Question

A total of $$\$ 35,000$$ is invested in two funds paying \(8 \%\) and \(10.5 \%\) simple interest. The total annual interest is $$\$ 3275 .$$ How much is invested at each rate?

Short answer

Therefore, the amount invested at 8% is \$16,000, and the amount invested at 10.5% is \$19,000.

Step-by-step solution

Setting up the equations

Let's denote the amount invested at 8% by \(x\) and the amount invested at 10.5% by \(y\). Then, by looking at the total amount invested, you can write the following equation: \(x + y = 35000\). The total annual interest gives you the second equation: \(0.08x + 0.105y = 3275\).

Solving the system of equations

Now, you need to solve this system of equations. A common method for this is substitution or elimination. In this case, it's easier to use elimination. Multiply the first equation by 0.08 to get: \(0.08x + 0.08y = 2800\). Then, subtract this equation from the second to isolate \(y\): \(0.025y = 475\). Thus, you find \(y = 19000\). Substitute \(y = 19000\) into the first equation to find \(x = 35000 - 19000 = 16000\).

Confirming the solution

It's always a good idea to check your solution by substituting it back into the original equations. When you do this, you'll confirm that your solution is correct.

Key concepts

Investment Strategies

Investing your money wisely is essential for growing wealth over time. When approaching an investment problem, like the one where a total of $35,000 is split between two funds, it's crucial to know the various strategies involved.
Different types of investments offer different interest rates, as seen with 8% and 10.5% in this exercise. Selecting the right mix is vital for maximizing returns while managing risks. Here are some key points to keep in mind when considering investment strategies:

• Diversify Your Portfolio: By spreading your investments across various options, you can manage risk more effectively. For instance, investing in both an 8% and a 10.5% fund can provide a blend of stability and potential higher returns.
• Analyze Your Financial Goals: Understand your risk tolerance and financial objectives. Whether you wish to preserve capital or achieve high growth, tailor your strategy accordingly.
• Evaluate the Interest Rates: As illustrated, different funds provide different returns. Carefully consider the offered rates and their impact on your overall gains.

Investment is not just about numbers—it's about making informed decisions that align with your financial future.

System of Equations

In mathematics, solving a system of equations is a vital skill, especially when dealing with investment scenarios. Our exercise presents two equations: one representing total investment (\(x + y = 35000\)) and another reflecting total interest (\(0.08x + 0.105y = 3275\)). Solving these gives insight into how much is placed in each fund.
Systems of equations allow us to solve for multiple unknowns simultaneously. Follow these steps to handle them effectively:

• Identify Variables: Designate variables for each unknown quantity—like \(x\) and \(y\) in this example—to clarify what you're solving for.
• Formulate Equations: Derive equations based on the relationships in the problem, as with the total money and interest here.
• Choose a Solution Method: Decide between substitution or elimination. In our example, elimination was used effectively to solve for \(y\) and then for \(x\).

Understanding and using systems of equations effectively opens doors to unravel complex problems and make informed decisions.

Interest Rate Calculations

Interest rate calculations play a crucial role in deciding how to allocate funds optimally. In this investment task, accurate calculations helped determine how much money should reside in funds offering different returns — 8% and 10.5%.
When calculating interest income, it's important to grasp a few foundational concepts:

• Simple Interest Formula: Simple interest can be calculated using the formula \(I = P \times r \times t\), where \(I\) is the interest, \(P\) is the principal amount, \(r\) is the rate, and \(t\) is the time period.
• Annual Rate of Return: This exercise focused on annual interest rates. It's important to compute the annual income generated to know if the investment meets your goals.
• Understanding Proportions: Knowing how different interest rates affect similar principal amounts aids in selecting investments with desired returns.

Mastering these calculations is fundamental as you delve deeper into economics and personal finance, empowering you to make savvy investment choices.