Question

A used car is offered for sale on an installment basis in which a down payment of RM5,000 has to be made, followed by a payment of RM1,000 at the end of the first month, RM950 at the end of the second month, RM900 at the end of the third month and so on for one year. What is the installment price of the car?

Answer 1

Answer: M13,693.76.R

Step-by-step explanation:

To determine the installment price of the car, you need to calculate the present value of the future payments. The present value is the current worth of a future sum of money, given a specified rate of return. In this case, the payments form an arithmetic sequence where each subsequent payment is reduced by a constant amount.

Let's denote:

- \( r \) as the constant reduction in monthly payments (in RM),

- \( n \) as the total number of monthly payments.

The present value (\( PV \)) of the future payments can be calculated using the formula for the sum of an arithmetic series:

\[ PV = \frac{a_1 \times (1 - (1 + r)^{-n})}{r} \]

where:

- \( a_1 \) is the first payment (RM1,000 in this case),

- \( r \) is the common difference (the reduction in payments),

- \( n \) is the total number of monthly payments.

Given that \( a_1 = 1000 \) (first payment), \( r = 50 \) (the reduction in each subsequent payment), and \( n = 12 \) (12 monthly payments), we can substitute these values into the formula to find the present value.

\[ PV = \frac{1000 \times (1 - (1 + 50)^{-12})}{50} \]

Now calculate \( PV \) to find the present value of the future payments. This value represents the installment price of the car.

\[ PV = \frac{1000 \times (1 - (1.05)^{-12})}{50} \]

\[ PV \approx \frac{1000 \times (1 - 0.315312)}{50} \]

\[ PV \approx \frac{1000 \times 0.684688}{50} \]

\[ PV \approx \frac{684.688}{50} \]

\[ PV \approx 13.69376 \]

Therefore, the installment price of the car is approximately M13,693.76.R

Answer 2

The installment price of the car, based on the initial down payment and the sequence of decreasing monthly payments, is RM13,700.

Step-by-step explanation:

Calculating the Installment Price of a Car

To find the installment price of the car, we need to calculate the total of all payments made over the span of one year. There is an initial down payment of RM5,000. Following that, there is a sequence of monthly payments starting at RM1,000 at the end of the first month and decreasing by RM50 each subsequent month for a year (12 months).

The sequence of monthly payments forms an arithmetic sequence where the first term (a) is RM1,000, the number of terms (n) is 12, and the common difference (d) is -RM50. The nth term of an arithmetic sequence is given by: a_n = a + (n - 1)d. The sum of an arithmetic sequence (S) can be found with the formula: S_n = n/2(2a + (n - 1)d).

Let’s calculate the sum of the monthly payments (S) using the formula. We have:
S = 12/2 [2(1,000) + (12 - 1)(-50)]
S = 6 [2,000 + 11(-50)]
S = 6 [2,000 - 550]
S = 6 [1,450]
S = 8,700

Now, to find the installment price of the car, we add the down payment to the sum of monthly payments:
Installment Price = Down Payment + Sum of Monthly Payments
Installment Price = RM5,000 + RM8,700
Installment Price = RM13,700

The installment price of the car is RM13,700.