Peter immediately deducts 5,000 euros for the new car, he will pay the rest of the purchase price in 60 equal monthly installments of 180 euros at the end of the month (p=4.8%). How much does the car cost him today

To determine the total cost of the car today, we need to calculate the present value of the 60 monthly installments and then add the 5,000 euros Peter paid upfront.

We are given:
- Initial payment: €5,000
- Monthly installment: €180
- Number of installments: 60
- Monthly interest rate: \( p = 4.8\% \) per year (which is 0.4% per month, or \( \frac{4.8}{12} \))

The present value of the monthly payments is given by the formula for the present value of an annuity:

\[
PV = C \times \frac{1 - (1 + r)^{-n}}{r}
\]

Where:
- \( PV \) is the present value of the installment payments
- \( C = 180 \) euros is the monthly payment
- \( r = \frac{4.8\%}{12} = 0.004 \) (monthly interest rate)
- \( n = 60 \) (number of payments)

### Step-by-step solution:

1. Calculate the present value of the installment payments.
2. Add the €5,000 initial payment to the present value of the installments.

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Here are a few related questions that might interest you:
1. How can we calculate the future value of monthly savings?
2. How does compound interest differ from simple interest?
3. What are annuities, and how are they calculated?
4. How do changing interest rates affect loan repayments?
5. What is the difference between present value and future value in financial math?

**Tip:** Understanding how interest rates affect loan repayments is key to managing long-term debt effectively!

Peter immediately deducts 5,000 euros for the new car, he will pay the rest of the purchase price in 60 equal monthly installments of 180 euros at the end of the month (p=4.8%). How much does the car cost him today?

Learn how to calculate the present value of 60 monthly installments of 180 euros with a 4.8% annual interest rate, and find out the total cost of a car after an initial payment of 5,000 euros. This detailed solution covers annuity concepts and interest rate calculations, suitable for Grades 10-12.

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