Problem 126 - College Algebra with Modeling and Visualization

Question

A student invests \(\$ 5000\) at two annual interest rates, \(5 \%\) and \(7 \% .\) After 1 year the student receives a total of \(\$ 325\) in interest. How much did the student invest at each interest rate?

Step-by-Step Solution

Answer

$1250 at 5% and $3750 at 7%.

Step 1: Understanding the Problem

We need to determine how much money was invested at each of the given interest rates (5% and 7%) such that the total interest received is $325 after one year.

Step 2: Setting Up Variables

Let the amount invested at 5% be denoted as \(x\). Then, the amount invested at 7% will be \(5000 - x\), since the total investment is $5000.

Step 3: Formulating the Interest Equations

The interest from the amount \(x\) invested at 5% is \(0.05x\). The interest from the amount \(5000 - x\) invested at 7% is \(0.07(5000 - x)\). The total interest from both investments is $325, so the equation is: \[ 0.05x + 0.07(5000 - x) = 325. \]

Step 4: Solving the Equation

First, simplify the equation: \(0.05x + 350 - 0.07x = 325\). Then combine like terms to get: \(-0.02x + 350 = 325\). Subtract 350 from both sides: \(-0.02x = -25\). Divide both sides by \(-0.02\) to find \(x\): \(x = 1250\).

Step 5: Determining Both Investment Amounts

Since \(x = 1250\), the amount invested at 5% is $1250. Therefore, the amount invested at 7% is \(5000 - 1250 = 3750\).

Step 6: Verification

Verify by calculating the interest: 5% of \(1250 is \)62.50 and 7% of \(3750 is \)262.50. The total interest is indeed \(62.50 + 262.50 = 325\), which matches the given condition.

Key Concepts

Interest Rates

Interest rates are a crucial concept in the realm of finance and mathematics, especially when it comes to investments. Simply put, an interest rate is the percentage of principal that is paid as a reward for lending or borrowing money. It can be considered akin to a rental fee for using money over time. There are different types of interest rates, most commonly categorized into simple and compound interest rates.

In this problem, we are dealing with simple interest, which is calculated as a straightforward percentage of the principal amount invested or loaned. The formula used for calculating simple interest is:

• Simple Interest = Principal × Rate × Time.

Here, the 'Rate' is expressed as a decimal. For example, a 5% interest rate is represented as 0.05.

Understanding how interest rates are applied helps investors decide how much to invest and at what rate to achieve a desired return. In scenarios where different rates are involved, like our original exercise, understanding how each rate contributes to the total interest can guide decision-making.

Investment Problems

Investment problems often involve calculating the amount invested across different options to reach a target outcome. These types of problems require understanding not only the interest rates but also how to set up equations to solve for unknowns. The problem from our exercise is a classic example where we need to determine the distribution of an initial investment across different rates.

To tackle investment problems effectively, one should:

• Identify the total amount of investment.
• Understand that the different investments will add up to the total initial investment.
• Calculate interest from each portion of the investment separately, based on the respective rates.
• Ensure the sum of these interests matches the total expected interest.

Breaking down the problem into these steps makes it easier to formulate a system of equations, which can then be solved to find the desired investment amounts at each rate. Being systematic is key to solving investment problems accurately and efficiently.

Linear Equations

Linear equations form the backbone of solving many mathematical problems, including those involving investments. In the given exercise, setting up a linear equation allows one to represent the relationship between the amounts invested at different interest rates.

Here's an important feature of linear equations:

• They can be represented in the form \( ax + b = c \), where \( x \) is the variable representing the unknown amount, and \( a \), \( b \), and \( c \) are constants.

For investment problems, the constants \( a \) and \( b \) are usually derived from the interest rates and total principal amounts, while \( c \) is the total interest received.

Solving linear equations involves:

• Simplifying the equation by combining like terms.
• Isolating the variable of interest on one side of the equation.
• Performing actions like addition, subtraction, multiplication, or division to solve for the variable.

In our exercise, simplifying and solving the linear equation helped us determine how much was invested at each rate, providing clarity and precision to the investment problem.