Problem 22 - Algebra and Trigonometry

Question

Investments If Ben invests \(\$ 4000\) at 4\(\%\) interest per year, how much additional money must he invest at 5\(\frac{1}{2} \%\) annual interest to ensure that the interest he receives each year is 4\(\frac{1}{2} \%\) of the total amount invested?

Step-by-Step Solution

Answer

Ben needs to invest an additional \(\$2000\).

Step 1: Define Variables

Let \( x \) be the additional amount Ben must invest at \(5\frac{1}{2} \% \) interest. The total amount invested is \(4000 + x\).

Step 2: Interest Calculation for Given Investments

The interest from the \(4000\) investment is \(0.04 \times 4000 = 160\), and the interest from the \(x\) investment is \(\frac{11}{200} \times x = 0.055x\).

Step 3: Equation Setup for Total Interest

The total interest Ben receives is \(160 + 0.055x\). We need this to equal \(4.5\%\) of the total investment, which is \(0.045 \times (4000 + x)\).

Step 4: Equation Formation

Set the total interest expression equal to \(4.5\%\) of the invested amount: \[160 + 0.055x = 0.045 \times (4000 + x)\]

Step 5: Simplify the Equation

Calculate the right side: \(0.045 \times (4000 + x) = 180 + 0.045x\). The equation becomes: \[160 + 0.055x = 180 + 0.045x\]

Step 6: Solve for \(x\)

Rearrange the equation to find \(x\): \[0.055x - 0.045x = 180 - 160\]\[ 0.01x = 20\] Thus, \(x = \frac{20}{0.01} = 2000\).

Step 7: Conclusion

Therefore, Ben needs to invest an additional \(\$2000\) at \(5\frac{1}{2} \%\) interest.

Key Concepts

Interest Rate

Interest rates are a fundamental concept in finance and investments. They represent the cost of borrowing money or the reward for saving or investing. In this exercise, Ben is dealing with fixed interest rates, which are percentages of the principal amount (the initial sum of money invested or loaned).

• A 4% interest rate on Ben's initial investment of $4000 means that each year, he earns 4% of $4000, which is $160.
• The second investment offers an interest rate of 5.5%, a higher percentage, which brings in more interest per dollar invested.

These rates allow Ben to receive rewards for his investments. By blending returns from both investments, Ben aims to meet his desired return rate of 4.5% on his entire capital. This process involves finding the right balance between two different interest rates to arrive at an average return that meets his financial goals.

Investment Strategy

An effective investment strategy involves planning how to allocate resources to achieve desired financial outcomes. In Ben's case, he aims to have a total average return of 4.5% on his total investment. Ben has several components to consider:

• He has a solid base investment of $4000 at a fixed 4% return rate.
• He needs to decide how much more to invest at a higher interest rate (5.5%) to reach his target average return.

Thus, Ben's strategy requires calculating the additional amount needed to invest at the higher rate to create a blended interest rate. This type of strategic decision-making ensures that he maximizes the potential for return while balancing risk and reward.

Algebraic Equations

Algebraic equations are essential tools in solving real-life problems, especially in finance. In this exercise, algebra helps us express the relationship between investments and interest earned in mathematical form.

• Step 1 involves introducing a variable, \( x \), to represent the unknown additional amount Ben needs to invest at 5.5% interest.
• Using known values and algebra, we set up an equation to determine \( x \): \[160 + 0.055x = 0.045 \times (4000 + x)\]

By solving this equation, Ben determines the exact amount he needs to invest additionally. It involves distributing, isolating variables, and solving for \( x \), resulting in an easy-to-follow series of steps that reflect how algebra aids financial decision-making.