Problem 51 - Precalculus with Limits

Question

Investment Portfolio A total of \(\$ 24,000\) is invested in two corporate bonds that pay 3.5\(\%\) and 5\(\%\) simple interest. The investor wants an annual interest income of \(\$ 930\) from the investments. What amount should be invested in the 3.5\(\%\) bond?

Step-by-Step Solution

Answer

To find the amount invested in the 3.5% bond, solve the equation to find the value of x.

Step 1: Set up the equation for the invested amounts

Let's call the amount to be invested in the 3.5% bond x. Therefore the amount to be invested in the 5% bond is ($24,000 - x).

Step 2: Set up the equation for the interest amounts

The total interest is the sum of the interest from both investments. This means 0.035*x (interest from the 3.5% investment) + 0.05*($24,000 - x) (interest from the 5% investment) = $930.

Step 3: Solve the equation

Now, solve the equation above. This will result in a linear equation in x that can be solved by standard methods.

Key Concepts

Interest Calculation

Understanding the concept of interest calculation is crucial when dealing with any form of investment or loan. The interest can be thought of as the cost of using someone else's money. In the context of investments, it's the amount earned on the invested capital.

There are two main types of interest: simple interest and compound interest. Simple interest is calculated only on the principal amount, or the initial sum of money that is put into the investment. In contrast, compound interest is calculated on the principal amount and also on the accumulated interest of previous periods.

The formula for calculating simple interest is given by: \[I = P \times r \times t\]where I represents the interest earned, P is the principal amount, r is the annual interest rate in decimal form, and t is the time the money is invested for, in years.

Applying this formula to solve textbook exercises, students need to understand each component. For instance, an annual interest rate of 3.5% should be converted into a decimal for calculation purposes as 0.035. This helps in setting up the correct equations for interest-related investment problems, like the one described in the exercise.

Linear Equation

In mathematics, a linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be represented in the form: \[ax + b = 0\]where a and b are constants, and x represents the variable. They are called 'linear' because when plotted on a graph, they produce a straight line.

Solving linear equations is a fundamental skill in algebra. The solution involves finding the value of the variable that makes the equation true. In the context of the original exercise, step 2 and 3 revolve around setting up and solving a linear equation through a process of isolating the variable: rearranging the terms, combining like terms, and finally dividing by the coefficient of the variable.

The problem provided is an example of how linear equations are used in financial mathematics to determine the distribution of funds within an investment portfolio. By solving the linear equation set up in step 2, one can determine how much money to invest in each bond to meet the desired interest income, showcasing the practical applications of algebra in real life.

Investment Portfolio

An investment portfolio is a collection of assets owned by an individual or institution designed to achieve a desired financial goal. This can encompass a wide range of asset classes, such as stocks, bonds, mutual funds, and real estate, among others. The choice of investments in a portfolio depends on various factors, including risk tolerance, time horizon, and financial goals.

When building an investment portfolio, diversification is a key concept. This involves allocating investments among different financial instruments, industries, and other categories to minimize the risk of loss. A well-diversified portfolio can provide a more stable return because various investments will react differently to the same economic event.

In our original exercise, the investment portfolio comprises two different corporate bonds with varying interest rates. The investor aims to achieve a target interest income, which requires strategic allocation of capital among the bonds. By calculating the appropriate amount to invest in the 3.5% and 5% bonds, the investor manages the portfolio to meet the desired financial outcome, illustrating how mathematical concepts directly assist in making informed investment decisions.