Problem 28 - Intermediate Algebra

Question

Suppose that Gus invested a total of $$\$ 8000$$, part of it at \(8 \%\) and the remainder at \(9 \%\). His yearly income from the two investments was $$\$ 690$$. How much did he invest at each rate?

Step-by-Step Solution

Answer

Gus invested $3000 at 8% and $5000 at 9%.

Step 1: Define Variables

Let \(x\) be the amount Gus invested at \(8\%\), and \(y\) be the amount invested at \(9\%\).

Step 2: Set Up Equations

The total amount invested is the sum of both investments: \(x + y = 8000\). The income from these investments is: \(0.08x + 0.09y = 690\).

Step 3: Solve First Equation for One Variable

From the equation \(x + y = 8000\), express \(y\) in terms of \(x\): \(y = 8000 - x\).

Step 4: Substitute into the Second Equation

Substitute \(y = 8000 - x\) into the second equation: \(0.08x + 0.09(8000 - x) = 690\).

Step 5: Simplify and Solve for x

Distribute \(0.09\) in the equation: \(0.08x + 720 - 0.09x = 690\). Simplify to: \(-0.01x = -30\). Solve for \(x\) to get \(x = 3000\).

Step 6: Solve for y

Use \(y = 8000 - x\). Substitute \(x = 3000\) into the equation: \(y = 8000 - 3000 = 5000\).

Key Concepts

Understanding Systems of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. In many investment problems, like Gus’s, we use systems of equations to find out how much someone invested at different interest rates.
For this exercise, the two key combinations are:

• Total investment: the sum of money Gus invested at both interest rates.
• Annual income from investment: the combined income from these investments.

The system is:

• First equation: Total amount, \(x + y = 8000\)
• Second equation: Total income, \(0.08x + 0.09y = 690\)

By solving these equations, we find out how much Gus put into each investment option. Breaking down these relationships into equations makes complex problems more manageable and reveals the exact amounts invested at each rate.

Deciphering Interest Rates

Interest rates represent the percentage of the invested amount that you earn as income over a period.
These rates are crucial in determining how investments grow over time.

• Gus had two interest rates: \(8\%\) and \(9\%\).

Understanding interest rates helps calculate the income generated. For example:

• Income from an investment at \(8\%\) is given by \(0.08x\).
• Income from an investment at \(9\%\) is given by \(0.09y\).

This shows how interest rates directly impact the calculation of annual earnings from an investment. In real-world situations, knowing how to calculate interest earned helps in making better financial decisions.

Manipulating Algebraic Expressions

When we talk about algebraic expressions, we refer to mathematical phrases made up of numbers, variables, and operators combined.

• For Gus's investment problem, expressions like \(0.08x + 0.09y = 690\) are used.

These expressions help us represent real-world problems in a mathematical form.
Let's dive into how they work:

• Distribution: Apply operations across terms, like \(0.09(8000 - x)\).
• Simplification: Combine or cancel out similar terms, such as rearranging the equation to solve for \(x\).

By manipulating these expressions, we bridge the gap between a written problem and a mathematical solution, ultimately solving for the unknown values. The power of algebra lies in transforming complex scenarios into solvable equations, providing clarity and precision in calculations.