Problem 24 - Algebra 2 and Trigonometry

Question

Every year, Javier makes a total contribution of \(\$ 125\) to two local charities. The two donations are in the ratio of \(3 : 2 .\) What contribution does Javier make to each charity?

Step-by-Step Solution

Answer

Javier contributes \(\$75\) to the first charity and \(\$50\) to the second charity.

Step 1: Define Variables

Let Javier's contribution to the first charity be \(3x\) and to the second charity be \(2x\). The contribution is split in a 3:2 ratio, so we use these expressions.

Step 2: Set Up the Equation

According to the problem, the total contribution Javier makes is \(\$125\). Therefore, the equation is: \[ 3x + 2x = 125 \]

Step 3: Solve for x

Combine the terms in the equation to find \(x\). \[ 5x = 125 \] Divide both sides by 5 to solve for \(x\): \[ x = 25 \]

Step 4: Calculate the Contributions

Now, find the contributions to each charity using \(x\). Calculate \(3x\) for the first charity and \(2x\) for the second charity.For the first charity: \[ 3x = 3 \times 25 = 75 \]For the second charity: \[ 2x = 2 \times 25 = 50 \]

Step 5: Verify the Solution

Add the calculated contributions to ensure they total \(\$125\): \[ 75 + 50 = 125 \]The solution is verified.

Key Concepts

Variable Definition

When solving problems involving ratios in algebra, the first step is defining the variables that represent the different components of the problem. In this exercise, Javier's total contribution to two charities is given, and we need to split it according to a specified ratio.

It's essential to clearly define what each variable stands for. Here, we let the contribution to the first charity be represented by \(3x\) and the second by \(2x\). The numbers 3 and 2 come from the given 3:2 ratio. The variable \(x\) acts as a common multiplier that helps maintain the given ratio in algebraic expressions.

Equation Setup

Once we have defined the variables, the next step is setting up an equation that represents the problem at hand. For Javier’s contributions, we are informed that the total amount he donates is \(\$125\).

We can use this information to form an algebraic expression combining his contributions to both charities: \(3x + 2x = 125\). By doing this, we translate the word problem into a mathematical equation, which is easier to work with when solving for unknowns. Setting up this equation correctly is crucial to finding a solution.

Problem Solving Steps

With the equation \(3x + 2x = 125\) set up, we can proceed with solving it through a series of systematic steps.

1. **Combine like terms:** Sum the terms \(3x\) and \(2x\) to simplify the equation to \(5x = 125\).
2. **Isolate \(x\):** Divide both sides by 5 to solve for \(x\).

\[ x = \frac{125}{5} = 25 \]
These steps show how to efficiently handle a linear equation, simplifying and solving it to find the value of \(x\), which is pivotal for solving the original problem.

Algebraic Expressions

The use of algebraic expressions allows us to handle real-world problems involving ratios conveniently. Once \(x\) is found, substituting back into our expressions \(3x\) and \(2x\) will give the actual numerical contributions.

For the first charity, calculate \(3x = 3 \times 25 = 75\). For the second charity, it is \(2x = 2 \times 25 = 50\). These calculations confirm that each step aligns with the defined ratio, ultimately helping drive our solution.

Expressions like \(3x\) and \(2x\) translate the raw data into a format where the relationship between variables is maintained and clarified.