Problem 79 - College Algebra with Modeling and Visualization

Question

Investment \(A\) sum of \(\$ 5000\) is invested in three mutual funds that pay \(8 \%, 11 \%,\) and \(14 \%\) annual interest rates. The amount of money invested in the fund paying \(14 \%\) equals the total amount of money invested in the other two funds, and the total annual interest from all three funds is \(\$ 595\) (a) Write a system of equations whose solution gives the amount invested in each mutual fund. Be sure to state what cach variable represents. (b) Solve the system of equations.

Step-by-Step Solution

Answer

Invest $1000 at 8%, $1500 at 11%, and $2500 at 14%.

Step 1: Define Variables

Let \( x \) be the amount invested at 8%, \( y \) be the amount invested at 11%, and \( z \) be the amount invested at 14%.

Step 2: Write Equations Based on Total Investment and Interest

The total amount invested is \(5000, so the first equation is:\[x + y + z = 5000\]The total annual interest from all three funds is \)595, so the second equation is:\[0.08x + 0.11y + 0.14z = 595\]

Step 3: Write Equation Based on Investment Condition

The amount invested in the 14% fund equals the amount invested in the other two funds combined, creating the third equation:\[z = x + y\]

Step 4: Solve the System of Equations

Substitute \(z = x + y \) from the third equation into the first two equations:1. \(x + y + (x + y) = 5000\) simplifies to:\[\2x + 2y = 5000 \ x + y = 2500\]2. \(0.08x + 0.11y + 0.14(x + y) = 595\) simplifies to:\[\0.08x + 0.11y + 0.14x + 0.14y = 595 \ 0.22x + 0.25y = 595\]Now we have two equations:\[\x + y = 2500 \0.22x + 0.25y = 595\]Solve the first equation for \(x\):\[x = 2500 - y\]Substitute \(x = 2500 - y\) into the second equation:\[\0.22(2500 - y) + 0.25y = 595 \550 - 0.22y + 0.25y = 595 \0.03y = 45 \y = 1500\]Substitute \(y = 1500\) back into \(x = 2500 - y\):\[\x = 2500 - 1500 = 1000\]Substitute \(x = 1000\) and \(y = 1500\) into \(z = x + y\):\[\z = 1000 + 1500 = 2500\]

Step 5: Verify the Solution

Check the solution to ensure it satisfies all original equations:1. \(x + y + z = 1000 + 1500 + 2500 = 5000\) is correct.2. \(0.08(1000) + 0.11(1500) + 0.14(2500) = 80 + 165 + 350 = 595\) is correct.3. \(z = x + y\) verifies as \(2500 = 1000 + 1500\). All equations are satisfied.

Key Concepts

Investment Problems

Investment problems often involve determining the allocation of funds across different assets to meet specific objectives, such as maximizing returns or meeting interest payment goals. In the given exercise, an amount of $5000 is distributed among three mutual funds, each offering different interest rates. The complexity arises from conditions such as keeping the sum allocated in one higher interest fund equal to the combined amount invested in others.

To solve such problems efficiently, forming a system of equations is essential:

• Define variables to represent investments in each fund.
• Translate the problem's conditions into mathematical equations.
• Ensure the equations reflect both the total investment amount and the constraints on individual allocations.

By solving these equations, the amount allocated in each fund can be determined. It's crucial to comprehend how these equations interrelate, giving insights into managing investments wisely.

Mutual Funds

Mutual funds are investment vehicles that pool money from multiple investors to invest in a diversified portfolio of assets like stocks and bonds. Each investor owns shares that represent a part of these holdings. The variety in mutual funds allows investors to optimize their portfolios based on risk tolerance and expected returns.

In solving the original exercise, understanding mutual funds helps appreciate the impact of distributing investments across funds with varying interest rates:

• An 8% fund is relatively low-risk with moderate returns.
• An 11% fund offers balanced risk and reward.
• A 14% fund is likely higher risk, promising higher returns.

Investors must assess their strategies to align with financial goals, using the diversity of mutual funds to mitigate risks while still aiming for fruitful returns. This exercise highlights how effectively managing mutual fund investments can generate desired income levels.

Interest Rates

Interest rates are the percentage return on investments over a given period and are pivotal in investment decision-making. They directly influence how much earnings one might expect from a mutual fund. In our exercise, different interest rates alter the total interest gained from the investment and illustrate choosing among potential returns:

• A lower interest rate, like 8%, means safer, but less lucrative investments.
• Mid-range rates, e.g., 11%, typically offer balanced options.
• High rates, like 14%, come with increased risk but also lucrative returns.

Understanding interest rates is key to formulating the system of equations in the exercise. Here, annual interests play a role in how funds are distributed among the available options, paving the way to achieve the total expected annual return ($595 in this case). With this understanding, investors can properly strategize fund distribution to maximize returns while considering their risk appetite.