Answer:
To calculate the value of each payment, we can set up an equation using the formula for the future value of an installment loan. In this case, we're dealing with two payments: one at 3 months and another at 10 months. The interest rate is 4.5% per annum, but we need to consider the ten-month period from now. We'll first calculate the future value (FV) of the loan at 10 months and then work backward to find the value of each payment.
Let's denote:
P1 = The first payment (settled at 3 months)
P2 = The second payment (settled at 10 months)
R = Annual interest rate = 4.5% = 0.045 (as a decimal)
n1 = Time to the first payment = 3 months
n2 = Time to the second payment = 10 months
We know that P2 is three times the amount of P1, so we can write:
P2 = 3P1
Now, let's calculate the future value (FV) of the loan at 10 months:
FV = P1(1 + R/12)^n1 + P2(1 + R/12)^n2
We'll use the value of FV, which is $5000 (the original loan amount), and solve for P1 and P2:
$5000 = P1(1 + 0.045/12)^3 + (3P1)(1 + 0.045/12)^10
Now, let's calculate the values:
For the first term:
P1(1 + 0.045/12)^3 = P1(1.00375)^3 ≈ 1.011282
For the second term:
(3P1)(1 + 0.045/12)^10 = 3P1(1.00375)^10 ≈ 1.04074 * 3P1 ≈ 3.1222P1
Now, we can rewrite the equation:
$5000 = 1.011282P1 + 3.1222P1
Combine like terms:
$5000 = 4.133482P1
Now, isolate P1 by dividing both sides by 4.133482:
P1 ≈ $5000 / 4.133482 ≈ $1,209.61 (rounded to two decimal places)
Now that we have the value of P1, we can find P2:
P2 = 3P1 ≈ 3 * $1,209.61 ≈ $3,628.83 (rounded to two decimal places)
So, the first payment (settled at 3 months) is approximately $1,209.61, and the second payment (settled at 10 months) is approximately $3,628.83.
Explanation:
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