Problem 38 - Algebra 2

Question

PERSONAL FINANCE For Exercises \(38-41,\) use the following information. Zach has purchased some home theater equipment for \(\$ 2000,\) which he is financing through the store. He plans to pay \(\$ 340\) per month and wants to have the balance paid off after six months. The formula \(B(x)=2000 x^{6}-\) 340\(\left(x^{5}+x^{4}+x^{3}+x^{2}+x+1\right)\) represents his balance after six months if \(x\) represents 1 plus the monthly interest rate (expressed as a decimal). Find his balance after 6 months if the annual interest rate is 12\(\% .\) (Hint: The monthly interest rate is the annual rate divided by \(12,\) so \(x=1.01 . )\)

Step-by-Step Solution

Answer

After 6 months, Zach's balance is approximately $31.70.

Step 1: Determine Monthly Interest Rate

Given the annual interest rate is 12%, calculate the monthly interest rate by dividing by 12. This gives us a monthly rate of \( \frac{12\%}{12} = 1\% \). Therefore, \( x = 1 + \frac{1}{100} = 1.01 \).

Step 2: Substitute x into Balance Formula

Substitute \( x = 1.01 \) into the balance formula: \[ B(1.01) = 2000 \cdot (1.01)^6 - 340 \cdot (1.01^5 + 1.01^4 + 1.01^3 + 1.01^2 + 1.01 + 1). \]

Step 3: Calculate (1.01)^6 and (1.01)^n for each n

Calculate \((1.01)^6 \), \((1.01)^5 \), \((1.01)^4 \), \((1.01)^3 \), and \((1.01)^2 \) using a calculator:- \( (1.01)^6 \approx 1.06152 \)- \( (1.01)^5 \approx 1.05101 \)- \( (1.01)^4 \approx 1.0406 \)- \( (1.01)^3 \approx 1.0303 \)- \( (1.01)^2 \approx 1.0201 \).

Step 4: Sum Constants in the Subtracted Term

Calculate \[ 1.01^5 + 1.01^4 + 1.01^3 + 1.01^2 + 1.01 + 1 \approx 1.05101 + 1.0406 + 1.0303 + 1.0201 + 1.01 + 1 = 6.151. \]

Step 5: Calculate Balance Formula

Substitute the calculated powers back into the balance formula:\[ B(1.01) = 2000 \times 1.06152 - 340 \times 6.151. \]Calculate each term:- \( 2000 \times 1.06152 = 2123.04 \)- \( 340 \times 6.151 = 2091.34 \).Finally, compute the balance:\[ B(1.01) = 2123.04 - 2091.34 = 31.70. \]

Step 6: Interpret the Result

The balance after six months, with a monthly interest rate of 1%, is approximately \( \\(31.70 \), indicating Zach still owes \( \\)31.70 \) after making six monthly payments of \( \$340 \).

Key Concepts

Interest Rate Calculation

Understanding interest rates is crucial in personal finance, especially when dealing with loans or financing. To calculate the monthly interest rate from an annual interest rate, simply divide the annual rate by 12. For example, if you have an annual interest rate of 12%, the monthly rate would be \( \frac{12\%}{12} = 1\% \), which is equivalent to 0.01 when expressed as a decimal. This means every month, the amount of interest added is based on 1% of the remaining balance. In equations or formulas, this rate is often represented as \( x = 1 + 0.01 = 1.01 \). This small transformation is vital as it allows us to incorporate interest into balance calculations seamlessly.

Balance Formula

The balance formula is a powerful tool used to track the remaining debt after payments and interest are applied. In our exercise, the formula \( B(x) = 2000 x^6 - 340(x^5 + x^4 + x^3 + x^2 + x + 1) \) calculates Zach's remaining balance. Here, \( x \) is 1 plus the monthly interest rate. The first part \( 2000 x^6 \) represents the compounded initial loan amount with interest over six months. The second part subtracts the effects of the six payments, each affected by the monthly interest. This formula, therefore, provides a comprehensive way to calculate outstanding debt after regular payments.

Monthly Payments

Monthly payments are crucial in determining how quickly a loan is repaid. In many cases, they consist of both principal and interest components. For Zach, the monthly payment is \( \$340 \), which is subtracted from the compounded amount in the formula. Each payment decreases what Zach owes, but because interest is applied to the balance, the reduction isn't a simple subtraction. Understanding this aspect is key to managing personal loans effectively. The goal is to ensure that payments exceed just the interest portion, progressively lowering the principal balance over time.

Loan Repayment

Completing loan repayment efficiently requires discipline and planning. Zach plans to pay off his \( \$2000 \) loan in six months, a timeline that impacts the interest accrued. Shorter repayment periods generally lead to less total interest paid. However, each loan is subject to its interest rate and conditions. Hence, using balance formulas allows borrowers to simulate different repayment scenarios. This helps in budgeting monthly expenses and keeping track of how much is left to pay at any point during the debt cycle.

Exponential Calculations

Exponential calculations are fundamental when dealing with interest calculations in finance. The powers of \( (1.01)^n \) represent how the initial loan balance compounds over time. For the exercise, you calculate \( (1.01)^2, (1.01)^3, \) up to \( (1.01)^6 \) to reflect how the balance changes monthly with interest. These calculations are central since they help in accurately estimating the compounded amount and understanding how interest grows over time. A good grasp of exponential calculations ensures that individuals can predict future financial obligations with greater precision.