Problem 34 - Introductory Algebra for College Students

Question

Involve dual investments. You invest 7200 dollar in two accounts paying \(8 \%\) and \(10 \%\) annual interest. At the end of the year, the accounts earn the same interest. How much was invested at each rate?

Step-by-Step Solution

Answer

$4000 was invested at \(8\%\) and $3200 was invested at \(10\%\)

Step 1: Define the variables

Let's denote the investment at a rate of \(8\%\) as \(x\) and the investment at a rate of \(10\%\) as \(y\). The total investment is $7200.

Step 2: Set up the equations

Based on the information given in the problem, two equations can be set up. The first equation will be from the total money invested, so: \(x + y = 7200\). The second equation will be set up from the interest earned. Since both accounts yielded the same interest, we can write: \(0.08x = 0.10y\).

Step 3: Solve the system of equations

We can solve this system of equations using a methodology such as substitution or elimination. To make things easier, let's solve the second equation for \(x\): \(x = 1.25y\). Substituting this into the first equation, we get: \(1.25y + y = 7200\), which simplifies to \(2.25y = 7200\). Solving for \(y\), we find \(y = 3200\). Substituting this value back into the equation for \(x\), we find \(x = 4000\).

Step 4: Interpret the results

The solution means that $4000 was invested at a rate of \(8\%\) and $3200 was invested at a rate of \(10\%\)

Key Concepts

Algebraic Investment Problems

Algebraic investment problems are a practical application of algebra where we solve for unknown variables based on given financial scenarios. These problems often involve calculating the amounts of money invested at different interest rates, the returns on those investments, or how these investments change over time.

In our exercise, two amounts with a total sum of $7200 are invested at two different interest rates. The key to solving these types of problems lies in setting up equations based on the provided data, which typically express relationships between the amounts invested, interest rates, and returns. It's important to clearly define your variables at the outset - in this case, we used x and y to represent the two unknown investment amounts. Once the variables are defined, the problem becomes much easier to manage, ensuring that students can follow along with the logical progression from problem to solution.

Solving Simultaneous Equations

Simultaneous equations, also known as systems of equations, consist of multiple equations that share common variables. Solving such a system means finding values for the variables that satisfy all equations simultaneously.

Methods of Solving

There are several methods to solve simultaneous equations, including substitution, elimination, and graphical methods. In our example, the substitution method was used: we first solved one equation for one variable and then substituted that expression into the other equation.

Importance in Investment Problems

Mastering simultaneous equations is crucial in solving algebraic investment problems as it equips students with the ability to dissect complex financial scenarios into simpler, solvable equations, providing clear insights into how different financial factors interact with each other.

Interest Rate Calculations

Interest rate calculations involve determining the amount of interest that will be earned or paid on an investment or loan. This concept is critical when it comes to managing personal finances or making business decisions.

There are different types of interest rates, such as simple interest and compound interest. In algebraic investment problems, you typically deal with simple interest, calculated as a percentage of the principal amount.

Application in Our Example

In the solved problem, we're looking at two investments with different interest rates yielding the same return. Understanding how to calculate interest - using the formula Interest = Principal × Rate × Time - allows students to set up the second key equation, 0.08x = 0.10y, thereby revealing the relationship between the two investments and their rates.