Math Problem Statement

You want to buy a new car. You can afford payments of $500 per month and can borrow the money at an interest rate of 6.3% compounded monthly for 5 years.

How much are you able to borrow?

$

How much interest do you pay?

$

Solution

To determine how much you can borrow and the total interest paid, we can use the formula for the present value of an annuity, since the payments are fixed, and the interest is compounded monthly. The formula is:

PV=P×1(1+r)nrPV = P \times \frac{1 - (1 + r)^{-n}}{r}

Where:

  • PVPV is the present value (amount you can borrow).
  • PP is the monthly payment ($500).
  • rr is the monthly interest rate (annual interest rate divided by 12).
  • nn is the total number of payments (5 years × 12 months/year = 60 payments).

Step 1: Determine the monthly interest rate rr:

r=6.3%12=0.06312=0.00525r = \frac{6.3\%}{12} = \frac{0.063}{12} = 0.00525

Step 2: Plug the values into the formula:

PV=500×1(1+0.00525)600.00525PV = 500 \times \frac{1 - (1 + 0.00525)^{-60}}{0.00525}

Let me calculate that for you.The amount you are able to borrow is approximately $25,677.10.

Step 3: Calculate the total interest paid:

The total interest paid can be determined by subtracting the borrowed amount from the total amount paid over the 5 years.

Total payments=P×n=500×60=30,000\text{Total payments} = P \times n = 500 \times 60 = 30,000 Total interest=Total paymentsAmount borrowed\text{Total interest} = \text{Total payments} - \text{Amount borrowed}

Let me calculate that for you.The total interest you will pay over 5 years is approximately $4,322.90.

Summary:

  • Amount you can borrow: $25,677.10
  • Total interest paid: $4,322.90

Would you like more details on how this is calculated, or have any further questions?

Here are 5 related questions to explore:

  1. How does a different interest rate affect the amount you can borrow?
  2. What if you increase the monthly payment to $600?
  3. How would the total interest change if the loan term is reduced to 3 years?
  4. How does a bi-weekly payment plan compare to a monthly one?
  5. How does making a larger initial payment affect the loan amount?

Tip: Always compare different loan terms and interest rates to minimize the total interest paid over time.

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Math Problem Analysis

Mathematical Concepts

Finance
Annuities
Compound Interest
Present Value Calculation

Formulas

Present Value of an Annuity: PV = P × (1 - (1 + r)^(-n)) / r
Total Interest Paid: Total Payments - Amount Borrowed

Theorems

Present Value Theorem for Annuities

Suitable Grade Level

Grades 10-12