Problem 70 - Precalculus Mathematics for Calculus

Question

A man invests his savings in two accounts, one paying \(6 \%\) and the other paying \(10 \%\) simple interest per year. He puts twice as much in the lower- yielding account because it is less risky. His annual interest is \(\$ 3520 .\) How much did he invest at each rate?

Step-by-Step Solution

Answer

He invested \( \$32000 \) at \( 6\% \) and \( \$16000 \) at \( 10\% \).

Step 1: Define Variables

Let \( x \) be the amount invested in the account paying \( 6\% \). Since he invests twice as much in the \( 6\% \) account, the amount invested in the \( 10\% \) account is \( x/2 \).

Step 2: Write Interest Equations

The interest from the \( 6\% \) account is \( 0.06x \) and from the \( 10\% \) account is \( 0.10(x/2) \).

Step 3: Set Up Total Interest Equation

The total interest from both accounts is given as \( \$3520 \), so we set up the equation: \( 0.06x + 0.10(x/2) = 3520 \).

Step 4: Simplify the Equation

Simplify the equation: \( 0.06x + 0.05x = 3520 \). Combining like terms gives \( 0.11x = 3520 \).

Step 5: Solve for \( x \)

Divide both sides by \( 0.11 \) to find \( x \): \( x = \frac{3520}{0.11} \approx 32000 \).

Step 6: Find the Amount Invested at Each Rate

Since \( x \approx 32000 \), the amount invested in the \( 6\% \) account is \( \\(32000 \) and in the \( 10\% \) account is \( \\)16000 \) (which is \( 32000/2 \)).

Key Concepts

Investment Strategy

When considering investment options, a balanced strategy is key to optimizing returns while minimizing risk. The investor in our exercise prioritized safety by allocating a larger share to a lower-yielding but less risky account.

Here's why this is beneficial:

• **Risk Mitigation:** By putting more funds into a safer account, the investor reduces potential losses should the higher-risk investment underperform.
• **Predictable Returns:** The lower-yielding account provides steadier returns, offering a reliable income stream.
• **Diversification:** This approach involves spreading investments across different accounts to balance out potential losses and gains.

This strategy of balancing security and profitability can be crucial for long-term financial planning. It ensures that while some funds are exposed to higher risks for higher potential returns, a buffer remains to shield against substantial losses.

Interest Rate Calculation

Calculating the interest rate accurately is vital to make informed investment decisions. In this case, the investor received different interest rates from two separate accounts.

Here's how interest is calculated:

• **Simple Interest Formula:** Simple interest is calculated by multiplying the principal amount by the interest rate and the time period. Formula: \( I = P \times r \times t \).
• For the lower-yield (6%) account, the interest is represented as \(0.06x\), where \(x\) is the amount invested.
• For the higher-yield (10%) account, it's expressed as \(0.10(x/2)\), reflecting that only half of \(x\) was invested.

By understanding these calculations, investors can better predict potential earnings and assess which account yields more benefit given their financial situation.

Algebraic Equations

Algebraic equations are essential tools for solving investment-related problems, providing a clear method to find unknown values.

In the given exercise, setting up and solving an equation was necessary to determine the investment amounts. Here's the breakdown:

• **Formulate the Equation:** The total interest from both accounts was known: \( \$3520 \). This led to the equation: \( 0.06x + 0.05x = 3520 \), where \(0.05x \) is the simplified representation of \(0.10(x/2)\).
• **Combine Like Terms:** Combine terms with \(x\) to simplify: \(0.11x = 3520\).
• **Solve for \(x\):** Divide by \(0.11\) to find \( x = \frac{3520}{0.11} \approx 32000 \). This calculation reveals the amount invested in the 6% account.

By comprehensively applying algebraic solutions, investors can logically determine the specifics of their investment allocations, clarifying complex financial decisions.