Problem 33 - Introductory Algebra for College Students

Question

Involve dual investments. You invest 6000 dollar in two accounts paying \(6 \%\) and \(9 \%\) 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

The amount invested at \(6\%\) is $3600 and the amount invested at \(9\%\) is $2400.

Step 1 Interpret the problem and establish the variables

Let's denote the amount invested at \(6\%\) as \(P_1\) and at \(9\%\) as \(P_2\). According to the problem, the total investment \(P_1 + P_2 = $6000\) and the earned interest is the same for both accounts, hence \(0.06 \cdot P_1 = 0.09 \cdot P_2\).

Step 2 Solve the system of equations

Now, we solve the system of two linear equations. We can rearrange the second equation to isolate one variable. Express \(P_2\) in terms of \(P_1\): \(P_2 = \frac{0.06}{0.09} \cdot P_1 = \frac{2}{3} \cdot P_1\).

Step 3 Substitute into the other equation

Next, substitute \(P_2\) from the derived equation into the first equation of the system: \(P_1 + \frac{2}{3} \cdot P_1 = $6000\). Solving this equation, we find that \(P_1 = $3600\).

Step 4 Find the second unknown

Finally, substitute \(P_1 = $3600\) into the total investment equation \( P_1 + P_2 = $6000\) to find \(P_2 = $2400\).

Key Concepts

Understanding Linear Equations

Linear equations are a fundamental concept in algebra that form the backbone of many types of mathematical problems. They are equations that form a straight line when graphed on a coordinate plane. The general form of a linear equation in two variables is:

• \( ax + by = c \)

where \( a \), \( b \), and \( c \) are constants. In our investment problem, we deal with two linear equations derived from the relationships between the amounts invested and the interest earned.
To solve these equations, we aim to find the values of the unknowns. This involves substitution or elimination methods. For example, by expressing one variable in terms of another, we can solve for both values accurately. The key point in such problems is to clearly identify the relationships expressed by the linear equations and methodically solve them.

The Basics of Interest Calculation

Interest calculation is crucial in understanding investment returns. It is the method of calculating the gain (or cost) of investing money in an account over a period of time. The interest is typically calculated as a percentage of the principal amount. There are two main types of interest calculations: simple and compound.

• Simple Interest: Calculated as \( I = P \, r \, t \), where \( I \) is the interest, \( P \) is the principal, \( r \) is the rate, and \( t \) is the time period.
• Compound Interest: Calculated on both the initial principal and the accumulated interest from previous periods.

In our problem, simple interest is used for both accounts, since each account earns interest at a fixed rate annually. The key takeaway is to always use the correct formula and apply the appropriate rate and time period to determine the earnings.

Deciphering Investment Problems

Investment problems often require setting up and solving equations based on given conditions. Before diving into calculations, clearly define your variables and understand relationships between them. In problems involving multiple investments, identify the sum of investments and specify how returns are calculated.

• Define your unknowns based on interest rates or returns.
• Check if the total investment is divided among accounts with different interest rates, as seen in this scenario.
• Formulate equations based on the stated relationships, such as the equality of interest earned here.

This problem illustrates that a clear understanding of both given data and what is being asked helps to effectively break it down into manageable parts. By organizing information into linear equations, you can solve for unknowns and determine precise investment allocations.