Problem 58 - Algebra for College Students

Question

Set up an algebraic equation and solve each problem. An inheritance of \(\$ 300,000\) is to be divided between a son and the local heart fund in the ratio of 3 to 1 . How much money will the son receive?

Step-by-Step Solution

Answer

The son will receive $225,000.

Step 1: Understand the Ratio

The problem states that the inheritance is to be divided between a son and the heart fund in the ratio of 3 to 1. This means for every 3 parts the son receives, the heart fund receives 1 part.

Step 2: Define Variables

Let the total parts be 4 (3 parts for the son + 1 part for the heart fund). Define the son's share as 3 parts and the heart fund's share as 1 part.

Step 3: Set Up the Equation

If the total inheritance is represented by 4 parts and these parts together equal $300,000, we can represent one part of the inheritance as \( x \). Therefore, the equation is: \[ 4x = 300,000 \]

Step 4: Solve for x

To find the value of one part, divide both sides of the equation by 4:\[ x = \frac{300,000}{4} \] Simplifying gives:\[ x = 75,000 \]

Step 5: Calculate the Son's Share

The son receives 3 parts of the inheritance. Multiply \( x \) by 3 to find the son's share:\[ 3x = 3 \times 75,000 = 225,000 \]

Key Concepts

Ratios in Algebra

Ratios are a fundamental part of algebra, especially when you need to divide quantities into specific proportions. In this exercise, we are working with a ratio of 3 to 1. The ratio dictates how the inheritance is divided between the son and the heart fund. A ratio of 3 to 1 means for every four parts, 3 parts are allocated to the son, and 1 part goes to the fund. Think of it as dividing a pie into 4 slices: 3 slices for one person and 1 slice for another. These ratios are expressed as fractions in the algebraic equation. To find the value of each part, we calculate it using the total amount available (in this case, $300,000). Once you understand the basics of ratios, you can apply them to various problems, ensuring you distribute values according to the specified proportion.

Financial Algebra Problems

Financial algebra problems often involve understanding and manipulating numbers that relate to real-world financial situations. In our inheritance problem, financial algebra allows us to use the ratio given to divide the inheritance accurately. Key components to identify in financial problems: * Total amount available: Here, it's $300,000. * Ratio of distribution: 3 to 1 between two parties. * Variables: Use variables to represent unknowns, like parts of the inheritance. By translating verbal problems into algebraic equations, you can solve complex financial problems step by step. This skill is crucial for real-world applications, such as budgeting and financial planning.

Solving Equations Step by Step

Solving equations is a step-by-step process. Breaking down problems makes them more manageable, preventing mistakes and confusion. For the inheritance issue, follow these steps: 1. **Understand the Problem**: Identify the total to be divided and the ratios given. 2. **Define the Variables**: Use variables like \( x \) to represent unknown parts. Here, \( 4x = 300,000 \) based on the ratio.3. **Set Up the Equation**: Write the equation to reflect the problem, ensuring it aligns with the ratio.4. **Solve for the Variable**: Isolate \( x \) to find the value of one part. In this example, \( x = 75,000 \).5. **Interpret the Results**: Multiply \( x \) by 3 to find the son's share, resulting in \( 225,000 \).Taking it slow and maintaining organization are key to successfully solving any algebraic equation.