QuestionAugust 12, 2025

The Johnson family buys furniture and appliances for a new house for 10,000.00 with a credit card that has a 15% APR. The Johnsons make a 325.00 monthly payment. How many months will it take to pay off the balance? Enter your answer as a whole number, such as: 20.

The Johnson family buys furniture and appliances for a new house for 10,000.00 with a credit card that has a 15% APR. The Johnsons make a 325.00 monthly payment. How many months will it take to pay off the balance? Enter your answer as a whole number, such as: 20.
The Johnson family buys furniture and appliances for a new house for 10,000.00 with a
credit card that has a 15%  APR.
The Johnsons make a 325.00 monthly payment. How many months will it take to pay off
the balance? Enter your answer as a whole number, such as: 20.

Solution
3.1(222 votes)

Answer

33 Explanation 1. Identify the monthly interest rate The APR is 15\%, so the monthly interest rate is \frac{15}{12} = 1.25\% or 0.0125. 2. Use the loan amortization formula The formula for the number of payments n is given by: \[ n = \frac{\log(\frac{P}{P - rA})}{\log(1 + r)} \] where P is the principal (10,000), r is the monthly interest rate (0.0125), and A is the monthly payment (325). 3. Calculate the number of months Substitute the values into the formula: \[ n = \frac{\log(\frac{10000}{10000 - 0.0125 \times 325})}{\log(1 + 0.0125)} \] 4. Simplify and solve Calculate the numerator: \[ \frac{10000}{10000 - 4.0625} = \frac{10000}{9995.9375} \approx 1.000406 \] Calculate the denominator: \[ \log(1.0125) \approx 0.005407 \] Solve for n: \[ n = \frac{\log(1.000406)}{0.005407} \approx \frac{0.000176}{0.005407} \approx 32.54 \]

Explanation

1. Identify the monthly interest rate<br /> The APR is $15\%$, so the monthly interest rate is $\frac{15}{12} = 1.25\%$ or $0.0125$.<br /><br />2. Use the loan amortization formula<br /> The formula for the number of payments $n$ is given by:<br />\[ n = \frac{\log(\frac{P}{P - rA})}{\log(1 + r)} \]<br />where $P$ is the principal ($10,000$), $r$ is the monthly interest rate ($0.0125$), and $A$ is the monthly payment ($325$).<br /><br />3. Calculate the number of months<br /> Substitute the values into the formula:<br />\[ n = \frac{\log(\frac{10000}{10000 - 0.0125 \times 325})}{\log(1 + 0.0125)} \]<br /><br />4. Simplify and solve<br /> Calculate the numerator:<br />\[ \frac{10000}{10000 - 4.0625} = \frac{10000}{9995.9375} \approx 1.000406 \]<br /><br /> Calculate the denominator:<br />\[ \log(1.0125) \approx 0.005407 \]<br /><br /> Solve for $n$:<br />\[ n = \frac{\log(1.000406)}{0.005407} \approx \frac{0.000176}{0.005407} \approx 32.54 \]
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