 1 into a fraction with 15 as the denominator: \(\frac{15}{15}\). Now subtract the pieces:

• \(\frac{15}{15} - \frac{11}{15} = \frac{4}{15}\)

This means Makayla received \(\frac{4}{15}\) of the votes. Using subtraction of fractions helps you see what part of a whole is left after something has been taken away.

Least Common Multiple

Finding the least common multiple (LCM) is crucial when you're dealing with adding or subtracting fractions. It's like finding a common language for them to "communicate" and combine or separate easily.
Imagine you and your friend each have boxes, but each box is of a different size. To compare and combine things from these boxes, it helps if the boxes are the same size. That's where the LCM steps in.
For the voting problem, we needed to add fractions with denominators 3 and 5; the LCM of 3 and 5 is 15. This meant we needed to express both fractions with the denominator of 15:

• Change \(\frac{1}{3}\) to \(\frac{5}{15}\)
• Change \(\frac{2}{5}\) to \(\frac{6}{15}\)

Once you have them in a common "language," you can easily add them and later subtract from the whole to find out what's left for Makayla. Understanding and finding the LCM makes operations with fractions much simpler and more straightforward.