Problem 2 - Technical Mathematics with Calculus

Question

A company has \(\$ 86,500\) invested in bonds and earns \(\$ 6751\) in interest annually. Part of the money is invested at \(7.4 \%,\) and the remainder at \(8.1 \%,\) both simple interest. How much is invested at each rate?

Step-by-Step Solution

Answer

The amount invested at 7.4% is \(\$45,500\) and at 8.1% is \(\$41,000\).

Step 1: Establish Variables

Let the amount invested at 7.4% be represented by the variable 'x'. Therefore, the amount invested at 8.1% will be \(\$86,500 - x\) because the total amount invested in bonds is \(\$86,500\).

Step 2: Set Up the Equation

Write the equation to represent the total interest earned from both investments. The interest from the first part is \(0.074x\) and from the second part is \(0.081(\$86,500 - x)\). The total interest earned is the sum of these two, which equals \(\$6,751\): \[0.074x + 0.081(\$86,500 - x) = \$6,751\]

Step 3: Simplify the Equation

Distribute the 8.1% across \(\$86,500 - x\) and combine like terms: \[0.074x + 0.081 \times \$86,500 - 0.081x = \$6,751\] \[ -0.007x + 0.081 \times \$86,500 = \$6,751\]

Step 4: Solve for x

Isolate the variable 'x' on one side of the equation to find its value: \[ -0.007x = \$6,751 - 0.081 \times \$86,500\] \[ x = \frac{ - (\$6,751 - 0.081 \times \$86,500) }{ -0.007} \]

Step 5: Calculate the Amount at 7.4%

Compute the value of 'x' to determine how much is invested at 7.4%.

Step 6: Calculate the Amount at 8.1%

Subtract the amount invested at 7.4% (x) from the total invested (\(\$86,500\)) to find how much is invested at 8.1%.

Key Concepts

Simple Interest Formula

Understanding the simple interest formula is essential in financial mathematics, particularly when calculating earnings on investments. Simple interest is calculated by multiplying the principal amount (the initial sum of money), the interest rate (as a decimal), and the time period involved.

Here's the formula for simple interest: \[ \text{Simple Interest} = P \times r \times t \]Where:

• \( P \) represents the principal amount,
• \( r \) stands for the interest rate per time period, and
• \( t \) represents the time the money is invested for.

The resulting figure gives you the total amount of interest earned over the investment period. It's important to note that simple interest does not consider compounding, where interest previously earned is added to the principal to calculate future interest.

Financial Mathematics

Financial mathematics encompasses the tools and techniques used to solve problems related to money, such as loans, annuities, mortgages, investments, and savings. It applies mathematical methods to financial problems, intertwining numbers with real-world financial scenarios.

For instance, when calculating investment earnings like in our textbook example, it's important to apply the concepts of financial mathematics to distribute investments in a way that maximizes returns or minimizes risk. This means understanding how to manipulate formulas, such as the simple interest formula, to determine optimal investment strategies.

Investment Distribution

Investment distribution involves allocating funds among various financial instruments or accounts to achieve a desired investment outcome. In the context of our exercise, the company distributed its investment between bonds with two different interest rates, 7.4% and 8.1%.

This strategy allows investors to balance their portfolio to manage risk, optimize returns, or meet other financial objectives. When distributing investments, it is crucial to perform financial calculations accurately to determine the precise amount to allocate to each investment to achieve the total desired interest earnings.

Solving Linear Equations

Linear Equations in Finance

Solving linear equations is a foundational skill in algebra that finds applications in financial mathematics. A linear equation is an equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be straightforward, such as \( ax + b = 0 \), where 'a' and 'b' are constants.

In our exercise, solving the linear equation is the method we use to find out how much is invested at each interest rate. When we set up the equation based on the provided interest rates and total interest, and then solve for 'x', we effectively distribute the total investment between the two bonds. This requires simplifying the equation and isolating the variable 'x' to find its value.

Steps in Solving

The steps often involve combining like terms, distributing factors, and manipulating the equation to get the variable alone on one side of the equality sign. Once 'x' is isolated, the solution gives us the investment amount for one rate, and subtracting from the total provides the investment for the other rate.