Problem 39 - College Algebra

Question

Your grandmother needs your help. She has \(\$ 50,000\) to invest. Part of this money is to be invested in noninsured bonds paying \(15 \%\) annual interest. The rest of this money is to be invested in a government-insured certificate of deposit paying \(7 \%\) annual interest. She told you that she requires \(\$ 6000\) per year in extra income from both of these investments. How much money should be placed in each investment?

Step-by-Step Solution

Answer

The grandmother should invest approximately \$25000 in noninsured bonds and \$25000 in a government-insured certificate of deposit.

Step 1: Define the Variables

Let's denote the amount invested in noninsured bonds as \(x\) and that in a government-insured certificate as \(y\).

Step 2: Formulate Equations

Using the fact that the total amount invested is $50000, and the interest from both sums up to $6000, two equations can be formed: \[x + y = 50000\] (Eqn 1) and \(0.15x + 0.07y = 6000\) (Eqn 2)

Step 3: Solve the Equations

To solve these equations, we can use the substitution or elimination method. For instance, solve Equation 1 for \(x\): \(x = 50000 - y\) and substitute this in Equation 2:

Step 4: Substitute and Solve

By substituting \(x\) into Equation 2, we get \(0.15(50000 - y) + 0.07y = 6000\). Solving this first equation gives \(y \approx 25000\). Substituting back this value in Equation 1 gives \(x = 50000 - 25000 = 25000\)

Step 5: Verify the Solution

We can substitute \(x=25000\) and \(y=25000\) into Equation 2 to check the validity. We can see that the left-hand side will indeed be $6000, which is the desired yearly return.

Key Concepts

Algebraic Equations

Algebraic equations are the foundation of many problem-solving scenarios, especially in financial contexts. They allow us to represent real-world situations with mathematical expressions where we can solve for unknown values. The problem at hand required us to find out how much money should be invested in two different types of investments to achieve a specific annual income. This was represented by two unknowns, designated as variables, which in this case were given the placeholders of x and y.

Understanding how to set up algebraic equations is critical. In our problem, we used the total amount of investment and the required annual income to form a system of equations. Remember, the key is to express the relationships and constraints given in the problem using algebraic expressions. If the total amount of money to invest is \$50000\$$, we set up one equation, and another to represent the desired annual income from the investments. Here, these became our guiding equations to find the solution. Solving algebraic equations usually involves isolating one variable and substituting its value into another equation, as demonstrated in the step-by-step solution.

Financial Mathematics

Financial mathematics encompasses the concepts and techniques used to solve problems related to money, investments, and finance. This field combines the logic of mathematics with the realities of financial decisions and planning. In the grandmother's investment scenario, financial mathematics enabled us to calculate the distribution of funds between different investment options offering varying interest rates.

Understanding the relationship between principal, interest rate, and time is crucial when solving such problems. In our exercise, we applied this knowledge to determine the appropriate amount to invest at different interest rates to achieve a specific amount of interest income per year. It's also important to note that financial mathematics is not just about computation; it also requires careful consideration of factors such as risk, return, and security, all of which play a role in deciding the best investment strategy.

Investment Strategies

An investment strategy is a planned approach to investing that takes into account various factors including the investor's financial situation, risk tolerance, and investment goals. In the textbook problem, the grandmother required a strategy to allocate her funds into two types of investments. One was a higher-risk, higher-interest noninsured bond and the other was a lower-risk, government-insured certificate of deposit.

Diversification is a key principle often recommended in investment strategies, which means spreading investments across different assets to minimize risks. By asking for a certain income from her total investments, grandma was essentially utilizing an income-focused strategy. This approach requires careful consideration of how much to invest in each option to not only reach the income goal but also to match her risk preference. Understanding different investment strategies can greatly assist individuals in making informed financial decisions, particularly when they are faced with various investment choices, as was in our solved problem.

System of Linear Equations

A system of linear equations is a set of two or more linear equations containing two or more variables. The goal is to find a common solution that satisfies all equations simultaneously. In our investment problem, we used a system of linear equations to figure out the distribution of funds to meet the desired interest income.

The system consisted of simple equations depicting the sum of the investments and the total interest yielded. These linear equations can be solved using methods such as substitution, elimination, or graphing. For ease and precision, we opted for the substitution method. The key part was to express one variable in terms of the other using one equation and then substituting into the second equation to find the values that satisfy both. The capability to solve such a system is an invaluable skill, particularly in dealing with financial decision-making problems where several variables and constraints are at play.