Problem 41 - College Algebra

Question

You inherit \(\$ 18,750\) with the stipulation that for the first year the money must be placed in two investments paying \(10 \%\) and \(12 \%\) annual interest, respectively. How much should be invested at each rate if the total interest earned for the year is to be \(\$ 2117 ?\)

Step-by-Step Solution

Answer

You should invest $6,650 at 10% interest and $12,100 at 12% interest to earn total interest of $2117 in a year.

Step 1: Set Up the Equations

Let's denote the amount to be invested at 10% as \( x \) and the amount to be invested at 12% as \( y \). We can setup the equations as follows: (1) \( x + y = 18750 \) (representing the total amount of money), (2) \( 0.10x + 0.12y = 2117 \) (representing the total interest earned).

Step 2: Solve the System of Equations

To solve the system of equations, multiply equation (1) by 0.10, which provides a new equation, (1'): \( 0.10x + 0.10y = 1875 \). Now, subtract equation (1') from equation (2) to find \( y \). This gives: \( 0.02y = 242 \). Dividing both sides by 0.02 gives \( y = 12100 \). Substituting \( y = 12100 \) into equation (1), we can solve for \( x \): \( x = 18750 - 12100 = 6650 \).

Step 3: Verify the Solution

Substitute these values \( x = 6650 \) and \( y = 12100 \) back into Equation 2 to verify. \( 0.10(6650) + 0.12(12100) = 2117 \). The expression equals the total amount of interest, 2117. This verifies that the solution is correct.

Key Concepts

Understanding the Inheritance Problem

When dealing with an inheritance problem like this, you're essentially managing a given sum of money with specific conditions that must be met. In this case, the inherited amount of $18,750 comes with a requirement to divide it into two separate investments, each yielding different interest rates. The objective is to determine how much should be invested at each interest rate to earn a desired total interest over a specified period.
This type of problem is common when dealing with financial stipulations in wills or trusts. It often involves balancing multiple conditions, such as interest rates and total earnings.

• Identify the total amount to be allocated, which is your inheritance.
• Recognize the stipulated conditions, such as different interest rates and a required total interest.
• Create equations to represent these conditions, which you'll solve to determine the specific amounts for each investment.

The key is to translate the conditions into mathematical equations, which leads us into the next concept of calculating interest rates.

Interest Rate Calculations in the Context

Interest rate calculations are pivotal in solving this exercise. To get started, understand that when you invest a sum of money at a particular interest rate, the formula to calculate the interest earned over a year is:\[ \text{Interest} = \text{Principal} \times \text{Interest Rate} \]In this case, two different investments involve interest rates of 10% and 12%. Here's how you can apply it:

• Let \( x \) be the amount invested at the 10% rate and \( y \) be the amount invested at the 12% rate.
• The interest from \( x \) would be \( 0.10x \), and from \( y \), it would be \( 0.12y \).
• Your goal is to adjust \( x \) and \( y \) so that the sum of these interests equals the total interest required, $2117.

This approach allows for a systematic method to compare and adjust the invested sums to meet any conditions for interest. Using equations to calculate interest in various scenarios enables you to make informed financial decisions, especially with constraints like those given in the problem.

Algebraic Problem Solving Techniques

Solving an algebraic problem involving a system of equations is a fundamental mathematical technique. There are typically two equations to consider:
1. The sum equation: \( x + y = 18750 \)
2. The interest equation: \( 0.10x + 0.12y = 2117 \)

• Begin by manipulating one of the equations, such as equation (1) above, to make it easier to solve the other.
• Using algebraic techniques like substitution or elimination, you can isolate one variable to solve for the other.
• For example, by multiplying equation (1) by 0.10 and subtracting from equation (2), you can solve for \( y \).

In this exercise, the strategy of elimination was employed by creating equation (1') \( 0.10x + 0.10y = 1875 \) from which \( 0.02y = 242 \), giving \( y = 12100 \). Once \( y \) is known, substitute it back into equation (1) to find \( x \). Doing so confirms that \( x = 6650 \). Employing these methods enables you to cross-check your solutions, ensuring accuracy and reliability.