Problem 64 - College Algebra with Corequisite Support

Question

For the following exercises, solve the system for \(x, y,\) and \(z\). You inherit one hundred thousand dollars. You invest it all in three accounts for one year. The first account pays \(4 \%\) compounded annually, the second account pays \(3 \%\) compounded annually, and the third account pays \(2 \%\) compounded annually. After one year, you earn $$\$ 3,650$$ in interest. If you invest five times the money in the account that pays \(4 \%\) compared to \(3 \%,\) how much did you invest in each account?

Step-by-Step Solution

Answer

Invest: $75,000 at 4%, $15,000 at 3%, $10,000 at 2%.

Step 1: Define the Variables

Let \( x \) be the amount invested in the account with 4% interest, \( y \) in the account with 3% interest, and \( z \) in the account with 2% interest. We need to find the values of \( x, y, \) and \( z \) that satisfy the given conditions.

Step 2: Set Up the Equation for Total Investment

The total amount invested is $100,000. Therefore, we have the equation: \[ x + y + z = 100,000 \]

Step 3: Set Up the Equation for Total Interest

The total interest earned after one year is $3,650. Therefore, we write the equation for the interest earned from each account: \[ 0.04x + 0.03y + 0.02z = 3,650 \]

Step 4: Relation Between Amounts in Accounts

According to the problem, the amount invested at 4% is five times the amount invested at 3%. Thus, we have: \[ x = 5y \]

Step 5: Substitute and Simplify

Substitute \( x = 5y \) into both previous equations:1. The total investment equation becomes \[ 5y + y + z = 100,000 \] which simplifies to \[ 6y + z = 100,000 \]2. The interest equation becomes \[ 0.04(5y) + 0.03y + 0.02z = 3,650 \] which simplifies to \[ 0.2y + 0.03y + 0.02z = 3,650 \], or \[ 0.23y + 0.02z = 3,650 \]

Step 6: Solve the System of Equations

From the equation \( z = 100,000 - 6y \), substitute \( z \) in the interest equation: \[ 0.23y + 0.02(100,000 - 6y) = 3,650 \]Simplify and solve for \( y \):\[ 0.23y + 2,000 - 0.12y = 3,650 \]\[ 0.11y = 1,650 \]\[ y = 15,000 \]Now find \( x \) using \( x = 5y \):\[ x = 5(15,000) = 75,000 \]And \( z \) using \( z = 100,000 - 6(15,000) \):\[ z = 100,000 - 90,000 = 10,000 \]

Step 7: Verify the Solution

Check if these values satisfy the interest equation: \( 0.04(75,000) + 0.03(15,000) + 0.02(10,000) = 3,000 + 450 + 200 = 3,650 \). The values satisfy all equations, confirming the solution is correct.

Key Concepts

Compound Interest

Compound interest is an essential concept in finance and investing, and it plays a crucial role in understanding how investments grow over time. Unlike simple interest, which is calculated only on the initial principal, compound interest accumulates on both the initial principal and the prior interest that has been added to the principal. This means that you earn interest on top of interest when it is compounded.

Here's a breakdown of how compound interest works:

• Initial Principal: This is the amount of money you start with in your investment.
• Interest Rate: The compound interest rate is the percentage of the principal that you earn over a specific period.
• Compounding Period: This defines how often the interest is calculated and added to the principal. For annual compounding, interest is added once a year.
• Future Value Formula: The formula for calculating the compound interest is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where:
  • A = the future value of the investment/loan, including interest
  • P = the principal investment amount (initial deposit)
  • r = the annual interest rate (decimal)
  • n = the number of times interest is compounded per year
  • t = the number of years the money is invested or borrowed for

The magic of compound interest lies in its ability to significantly increase the value of investments over time, especially when the interest is compounded more frequently.

Investment Allocation

Investment allocation refers to the strategic distribution of financial resources among various asset classes or investment vehicles. The goal is to maximize returns while minimizing risk by diversifying where your money is invested. In our exercise, the allocation is across three accounts with different interest rates.

Key points about investment allocation:

• Diversification: Spreading investments across different accounts or asset classes helps manage risk. Different financial instruments can perform differently under varying market conditions.
• Risk Tolerance: Determines how much risk an investor is willing to take. Some prefer a conservative approach with lower returns but less risk, while others might seek higher returns with more exposure to risk.
• Return Expectations: Allocation involves anticipating expected returns based on historical and predicted performance. Our scenario uses different interest rates to illustrate the potential return on investment.
• Balance and Reallocation: Regularly reviewing and adjusting allocations ensures alignment with financial goals and market changes.

In this exercise, finding out how much to invest in each account was crucial, driven by a condition that one account has five times the investment of another. This illustrates the importance of not only the percentage allocated but also the conditions or rules that guide those allocations.

Interest Calculation

Interest calculation is a fundamental financial concept used to determine how much interest has been earned or paid on a principal sum of money over a particular period. It is calculated based on the principal amount, interest rate, and the time period involved.

For understanding interest calculation, consider:

• Principal Amount: The original sum of money on which interest is calculated.
• Interest Rate: A percentage that represents the cost of borrowing money or the reward for saving, typically expressed annually.
• Time Period: The duration for which the money is invested or borrowed. This period greatly influences the amount of interest earned or paid.
• Interest Formula: In our exercise, the formula used is a linear combination of interest from different accounts: \[ 0.04x + 0.03y + 0.02z = 3,650 \] This shows how each account contributes to the total interest based on its allocated principal and interest rate.

Understanding interest calculation helps investors to plan and predict the performance of their investments, ensuring that the financial goals are met effectively.