Problem 202 - Elementary Algebra

Question

Avery and Caden have saved \(\$ 27,000\) towards a down payment on a house. They want to keep some of the money in a bank account that pays \(2.4 \%\) annual interest and the rest in a stock fund that pays \(7.2 \%\) annual interest. How much should they put into each account so that they earn \(6 \%\) interest per year?

Step-by-Step Solution

Answer

Put \( 6750 \) in the bank account and \( 20250 \) in the stock fund.

Step 1: Define Variables

Let the amount of money placed in the bank account be denoted as \( x \). Thus, the amount placed in the stock fund would be \( 27000 - x \).

Step 2: Set Up Interest Equations

The interest earned from the bank account will be \( 0.024x \), and the interest earned from the stock fund will be \( 0.072(27000 - x) \).

Step 3: Calculate Total Interest

The total interest earned should be equal to \( 6 \% \) of \( 27000 \), which can be written as \( 0.06 \times 27000 = 1620 \). Thus, the equation becomes: \( 0.024x + 0.072(27000 - x) = 1620 \).

Step 4: Simplify the Equation

First, distribute \( 0.072 \): \( 0.024x + 0.072 \times 27000 - 0.072x = 1620 \). Simplifies further to: \( 0.024x + 1944 - 0.072x = 1620 \).

Step 5: Combine Like Terms

Combine the \( x \) terms: \( -0.048x + 1944 = 1620 \).

Step 6: Solve for \( x \)

Isolate \( x \) by subtracting 1944 from both sides: \( -0.048x = 1620 - 1944 \). This simplifies to: \( -0.048x = -324 \). Divide both sides by \( -0.048 \): \( x = 6750 \).

Step 7: Determine Amounts for Each Account

Thus, \( x = 6750 \) is the amount to be placed in the bank account. The amount in the stock fund is \( 27000 - 6750 = 20250 \).

Key Concepts

Interest Rate

Interest rate is the percentage at which money grows over a period. In this problem, we deal with two types of interest rates: one from a bank account and another from a stock fund. The bank account offers an annual interest rate of 2.4%, which means that for every dollar in the bank, you will earn 2.4 cents per year. The stock fund offers a higher annual interest rate of 7.2%, providing 7.2 cents for every dollar invested per year. Understanding these rates allows us to calculate the total interest generated from different investment distributions. It's important to know how to convert these percentages into decimal form: 2.4% becomes 0.024 and 7.2% becomes 0.072. By multiplying these rates with the respective invested amounts, we find out how much interest each investment generates.

Algebraic Equations

Algebraic equations help us solve for unknown variables. In this exercise, we use the variable \( x \) to represent the amount of money placed in the bank account. We derive the equation from two interest components: one from the bank and one from the stock fund. The total interest must equal 6% of the entire investment, or \(27,000. The key equation becomes: \[ 0.024x + 0.072(27000 - x) = 1620 \]
This equation includes both the bank's and stock fund's interests. Distributing and combining like terms allows us to isolate \( x \). By simplifying further:
\[ 0.024x + 1944 - 0.072x = 1620 \]
\[ -0.048x + 1944 = 1620 \]
\[ x = 6750 \]
Therefore, \( x \) equals \)6,750, the amount for the bank account.

Investment Distribution

Understanding investment distribution is key to managing finances effectively. In this scenario, allocating \(27,000 between a bank account and a stock fund ensures an optimal return. The goal is to earn an overall annual interest rate of 6%. We've determined that \)6,750 should be placed in the bank account. The remaining amount, calculated by subtracting this from the total, results in $20,250 for the stock fund. This division strikes a balance, utilizing the different interest rates of the two investments while achieving the desired average return rate. Distributing investments in this manner can help diversify risk and potentially increase overall earnings, crucial for financial planning.