Barton took out a car loan of 2,625. He makes monthly payements of 175. The graph represents the remaining balance after each payment. Which example models the correct slope?

Question

m=9-8/1050-1225
m=2450-2625/0-1
m=1050-1225/9-8
m=525-700/11-12

Answer 1

To determine the correct slope for Barton’s car loan payments, we first need to understand what the slope represents in this context. The slope (\(m\)) represents the change in the remaining balance of the loan for each monthly payment made.

Here, let's denote:

• The initial loan amount: \(2,625\)
• Monthly payment: \(175\)

To find the slope, we can calculate the change in the balance over time as Barton's payments reduce the loan amount.

Assuming he makes a payment every month, the balance decreases by \(175\) each month. Therefore, the slope should reflect the change in balance over time.

The remaining balance after each payment can be expressed as: \[ \text{New Balance} = \text{Old Balance} - \text{Monthly Payment} \] Thus the change in balance for each payment is negative; specifically: \[ m = \frac{\text{Change in Balance}}{\text{Change in Time}} = \frac{-175}{1} \] However, since the options given do not specifically relate to this exact calculation in simple terms of \( -175 \), we will analyze the provided options based on the concept of change in balance:

Analyzing the Given Slope Options:

1. \(m = \frac{9 - 8}{1050 - 1225}\):

   • Change in balance = \(1050 - 1225 = -175\) (where \(1050\) and \(1225\) doesn't represent the interval after payments)
   • Time Change = \(1\)
   • So, this representation does not seem valid as there's no interval for remaining balance.

2. \(m = \frac{2450 - 2625}{0 - 1}\):

   • Change in balance = \(2450 - 2625 = -175\)
   • Time Change = \(1\)
   • Thus this slope models the correct rate of change: \[ m = \frac{-175}{1} = -175 \]

3. \(m = \frac{1050 - 1225}{9 - 8}\):

   • Change in balance = \(1050 - 1225 = -175\)
   • Change in time = \(1\)
   • As these numbers also don’t correspond to any realistic scenario of payment, they can’t also be justified.

4. \(m = \frac{525 - 700}{11 - 12}\):

   • Change in balance = \(525 - 700 = -175\)
   • Change in time = \(-1\)
   • This would yield \(m = \frac{-175}{-1} = 175\), though the interpretation diverges.

In conclusion, the most suitable choice correlating to the slope of payments that decreases the loan balance effectively would be:

\(m = \frac{2450 - 2625}{0 - 1}\), which correctly denotes the change in balance of \(-175\) for 1 month, which is indicative of the loan repayment movement.