Problem 39 - 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 9.6\(\%\)

Step-by-Step Solution

Answer

His balance after 6 months is approximately \(\$3.39\).

Step 1: Convert Annual Interest Rate to Monthly Interest Rate

The annual interest rate is 9.6\(\%\), which needs to be converted to a monthly interest rate. Divide the annual rate by 12 to get the monthly rate: \(\frac{9.6}{12} = 0.8\%\). Convert this percentage to a decimal by dividing by 100: \(\frac{0.8}{100} = 0.008\). Thus, the monthly interest rate is \(0.008\).

Step 2: Calculate \(x\)

Since \(x\) represents 1 plus the monthly interest rate, add 1 to the monthly rate we calculated: \(x = 1 + 0.008 = 1.008\).

Step 3: Compute Each Term of the Balance Formula

Substitute \(x = 1.008\) into the balance formula: \[B(x) = 2000 \times (1.008)^6 - 340 \times ((1.008)^5 + (1.008)^4 + (1.008)^3 + (1.008)^2 + (1.008) + 1)\].Calculate each power of \(x\) individually:- \((1.008)^6\)- \((1.008)^5\)- \((1.008)^4\)- \((1.008)^3\)- \((1.008)^2\)Substitute these back into the formula.

Step 4: Simplify the Expression

Plug in the values from the previous step into the formula:- Calculate \((1.008)^6 \approx 1.04889\)- Calculate \(340 \times ((1.008)^5 + (1.008)^4 + (1.008)^3 + (1.008)^2 + (1.008) + 1) = 340 \times (6.1299)\)Thus, the balance calculation becomes:\[B(x) = 2000 \times 1.04889 - 340 \times 6.1299\].

Key Concepts

Monthly Interest Rate Calculation

When dealing with personal finance, understanding how to calculate the monthly interest rate is critical. If you have an annual interest rate, like Zach does with his financing at 9.6%, you must convert this to a monthly interest rate to accurately compute monthly costs and payments. This conversion is quite simple.
To get the monthly interest rate from an annual rate:

• First, divide the annual rate by 12 (the number of months in a year). For Zach's case: \(\frac{9.6\%}{12} = 0.8\%\).
• Next, transform this percentage into a decimal by dividing by 100. Thus, \(\frac{0.8}{100} = 0.008\).

This decimal is your monthly interest rate. Here it indicates that every month, the interest added to the balance is 0.008 of the current balance.

Balance Formula

The balance formula is crucial for understanding how much you will owe over time when financing a purchase. In our example, it helps Zach plan his payments to clear his debt in six months. The formula used for Zach is:
\[B(x) = 2000 \cdot x^6 - 340 \cdot (x^5 + x^4 + x^3 + x^2 + x + 1)\]Here is what each part represents:

• The first term, \(2000 \cdot x^6\), represents the growth of the initial borrowed amount, compounded over six months.
• The second term, involving 340 multiplied by powers of \(x\), accounts for his monthly payments and how they reduce his total owed amount monthly.

Understanding the interplay of these terms is key to mastering balance formulas, letting you see how different interest rates and payments affect your outstanding balance.

Interest Rate Conversion

Interest rate conversion is a fundamental concept in personal finance, especially when dealing with compound interest over different periods.
In Zach's scenario, the fixed-point because the rate is annual, which must be converted to monthly before calculating the balance.

• Always begin by adjusting an annual rate to the period required (here, it's monthly), which involves dividing by the appropriate factor (12 for months).
• Converting interest percentages to decimals youngers accurate calculations and helps avoid common pitfalls in financial math.

Without converting interest rates correctly, calculations like Zach’s would yield incorrect values, potentially leading to over or underestimations of the amount owed.

Exponentiation for Finance Calculations

Exponentiation plays a prominent role in understanding financial calculations, particularly in dealing with compounded interest. Each power in our balance formula reflects the compound nature of interest over time, crucial when working out how debts and investments grow.
Why is exponentiation important?

• In finance, raising a base to a power (like \( x^6 \)) shows how principal amounts grow with compounding periods. Here, \(x\) is 1 plus the monthly interest rate.
• The multiple powers in Zach’s balance calculation show how payments applied at different times impact the total due. Each power decrease represents one less month of effective growth because of a payment.

Correctly applying these principles in computations can make a significant difference in evaluating the true cost of loans or the potential growth of investments.