Problem 39 - Elementary Algebra

Question

If \(\$ 3000\) is invested at \(9 \%\) interest, how much money must be invested at \(12 \%\) so that the total return for both investments averages \(11 \%\) ?

Step-by-Step Solution

Answer

Invest \$6000 at 12\%.

Step 1: Understand the Problem

We need to find how much money, say \(x\) dollars, should be invested at \(12\%\) such that the average return on both investments equals \(11\%\). We are given: \$3000 is invested at \(9\%\).

Step 2: Set Up the Equation for Total Interest

Find the interest from both investments. The interest from the \$3000 investment is \(3000 \times \frac{9}{100} = 270\). The interest from the \(x\) dollar investment at \(12\%\) is \(x \times \frac{12}{100}\).

Step 3: Formulate the Average Total Return Equation

Let the total invested amount be \(3000 + x\). The desired total return is \(11\%\) of this total investment: \((3000 + x) \times \frac{11}{100}\). Equate this to the sum of the individual interests: \[270 + \frac{12}{100}x = \frac{11}{100}(3000 + x)\]

Step 4: Solve the Equation

Distribute and simplify the right side: \[270 + 0.12x = 330 + 0.11x\]Subtract \(0.11x\) from both sides: \[270 + 0.01x = 330\]Subtract 270 from both sides: \[0.01x = 60\]Finally, solve for \(x\): \[x = \frac{60}{0.01} = 6000\]

Step 5: Conclusion

The amount of money that needs to be invested at \(12\%\) to achieve an average return of \(11\%\) is \$6000.

Key Concepts

Understanding Average Return

Understanding the concept of average return is essential when dealing with multiple investments. Average return represents the mean percentage of profit you gain from different investments over a set period. It's like finding an average of numbers. When you average percentages from two investment sources, you're balancing their individual contributions to meet a target goal.
For example: if you invest in two different accounts with different interest rates, the average return is the sum of the returns divided by the total investment. This is useful to determine how well your combined investment strategies are performing.

Calculating Interest Rates

Interest rates are basically the cost of money. These rates tell you how much extra you earn on your invested amount over a period, usually annually. Knowing how to calculate interest is a key skill in financial planning.

The formula to find simple interest is:

• Interest = Principal x Rate x Time

In our exercise, we observe the interest on $3000 invested at 9% and $x at 12%. Calculating these is about multiplying the principal by the rate to find how much return you receive from each investment.

Basics of Financial Mathematics

Financial mathematics involves using math to solve problems related to money, such as investment growth or loans. It relies on formulas and algebra to predict or calculate future financial scenarios.
This area of math helps us understand how different rates and investments can affect income over time. It allows you to strategically plan for better returns on your investments by understanding how different factors like time and interest rates influence growth.

Solving Algebra Problems in Finance

Algebra problem-solving in finance is like uncovering the unknowns in your investment story. You formulate equations based on known facts to solve for unknown variables. This exercise serves as a perfect example.
When figuring out the needed investment amount for a desired average return, we set up an equation that balances the contributions from two different rates. By using algebraic manipulations like distribution, simplification, and solving for variables, we determined that investing $6000 at 12% was necessary to reach our target.