Question

Michelle has a \(\$ 421.55\) per month car payment. If she needs to pay a total of \(\$ 19,500\), including interest, how many months from now will she make her last payment? A. 45 months B. 46 months C. 47 months D. 48 months

Short answer

C. 47 months

Step-by-step solution

- Understand the Problem

Michelle pays \(421.55 per month for her car, and she needs to pay a total amount of \)19,500. We need to find out how many months it will take her to pay off the total amount.

- Set Up the Equation

To find the number of months, set up the equation: \[ 421.55 \times n = 19,500 \]where \( n \) is the number of months.

- Solve for the Number of Months

Rearrange the equation to solve for \( n \): \[ n = \frac{19,500}{421.55} \]

- Perform the Division

Calculate the division: \[ n \rightarrow \frac{19,500}{421.55} \rightarrow \text{Approximately } 46.25 \]

- Interpret the Result

Since \( n \) represents the number of months and it must be a whole number, Michelle will make her last payment in the 47th month.

Key concepts

monthly payments

Understanding monthly payments is crucial when managing a loan. Monthly payments are the consistent amount you pay each month until your total loan amount, including interest, is fully paid off. For Michelle's car loan, her monthly payment is set at \( 421.55 \). This means every month she pays \( 421.55 \) towards her total debt until it's cleared. It is essential to know this fixed number because it helps in planning your budget and ensuring you don't fall behind on your payments.

division

Division is a fundamental mathematical operation you need to know to solve this type of problem. When you have a total amount to be paid and the monthly payment amount, you divide the total loan amount by the monthly payment to find out how many months it will take to repay the loan. In Michelle’s case, the total loan amount is \( 19,500 \), and the monthly payment is \( 421.55 \). The equation for finding the number of months is: \[ n = \frac{19,500}{421.55} \]. Performing the division: \( n \rightarrow \frac{19,500}{421.55} \rightarrow \text{Approximately } 46.25 \). This calculation shows that it would take just over 46 months. However, since you can't make a fraction of a payment, Michelle will need to make her final payment in the 47th month.

interest calculation

Interest calculation is also key when understanding loans. Interest is the extra amount paid to the lender for borrowing money. The total amount Michelle needs to pay back is not just the cost of the car but includes interest. Therefore, she pays more than the original price of the car. Given the total loan amount \( 19,500 \) includes the interest, knowing how to calculate interest can help you understand how much you are truly paying for borrowing. To find the interest, you can subtract the principal (initial loan amount without interest) from the total repayment amount, but in this problem, it's already included in the total debt.