Problem 3 - 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. Two-thirds of the children in the fourth-grade class are girls. If there are 20 girls, what is the total number of children in the class?

Step-by-Step Solution

Answer

The total number of children in the fourth-grade class is 30.

Step 1: Understand the problem

We know that two-thirds of the children in the fourth-grade class are girls and there are 20 girls in total. We need to find the total number of children in the class.

Step 2: Set up the equation

Let's denote the total number of children in the class as \( T \). According to the problem, two-thirds of \( T \) equals 20. This can be written as the equation: \[ \frac{2}{3} T = 20 \]

Step 3: Solve the equation

To find \( T \), we need to isolate it on one side of the equation. Multiply both sides by \frac{3}{2} \ to get: \[ T = 20 * \frac{3}{2} \]

Step 4: Simplify the right-hand side

Calculate the value on the right-hand side: \[ T = 20 * 1.5 \] \[ T = 30 \]

Step 5: Write the answer in a complete sentence

The total number of children in the fourth-grade class is 30.

Key Concepts

fractional equations

Fractional equations are equations that include fractions with variables in their numerators, denominators, or both. In this problem, we encounter the fraction \( \frac{2}{3} \) which represents two-thirds of the total number of children in the class. To solve a fractional equation, it's necessary to clear the fraction by performing operations that eliminate the fraction. Here, we solve \( \frac{2}{3} T = 20 \) by multiplying both sides by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \). This helps in isolating the variable on one side of the equation.

problem solving strategy

A good problem solving strategy involves several systematic steps. In this particular problem, the steps followed are:

• Understanding the problem: Carefully read and identify what's known and what needs to be found.
• Setting up the equation: Translate the word problem into a mathematical equation using a variable to represent the unknown.
• Solving the equation: Use algebraic methods to isolate the variable and solve for it.
• Verification: Always check if your solution makes sense in the context of the problem.
• Answering in complete sentence: A complete sentence shows clarity and thorough understanding of the problem.

elementary algebra

Elementary algebra involves basic algebraic operations and concepts such as variables, constants, equations, and simple transformations. In this exercise, setting up the equation \( \frac{2}{3} T = 20 \) and isolating \(T\) using multiplication illustrates fundamental algebra principles. These include understanding ratios (two-thirds in this case), writing equations based on word problems, and performing operations like multiplication to find the solution.

equation isolation

Equation isolation is the process of manipulating an equation to get the unknown variable by itself on one side of the equation. In our problem, we start with the equation \( \frac{2}{3} T = 20 \). To isolate \(T\), multiply both sides by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \). This gives: \[ T = 20 \times \frac{3}{2}\]. Simplifying this, we find \(T = 30\). By performing these operations step-by-step, we isolate \(T\) and solve for it effectively, demonstrating the key concept of equation isolation.