Problem 70 - Algebra and Trigonometry

Question

Investments 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 be- cause it is less risky. His annual interest is \(\$ 3520 .\) How much did he invest at each rate?

Step-by-Step Solution

Answer

Invested $16000 at 10% and $32000 at 6%.

Step 1: Understanding the Problem

Let's define the unknowns. Let \( x \) be the amount invested at 10\(\%\). Since the problem states the man invests twice as much in the 6\(\%\) account, the amount there is \( 2x \). We need to find both \( x \) and \( 2x \).

Step 2: Setting Up the Interest Equations

Interest earned from the 10\(\%\) account is \( 0.10x \) and interest earned from the 6\(\%\) account is \( 0.06(2x) \). The total interest is \( \$3520 \). The equation is: \( 0.10x + 0.06(2x) = 3520 \).

Step 3: Simplifying the Equation

First, expand and simplify \( 0.06(2x) \) to get \( 0.12x \). Substitute in the equation to get \( 0.10x + 0.12x = 3520 \). This simplifies to \( 0.22x = 3520 \).

Step 4: Solving for x

Divide both sides of the equation \( 0.22x = 3520 \) by 0.22 to isolate x. Thus, \( x = \frac{3520}{0.22} = 16000 \). The amount invested at 10\(\%\) is \( \$16000 \).

Step 5: Finding the Amount in the 6% Account

Since \( x = 16000 \), the amount in the 6\(\%\) account is \( 2x = 2(16000) = 32000 \). The amount invested at 6\(\%\) is \( \$32000 \).

Key Concepts

Investment Strategy

When thinking about how to invest money, one of the key considerations is the risk associated with each option. In the problem, the man chose to place twice as much money in a 6\(\%\) account because it was less risky. This demonstrates a basic investment strategy.

An investment strategy involves deciding how to allocate savings across different investment opportunities. This strategy can vary widely based on personal circumstances, financial goals, and risk tolerance.

• Diversification: In the problem, investing in two accounts with different interest rates showcases diversification. This can help balance risk and potential return.
• Risk Assessment: Consider how much risk you are willing to take. Less risky investments, like the 6\(\%\) account, often offer a lower return but more security.
• Allocation: Decide how much to allocate to each option. The man invested more in the lower-risk account, which could indicate a desire for stable returns.

Considering these factors can lead to an effective investment strategy that aligns with one's financial goals.

Interest Rate Calculation

Interest rate calculation is essential for understanding how much your money will grow over time. In the problem, simple interest is used to calculate the interest for each account.

Simple interest is calculated using the formula:\[I = P \cdot r \cdot t\]where \(I\) is the interest earned, \(P\) is the principal amount (initial investment), \(r\) is the rate of interest per year, and \(t\) is the time in years.

• For the 10\(\%\) account, the interest calculation is \(0.10 \times x\).
• For the 6\(\%\) account, the calculation is \(0.06 \times 2x\).

This simple method is easy to use, which makes it a common tool for quick calculations. Understanding these basics will help you determine future savings potential.

Problem-Solving Steps

Solving problems involving investments and interest can be straightforward if broken down into clear, manageable steps.

In the exercise, the solution follows a logical process:

• Define the Variables: Identify what you need to find. In this case, it's the different amounts invested at each rate.
• Set Up Equations: Use the given information to form equations. Here, the interest equations were set up for both accounts.
• Simplify and Solve: Simplify the equations as much as possible before solving them. This leads to finding the value of \(x\), the amount invested at 10\(\%\).
• Check Your Work: Always verify your solutions by plugging them back into the original problem to ensure consistency.

By following these steps, you can approach similar problems with confidence.