Problem 23 - Algebra 2 and Trigonometry

Question

In the chess club, the ratio of boys to girls is 6 : 5. There are 3 more boys than girls in the club. How many members are in the club?

Step-by-Step Solution

Answer

There are 33 members in the chess club.

Step 1: Understand the Ratio

We start by understanding the given ratio 6:5, which means for every 6 boys, there are 5 girls. Let the number of boys be represented as \(6x\) and the number of girls as \(5x\).

Step 2: Express the Relationship in an Equation

According to the problem, there are 3 more boys than girls. We can express this relationship as an equation: \(6x = 5x + 3\).

Step 3: Solve for x

Subtract \(5x\) from both sides of the equation to solve for \(x\):\[ 6x - 5x = 3 \]\[ x = 3 \]

Step 4: Calculate the Number of Boys and Girls

Substitute \(x = 3\) back into the expressions for boys and girls:\[ \text{Number of boys} = 6x = 6(3) = 18 \]\[ \text{Number of girls} = 5x = 5(3) = 15 \]

Step 5: Find the Total Number of Members

Add the number of boys and girls to find the total number of members in the club:\[ 18 + 15 = 33 \]

Key Concepts

Algebraic Equations

Algebraic equations are mathematical sentences that involve unknowns or variables. In the realm of algebra, equations provide a means to articulate relationships among quantities that need to be determined. These are written using symbols like \( +, -, \, \) and \( = \), with the main purpose being to solve these relationships for unknowns, often denoted by letters such as \( x \) or \( y \). In our specific example, we used an algebraic equation to express a relationship between boys and girls in a club. The equation \( 6x = 5x + 3 \) describes how the number of boys exceeds the number of girls by 3. Here, \( x \) represents a common multiplier that applies to both quantities, allowing for a straightforward solution to find actual numbers.

Solving Equations

Solving equations is a fundamental skill in algebra, and it involves finding the value(s) of the variable(s) that satisfy the condition of the equation. The process generally includes rearranging the equation to isolate the variable on one side. Consider the equation we derived: \( 6x = 5x + 3 \). The goal here is to find what \( x \) equals. We start by moving all terms involving \( x \) to one side. This is achieved by subtracting \( 5x \) from both sides, leading to \( x = 3 \).

• Ensure both sides of the equation are balanced as you isolate the variable.
• Check your solution by plugging it back into the original equation.

Using these steps helps ensure that students correctly identify the values that solve the equation. They verify their understanding and compliance with algebraic principles.

Word Problems

Word problems are a common way to apply math concepts to real-world situations. They require students not only to perform calculations but also to translate text into mathematical equations. In this problem, we interpreted a narrative about boys and girls in a chess club and used it to form algebraic equations. The key to solving word problems is:

• Understanding the problem: Identify what is being asked.
• Translating the scenario into mathematical expressions.
• Solving the equations using appropriate algebraic methods.
• Interpreting the solution in the context of the original problem.

In the chess club problem, understanding that each part of the ratio represented a multiple of boys and girls helped translate the situation into an algebraic equation that solved for the number of members. Learning to balance these problems with proper equation formation and solution checking is vital, developing skills crucial for academic and daily problem-solving.