(second question of 2 that i need help with)

A $25000 loan is repaid over 8 years with monthly instalments of $391. What was the interest rage (p.a.) changed on the loan?

Po = $25,000.

t = 8yrs. * 12mo./yr. = 96 mo.

P = 391/mo. * 96mo. = $37,536.

Int./yr. = (P-Po)/t = (37,536-25,000)/8yrs. = $1567.

APR

APR = (1567/37,536)*100 = 4.17%.

To calculate the interest rate, we can use the formula for the monthly installment of a loan:

monthly installment = (loan amount * monthly interest rate) / (1 - (1 + monthly interest rate)^(-number of months))

Given:
- Loan amount (P) = $25000
- Monthly installment (A) = $391
- Number of months (n) = 8 * 12 = 96 (since there are 12 months in a year)

Let's assume the monthly interest rate is r.

Using the given formula, we can rearrange it to find the monthly interest rate:

monthly interest rate = 2 * (A / P) / [(sqrt(1 + (2 * (A / P) * n)) - 1)]

Plugging in the values we know:

monthly interest rate = 2 * ($391 / $25000) / [(sqrt(1 + (2 * ($391 / $25000) * 96)) - 1)]

Calculating this will give us the interest rate per month. To get the annual interest rate, we multiply the monthly interest rate by 12:

annual interest rate = monthly interest rate * 12

Let's calculate it step by step:

Step 1: Calculate the monthly interest rate
monthly interest rate = 2 * ($391 / $25000) / [(sqrt(1 + (2 * ($391 / $25000) * 96)) - 1)]
monthly interest rate ≈ 0.0095833

Step 2: Calculate the annual interest rate
annual interest rate = monthly interest rate * 12
annual interest rate ≈ 0.115

Therefore, the interest rate per annum for the loan is approximately 11.5%.

To find the interest rate (p.a.) charged on the loan, we can use the loan repayment formula:

Monthly Payment = (Loan Amount * Monthly Interest Rate) / (1 - (1 + Monthly Interest Rate) ^ (-n))

Where,
Monthly Payment = $391
Loan Amount = $25000
n = 8 years * 12 months/year = 96 months

We can rearrange the formula to solve for the Monthly Interest Rate:

Monthly Interest Rate = ((Monthly Payment / Loan Amount) - 1) / (1 + (1 / ((1 + Monthly Interest Rate) ^ (-n))))

Since the Monthly Interest Rate appears on both sides of the equation, we will need to use an iterative approach to find the solution.

Let's start by assuming an initial Monthly Interest Rate value, such as 0.01 (1%). We'll use this value to calculate a new Monthly Interest Rate using the formula. If the calculated Monthly Payment matches the given value of $391 within an acceptable margin of error, we can consider the calculated Monthly Interest Rate as the answer.

Here's the step-by-step process:

1. Assume an initial Monthly Interest Rate value, such as 0.01 (1%).
2. Substitute the assumed Monthly Interest Rate value into the formula.
3. Calculate the value of the left side of the equation.
4. Check if the calculated Monthly Payment matches the given value of $391 (within an acceptable margin of error, considering rounding).
5. If the calculated Monthly Payment matches the given value, the assumed Monthly Interest Rate is the answer.
6. If the calculated Monthly Payment does not match the given value, adjust the assumed Monthly Interest Rate value (e.g., increase or decrease by a small increment) and repeat steps 2-5 until a close match is found.

Using this iterative approach, you can find the Monthly Interest Rate charged on the loan.