Problem 74 - Intermediate Algebra

Question

\(\quad\) A retired teacher has a choice of two investment plans: An insured fund that pays \(11 \%\) interest A risky investment that pays a \(13 \%\) return If the same amount invested at the higher rate would general an extra \(\$ 150\) per year than it would at the lower rate, how much does the teacher have to invest?

Step-by-Step Solution

Answer

The teacher needs to invest $7500.

Step 1: Understanding the Problem

The teacher has two investment options: one with an interest rate of 11% and another with a 13% interest rate. If the teacher invests the same amount in both options, the investment at the higher rate yields an extra $150 annually compared to the lower rate. Our goal is to find the amount the teacher invests.

Step 2: Setting Up the Equations

Let's denote the amount to be invested by the teacher as \( x \). The return from the 11% option is \( 0.11x \) and from the 13% option is \( 0.13x \). The difference in income from these two investments is given to be $150. Thus, we can set up the equation \( 0.13x - 0.11x = 150 \).

Step 3: Simplifying the Equation

Simplify the equation \( 0.13x - 0.11x = 150 \) to find the unknown. You get \( 0.02x = 150 \).

Step 4: Solving for the Investment Amount

To solve for \( x \), divide both sides of the equation by 0.02: \( x = \frac{150}{0.02} \).

Step 5: Calculating the Result

Calculate \( x = \frac{150}{0.02} = 7500 \). Thus, the teacher needs to invest \( \$7500 \) in either investment to meet the condition given in the problem.

Key Concepts

Investment Calculations

Investment calculations are the foundation of understanding how much you should invest and what returns to expect. In our problem, we dealt with two investment options: - An insured fund with an 11% interest rate - A risky investment with a 13% interest rate To figure out how much a retired teacher should invest, the problem sets a scenario where the extra income from the higher interest rate needs to be $150 more than the lower one. This involves selecting an appropriate amount to reach this goal of additional earnings. The core of investment calculations is accurately determining potential returns. This involves: - Identifying the potential interest rate or return - Computing the actual dollar gain from that percentage - Comparing the differing outcomes to find extra income, how long it might take to achieve goals, or in our case, how much must be invested. This process helps you not just solve word problems, but also make informed decisions about personal or business investments.

Algebraic Equations

Algebra provides a powerful tool for solving interest rate problems, like the one faced by our teacher. In this example, algebraic equations are used to set up and solve the problem.We began by letting the amount the teacher invests be represented by a variable, such as \( x \). Using the interest rates:- The return from the 11% interest is represented as \( 0.11x \).- Similarly, the return from the 13% rate is \( 0.13x \).The difference equation \( 0.13x - 0.11x = 150 \) demonstrates the equation setup. Simplifying this step-by-step, we derived \( 0.02x = 150 \).Algebra doesn't just give us a way to express the problem; it allows us to systematically find solutions:- We isolate the variable \( x \) by performing operations on both sides.- Finally, dividing by 0.02 gives us the amount to invest, which resolves the given scenario.This shows how algebra seamlessly helps turn a word problem into a solvable equation.

Financial Mathematics

Financial mathematics connects algebraic skills to real-life financial decisions. Here, we used it to solve a practical problem, ensuring our finished calculations were correct and meaningful. By setting differences in expected returns, financial mathematics lets you:

• Calculate necessary investments to achieve desired returns
• Plan for future financial goals with precision
• Compare different investment options to find the most suitable one

In our example, understanding financial math allowed insights into: - The impact of small percentage differences over time - How these small differences can roll into significant earnings or shortfalls Ultimately, mastering financial mathematics is crucial as it serves as a lens through which we can view and analyze financial scenarios, ensuring informed decision-making no matter the context.