Problem 55 - Precalculus: Mathematical 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's define two variables to solve this exercise. Let \( x \) be the amount invested at \( 6\% \). Since the man invested twice as much in this account as compared to the \( 10\% \) account, the amount invested at \( 10\% \) is \( \frac{x}{2} \).

Step 2: Set Up Interest Equation

The total interest earned from the \( 6\% \) account is \( 0.06 \times x \) and from the \( 10\% \) account is \( 0.10 \times \frac{x}{2} \). The total interest earned is \$3520, so we can write the equation as: \[ 0.06x + 0.10 \times \frac{x}{2} = 3520 \]

Step 3: Simplify Interest Equation

Simplify the equation by calculating \( 0.10 \times \frac{x}{2} \), which simplifies to \( 0.05x \). Now the equation becomes:\[ 0.06x + 0.05x = 3520 \] Combine like terms:\[ 0.11x = 3520 \]

Step 4: Solve for x

Solve for \( x \) by dividing both sides of the equation by \( 0.11 \):\[ x = \frac{3520}{0.11} \]Calculating this gives \( x = 32000 \).

Step 5: Find Investments in Each Account

Now that we have \( x = 32000 \), which is the amount invested at \( 6\% \), the amount invested at \( 10\% \) is \( \frac{32000}{2} = 16000 \).

Key Concepts

Investment Distribution

Investment distribution is about how and where you decide to place your money. It's a crucial part of managing wealth. When investing in accounts with different interest rates, one aims to balance risk and reward. In our exercise, there is an account with a lower 6% interest rate and another with a higher 10% interest rate. The man decided to put twice as much money into the 6% account because it is less risky than the 10% one.
The reasoning behind placing a larger amount in a lower-yield account, despite its smaller returns, is all about security. Safer investments usually mean less risk, and individuals may prefer security if they rely on this money for future needs.
In our example, the solution shows that the investor placed $32,000 in the 6% account and $16,000 in the 10% account. This strategic distribution allows benefiting from both safety and growth.

Interest Rate Calculation

Calculating interest rate allows you to figure out how much money you make from your investments. Simple interest is one of the most straightforward methods for determining returns.
For the 6% account, the interest earned is calculated by multiplying the invested amount by 0.06. For the 10% account, it's done similarly but with its respective interest rate.
In this problem, we compute two interests:

• 6% interest from the first account: \( 0.06 \times 32000 \)
• 10% interest from the second account: \( 0.10 \times 16000 \)

This goes to show that calculating interest accurately helps in assessing the overall return of your investment strategy.

Solving Equations

Solving equations is a fundamental skill in mathematics. It helps in figuring out unknown variables by setting up relationships between known facts.
In our exercise, we set up an equation to find how much was invested at each rate. The equation takes into account the total annual interest earned, which was $3520.
The approach to solving this involves:

• Setting up the equation with known interest rates: \( 0.06x + 0.05x = 3520 \)
• Combining like terms to simplify: \( 0.11x = 3520 \)
• Solving for \( x \) by dividing both parts of the equation: \( x = \frac{3520}{0.11} \)

By following these steps, we find \( x = 32000 \), indicating the amount invested at 6%. This highlights how equations can not only determine amounts but also clarify the broader financial strategy in investments.