Problem 45 - Algebra for College Students

Question

Solve each equation. A sum of \(\$ 95,000\) is split between two investments, one paying \(6 \%\) and the other \(9 \%\). If the total yearly interest amounted to \(\$ 7290\), how much was invested at \(9 \%\) ?

Step-by-Step Solution

Answer

$53,000 was invested at 9%.

Step 1: Define Variables

Let \( x \) be the amount invested at 9%. Then, \( 95000 - x \) is the amount invested at 6%.

Step 2: Set Up the Equation

The total interest from both investments is the sum of the interest from each part: \( 0.09x + 0.06(95000 - x) = 7290 \).

Step 3: Simplify the Equation

Expand the equation: \( 0.09x + 5700 - 0.06x = 7290 \).

Step 4: Combine Like Terms

Combine the \( x \) terms: \( 0.03x + 5700 = 7290 \).

Step 5: Solve for \( x \)

Subtract 5700 from both sides: \( 0.03x = 1590 \). Divide both sides by 0.03 to solve for \( x \): \( x = 53000 \).

Key Concepts

Investment Calculations

Investment calculations are essential for managing finances effectively. They help in determining where to allocate money for optimal returns. Understanding how to calculate interest and returns allows investors to make informed decisions.

• Splitting Funds: Investors often split money between various investments to manage risk and increase potential earnings.
• Interest Rates: Knowing the percentage of returns on investment is crucial.
  
• Allocation: By determining how much to put in each investment, one can maximize their returns based on different interest rates.
  

In this problem, the total investment amount was divided into two parts, one part invested at 6% interest and the other at 9%. Calculating each portion's contribution to the total interest helps decide where more money should be invested to get maximum returns.

Interest Rate Problems

Interest rates are a core aspect of financial literacy. They determine how much earnings you will generate from an investment over time.

• Simple Interest: Interest is the extra money earned on an investment, calculated as a percentage of the original sum. For instance, a 9% interest rate means you earn $9 for every $100 invested.


• Compounding: However, in this problem, we deal with simple interest, which is straightforward because it calculates returns only from the original principal. The investment was split, leading to varying returns depending on allocated percentages.

Understanding these rates and their calculations helps in setting financial goals and expectations from investments. It showcases how varied investments perform based on their respective interest rates.

Solving Equations

Solving equations can seem challenging but is a methodical process. It is about finding unknown values using available information.

• Define Variables: First, determine what you don't know and assign a variable to it, like in our exercise, where 'x' was the amount invested at 9%.


• Set Up Equations: Use information provided to set up an equation that represents the situation. Here, the equation represents total interest from the investments.


• Simplify: Break down the equation step by step, combining like terms and breaking complex parts into simpler components.
  
• Solve: Finally, manipulate the equation to isolate the variable, revealing the unknown value, as we did by finding that $53,000 was invested at 9%.

This systematic approach is powerful for solving any equation, providing a framework to solve real-life problems efficiently.