Problem 33 - Elementary Algebra

Question

Nina received an inheritance of \(\$ 12,000\) from her grandmother. She invested part of it at \(6 \%\) interest, and she invested the remainder at \(8 \%\). If the total yearly interest from both investments was \(\$ 860\), how much did she invest at each rate?

Step-by-Step Solution

Answer

Invested $5000 at 6% and $7000 at 8%.

Step 1: Define Variables

Let us define two variables: \( x \) as the amount Nina invested at 6% interest and \( y \) as the amount invested at 8% interest. We know that the total inheritance is \( x + y = 12000 \).

Step 2: Set Up Interest Equations

The total interest from the 6% investment is \( 0.06x \) and from the 8% investment is \( 0.08y \). Given that the total yearly interest from both investments is \$860, we have the equation \( 0.06x + 0.08y = 860 \).

Step 3: Solve the System of Equations - Substitution

From Step 1, express \( y \) in terms of \( x \): \( y = 12000 - x \). Substitute \( y \) in the interest equation: \( 0.06x + 0.08(12000 - x) = 860 \).

Step 4: Simplify the Equation

Distribute the 0.08: \( 0.06x + 960 - 0.08x = 860 \). Combine like terms to get \( -0.02x + 960 = 860 \).

Step 5: Solve for \( x \)

Subtract 960 from both sides: \( -0.02x = -100 \). Divide both sides by -0.02 to find \( x \): \( x = 5000 \).

Step 6: Solve for \( y \)

Now that \( x = 5000 \), substitute back to find \( y \): \( y = 12000 - 5000 = 7000 \).

Step 7: Conclusion

Nina invested $5000 at 6% and $7000 at 8%. Check calculations with original conditions to verify.

Key Concepts

Inheritance Investment

When someone receives an inheritance, it often presents a unique opportunity for financial growth. In Nina's case, inheriting $12,000 allowed her the chance to potentially earn interest over time. Investing an inheritance wisely can maximize the value of the initial sum over the years. - When considering how to invest such funds, it’s essential to analyze different interest rates that financial institutions offer. - The goal is to balance between risk and return to achieve personal financial goals. Generally, higher interest rates can offer a higher return, but may also come with higher risk. In Nina's scenario, she chose to invest in accounts with guaranteed returns, at 6% and 8%, to ensure steady growth. Understanding options for inheritance investment such as bonds, savings accounts, or stock market can help in making informed decisions.

Interest Calculations

Calculating interest is crucial to understanding how much an investment will grow over time. In Nina's case, she invested part of her inheritance at different interest rates and needed to calculate the total interest earned. - Interest can be calculated using the formula: \[ \, \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \, \] For Nina:

• The interest from the portion invested at 6% is \(0.06x\).
• The interest from the portion invested at 8% is \(0.08y\).
• Combined, these investments yielded a yearly interest of $860.

Understanding these calculations helps to plan how much should be invested to meet future financial needs. It's a core part of financial literacy, enabling individuals to grow their wealth responsibly.

Algebraic Equations

Algebraic equations are mathematical statements that use variables to represent numbers. They are crucial in solving for unknown values. In the context of Nina's investment problem, algebraic equations were used to determine how much was invested at each interest rate. The problem was set up as:

• \(x + y = 12000\) - this equation ensures that the total amount invested was all of Nina's inheritance.
• \(0.06x + 0.08y = 860\) - this represents the total interest from both investments.

These equations need to be manipulated in order to isolate the variable, making it possible to solve for unknown quantities. Algebraic equations serve as a powerful tool to model real-world financial situations like investments.

System of Equations

A system of equations is a set of equations with multiple variables, which are solved simultaneously. When dealing with investments like Nina's, systems of equations can help determine how funds should be allocated. In Nina's case:

• The two main equations are \(x + y = 12000\) and \(0.06x + 0.08y = 860\).
• We used substitution: expressing one variable in terms of another, \(y = 12000 - x\), to solve these equations.
• By substituting \(y\) in the interest equation, we find the value of \(x\) and subsequently \(y\).

Learning to solve systems of equations is essential, not just for this investment problem, but also for many other disciplines, like physics, economics, and engineering. It allows us to understand and work through complex interdependencies in a structured way.