Problem 41 - 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). Suppose he finances his purchase at 10.8\(\%\) and plans to pay \(\$ 410\) every month. Will his balance be paid in full after five months?

Step-by-Step Solution

Answer

Zach won't fully pay off his balance in five months with $410/month payment at a 10.8% annual interest rate.

Step 1: Identify the Variables

Zach is financing equipment costing \(\\(2000\). He initially plans to pay \(\\)340\) every month for six months. We need to check if paying \(\$410\) monthly for five months will clear the balance with an interest rate of 10.8\%.

Step 2: Convert Percentage to Decimal

The interest rate is 10.8\% annually, which needs to be converted to a monthly interest rate. As a monthly interest rate, it is \(\frac{10.8}{100}\div 12\). Calculating this gives us a monthly rate of \(\frac{0.108}{12} = 0.009\). Thus, \(x = 1 + 0.009 = 1.009\).

Step 3: Use the Formula for Balance after Five Months

The formula for the remaining balance after five months, given he's paying \(\$410\) monthly, is \(B(x) = 2000x^5 - 410(x^4 + x^3 + x^2 + x + 1)\). Substitute \(x = 1.009\) into the formula.

Step 4: Calculate Power Terms

First, calculate each power of \(x\):- \(x^1 = 1.009\)- \(x^2 = 1.009^2\)- \(x^3 = 1.009^3\)- \(x^4 = 1.009^4\)- \(x^5 = 1.009^5\)Compute these values to be used in the next step.

Step 5: Substitute and Solve

Substitute the computed values into the balance formula:\[B(x) = 2000 \times x^5 - 410(x^4 + x^3 + x^2 + x + 1)\]Calculate each term and sum them up to find \(B(x)\). Ensure you perform multiplication and summation accurately.

Step 6: Evaluate the Remaining Balance

Check if \(B(x) = 0\) (or very close to 0). If the calculation results in a balance of zero, then Zach will have paid off his balance fully. If \(B(x)\) is positive, there remains an unpaid balance.

Key Concepts

Understanding Monthly Interest Rate

When dealing with loans, the monthly interest rate plays a crucial role in determining how much extra you'll pay over the life of the loan. In Zach's scenario, the annual interest rate is 10.8\(\%\). To find the monthly interest rate, this annual rate has to be converted into a decimal and then divided by 12 (the number of months in a year).
This looks like this:

• Convert percentage to decimal: \(10.8/100 = 0.108\)
• Divide by 12 months: \(0.108 / 12 = 0.009\)

Thus, the monthly interest rate Zach faces is 0.009, or 0.9\(\%\) per month. This monthly rate is crucial as it impacts how much interest accumulates on any remaining balance each month.

Balance Calculation Explained

The balance calculation is the crux of knowing whether a debt is paid off or if any amount remains. For Zach, calculating the balance involves inputting his monthly payment and the monthly interest rate into the provided balance formula.
Here's the formula: \[ B(x) = 2000x^5 - 410(x^4 + x^3 + x^2 + x + 1) \]In this formula:

• \(2000\) is the initial loan amount.
• \(x\) represents \(1 + \text{monthly interest rate}\), which we've identified as \(1.009\).
• \(410\) is the monthly payment Zach makes over the five months.

Plug \(x = 1.009\) into the equation and compute the powers of \(x\):

• \(x^1, x^2, x^3, x^4, x^5\)

These terms are calculated to determine the total amount that needs to be deducted by the payments Zach made. Calculating these precisely helps you understand if Zach's final balance is zero or if he still owes money.

Navigating Loan Repayment

Loan repayment involves careful planning and calculations to ensure all debt is paid off. For Zach, choosing to pay \(\$410\) monthly while managing a monthly interest rate can influence whether he covers his balance in time.
Key considerations for loan repayment include:

• Ensuring monthly payments cover both the principal balance and accumulating interest.
• Understanding how changes in payments (like paying more or less than initially planned) can impact the remaining balance.
• Ensuring calculations are up-to-date and precise to avoid underpaying.

In Zach's case, after calculating with the formula and payments, if \(B(x) = 0\) or very near, it means he has paid off his loan entirely within five months. If \(B(x)\) is positive, he still owes some amount. Calculating and confirming this ensures Zach's loan is managed effectively.