Ronan takes a loan of K30 000. He repays the loan with 15% p.a. simple interest in one year. What is the loan balance after the first repayment has been made fortnightly?
Answer (2)
Ronan's loan balance after making his first fortnightly repayment of K1326.92 will be approximately K28846.16. This amount is calculated after accounting for the interest paid during that period. Simple interest calculations were used to derive the total interest and repayment amounts.
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Calculate the annual interest: I n t eres t = 30000 × 0.15 = K 4500 .
Determine the total amount to repay: T o t a l = 30000 + 4500 = K 34500 .
Calculate the fortnightly repayment: F or t ni g h tl y P a y m e n t = 26 34500 ≈ K 1326.92 .
Find the remaining balance after the first repayment: R e mainin g B a l an ce = 30000 − ( 1326.92 − 26 4500 ) ≈ K 28846.16 .
K 28846.16
Explanation
Understanding the Problem Let's break down this loan problem step by step so we can understand how the loan balance changes after the first repayment.
Calculating Annual Interest First, we need to calculate the annual interest on the loan. The formula for simple interest is:
I n t eres t = P r in c i p a l × R a t e × T im e
Where:
Principal is the initial loan amount (K30 000)
Rate is the annual interest rate (15% or 0.15)
Time is the loan term in years (1 year)
Calculating Total Amount to Repay Plugging in the values, we get:
I n t eres t = 30000 × 0.15 × 1 = K 4500
So, the annual interest is K4500.
Determining the Number of Fortnightly Repayments Next, we find the total amount to be repaid, which is the sum of the principal and the interest:
T o t a l = P r in c i p a l + I n t eres t = 30000 + 4500 = K 34500
Calculating the Fortnightly Repayment Amount Since the repayments are made fortnightly (every two weeks), we need to find out how many fortnights there are in a year. There are approximately 52 weeks in a year, so there are 26 fortnights:
F or t ni g h t s = 2 weeks/fortnight 52 weeks = 26 fortnights
Calculating the Interest Paid in the First Fortnight Now, we calculate the amount of each fortnightly repayment by dividing the total amount to be repaid by the number of fortnights:
F or t ni g h tl y P a y m e n t = F or t ni g h t s T o t a l = 26 34500 ≈ K 1326.92
Calculating the Principal Repaid in the First Fortnight To find out how much of the first repayment goes towards interest, we calculate the interest accrued in the first fortnight. Since it's simple interest, we can divide the annual interest by the number of fortnights:
I n t eres t f or t ni g h t = F or t ni g h t s A nn u a l I n t eres t = 26 4500 ≈ K 173.08
Calculating the Remaining Loan Balance The amount of the first repayment that goes towards reducing the principal is the difference between the fortnightly payment and the interest paid in the first fortnight:
P r in c i p a l re p ai d = F or t ni g h tl y P a y m e n t − I n t eres t f or t ni g h t = 1326.92 − 173.08 ≈ K 1153.84
Final Answer Finally, we calculate the remaining loan balance after the first repayment by subtracting the principal repaid from the initial loan amount:
R e mainin g B a l an ce = P r in c i p a l − P r in c i p a l re p ai d = 30000 − 1153.84 ≈ K 28846.16
Therefore, the loan balance after the first repayment is approximately K28846.16.
Final Answer The loan balance after the first repayment is approximately K28846.16.
Examples
Understanding loan repayments is crucial in personal finance. For instance, when buying a car or a house, knowing how each payment affects the remaining balance helps in planning your budget and understanding the true cost of borrowing. This calculation shows how the principal is reduced with each payment, allowing you to foresee when the loan will be fully paid off and how much interest you will pay in total. This knowledge empowers you to make informed financial decisions.
