Problem 4 - Elementary Algebra

Question

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question. Three-fifths of the members of the school choir are women. If there are 24 women, what is the total number of choir members?

Step-by-Step Solution

Answer

The total number of choir members is 40.

Step 1 - Understand the Problem

We are told that three-fifths of the choir members are women and there are 24 women in the choir. We need to find the total number of choir members.

Step 2 - Assign Variables

Let the total number of choir members be denoted by the variable \( x \).

Step 3 - Set Up the Equation

We know that three-fifths of the choir members are women. This can be written as: \( \frac{3}{5}x = 24 \).

Step 4 - Solve the Equation

To find \( x \), we need to isolate \( x \). Multiply both sides of the equation by the reciprocal of \( \frac{3}{5} \), which is \( \frac{5}{3} \): \[ x = 24 \times \frac{5}{3} \]Calculating this gives: \[ x = 40 \]

Step 5 - Write the Conclusion

Therefore, the total number of choir members is 40.

Key Concepts

fractions

In algebra word problems, fractions often represent parts of a whole. In this problem, three-fifths \(\frac{3}{5}\) of the choir members are women. This fraction helps simplify the calculation and understand the relationship between the parts and the whole.

Key points to remember:

• The numerator (top number) indicates how many parts we have.
• The denominator (bottom number) indicates the total number of equal parts.

In our example, \(\frac{3}{5}\) means that out of every 5 choir members, 3 are women. Always relate fractions to the given context to understand the problem better.

Practice by identifying fractions in other word problems and calculating their corresponding values.

solving equations

Solving equations is a critical step in finding the value of unknown variables. In this exercise, the equation \(\frac{3}{5}x = 24\) represents the relationship between the total choir members \(x\) and the number of women.

Steps to solve the equation:

• Isolate the variable on one side of the equation.
• Use inverse operations, such as adding, subtracting, multiplying, or dividing both sides by the same number.

Here, we used the reciprocal of \frac{3}{5}\ to isolate \(x\): \[ x = 24 \times \frac{5}{3} \] This calculation leads to \(x = 40\). Practice solving different types of equations to become more comfortable with this process.

variable assignment

Variable assignment is the process of representing unknown values with symbols, usually letters like \(x\) or \(y\). In our problem, we assigned \(x\) to represent the total number of choir members.

Why assign variables?

• They make it easier to work with unknown quantities.
• They help set up equations based on given relationships and conditions

Consider the relationships in the problem and assign variables logically. This will make it easier to set up and solve equations. Practice variable assignment in different contexts to master this skill.

ratio and proportion

Ratios and proportions are used to compare quantities and their relationships. In this problem, the ratio of women to total choir members is three-fifths \(\frac{3}{5}\). This tells us that for every 5 choir members, 3 are women.

To solve problems using ratios and proportions:

• Set up the ratio based on the given relationships.
• Formulate a proportion involving the known and unknown quantities.

Our problem statement \(\frac{3}{5}x = 24\) is an example of using a proportion. This concept is crucial for understanding many algebra word problems. Practice by identifying and setting up ratios and proportions in different scenarios.