Problem 73 - Algebra for College Students

Question

Solve each problem by using a system of equations. Melinda invested three times as much money at \(11 \%\) yearly interest as she did at \(9 \%\). Her total yearly interest from the two investments was \(\$ 210\). How much did she invest at each rate?

Step-by-Step Solution

Answer

Melinda invested $500 at 9% and $1500 at 11%.

Step 1: Understanding the Problem

We are given that Melinda invested three times as much money at 11% as she did at 9%. We need to find out how much she invested at each rate, given that the total interest from both investments is $210.

Step 2: Assign Variables

Let's assign the variables: let \( x \) be the amount invested at 9%, and \( 3x \) be the amount invested at 11%.

Step 3: Write Interest Equations

Write the equations for the interest from each investment: the interest from 9% is \( 0.09x \), and the interest from 11% is \( 0.11(3x) \). The total interest from both investments is $210.

Step 4: Set Up the Equation

Combine the two interest expressions into one equation: \[ 0.09x + 0.11(3x) = 210 \]

Step 5: Expand and Simplify

Expand the equation: \[ 0.09x + 0.33x = 210 \]. Simplify by combining like terms: \[ 0.42x = 210 \].

Step 6: Solve for x

To solve for \( x \), divide both sides by 0.42: \[ x = \frac{210}{0.42} \]. Calculate \( x \).

Step 7: Calculate Amounts Invested

After calculating, \( x = 500 \). Therefore, Melinda invested $500 at 9%. Since she invested three times as much at 11%, she invested \( 3 \times 500 = 1500 \).

Step 8: Verification

Verify by checking if the total interest of $210 is correct: \[ 0.09 \times 500 + 0.11 \times 1500 = 45 + 165 = 210 \], which matches the given total interest.

Key Concepts

Investment Problems

Investment problems often involve determining how much money is put into different accounts or options, usually with varying interest rates. These types of problems are common in algebra and finance courses because they require an understanding of systems of equations to find solutions.

Typically, the goal is to determine the investments' amounts by looking at the total interest earned from the combined investments. You start by assessing the investments made at each rate. In our example, Melinda invested money at two different rates: 9% and 11%.

To tackle such problems effectively:

• Identify the different investment accounts or options involved.
• Understand the relationships between different investments - sometimes one is a multiple of the other.
• Gather all numerical data provided, such as the interest rates and total gains.
• Create mathematical equations based on these relationships and details.

Investment problems are an excellent way to practice applying mathematical concepts to real-world situations.

Interest Calculation

Interest calculation is crucial in solving investment problems as it helps determine the earnings from an investment over a particular period. The basic concept is to calculate the product of the principal amount (the money invested), the interest rate, and the time period (usually one year for simple interest).

The formula for calculating interest (I) is given by:

• Interest (I) = Principal (P) × Interest Rate (r) × Time (t).

In most investment problems, the time is assumed to be one year unless stated otherwise. Hence, the formula simplifies to I = P \cdot r.

For example, the interest on the amount Melinda invested at 9% can be calculated as:

• Interest = Principal × 9% = \(0.09x\).

Similarly, the interest from the amount invested at 11% is calculated by:

• Interest = Principal × 11% = \(0.11(3x)\).

This straightforward process helps in understanding and solving the investment systems of equations challenges.

Problem-Solving Steps

Solving word problems, especially involving investments and interest calculations, requires a systematic approach. Following clear steps ensures that each part of the problem is logically connected and that the solution is accurate.

Here are key problem-solving steps to keep in mind:

• Understand the Problem: Carefully read the problem to determine what you need to find and what information is given.
• Assign Variables: Assign variables to unknown amounts. For instance, in our example, we used \(x\) for the amount invested at 9%.
• Write Equations: Develop equations based on the relationships between the variables. Break down the interest calculations into algebraic expressions.
• Set Up the System of Equations: Combine all expressions into one or more equations that model the total situation.
• Solve the Equations: Use algebraic techniques to solve for the variables. Simplifying and solving equations are critical skills.
• Verify Your Solution: Accurately plugging your solution back into the context of the original problem confirms its correctness.

By following these steps, solving investment problems becomes a systematic and manageable task.

Algebraic Equations

Algebraic equations are the foundation for solving investment problems with systems of equations. They represent the relationships between the amounts invested and the total interest generated.

For investment problems like the one presented, you'll often set up equations using interest calculations:

• Start with the individual interest expressions, for example: \(0.09x\) for the 9% rate and \(0.11(3x)\) for the 11% rate.
• Add these amounts to reflect their total contribution to the final problem's condition, such as a total interest of $210: \[0.09x + 0.11(3x) = 210\].

Solving these algebraic equations involves:

• Expanding terms where necessary, such as distributing multipliers within expressions.
• Simplifying the equation by combining like terms.
• Isolating the variable on one side of the equation by performing arithmetic operations.

The solution of these algebraic equations yields the values of variables. These values provide the answer to how much was invested at each interest rate. Mastery of algebraic techniques in forming and solving equations is pivotal in tackling real-world financial problems with confidence.