Determining the Duration of a Loan with Simple Interest: A Comprehensive Guide

When it comes to understanding loan repayments, particularly those involving simple interest, accurately calculating the duration is crucial. This guide will walk you through the process using a specific scenario: a loan of $65,000 in March 2014, where the monthly interest rate is 3% per year, and the total amount paid back is $72,000. We will explore the step-by-step process, the formula for simple interest, and the key factors involved.

Understanding Simple Interest and Loan Repayments

Simple interest is a type of interest where the interest is calculated only on the original principal amount every year. The formula for simple interest is:

A P(1 rt)

where:

A is the total amount repayable, including the principal and the interest. P is the principal amount of the loan. r is the annual interest rate in decimal form. t is the time in years.

Application of the Formula: Determining the Duration of the Loan

In the given scenario, we have the following details:

Principal (P) $65,000 Total paid back (A) $72,000 Annual interest rate (r) 3%, or 0.03 in decimal form

We can use the formula to find the duration (t):

A P I

Where:

I is the total interest accrued. I A - P 72,000 - 65,000 7,000

The formula for simple interest is:

7,000 65,000 * 0.03 * t

Solving for t:

t 7,000 / (65,000 * 0.03) 7,000 / 1,950 ≈ 3.59 years

To convert this duration into a more understandable format, we can break it down into years and months:

3 full years About 0.59 of a year, which is approximately 0.59 * 12 7 months

Adding this to the start date of March 2014, the loan duration would end in October 2017.

Revisiting the Calculation with Python Code

To ensure our calculations are accurate, we can also use Python code. Here is a simple Python script to perform these calculations:

total_paid  72000
principal  65000
annual_interest_rate  0.03
# Calculate the interest
interest  total_paid - principal
def calculate_duration(total_paid, principal, interest_rate):
    return (total_paid - principal) / (principal * interest_rate)
time_in_years  calculate_duration(total_paid, principal, annual_interest_rate)
print(f'The duration of the loan is approximately {time_in_years:.2f} years, or about 3.6 years.')

Running this script, we get:

The duration of the loan is approximately 3.59 years, or about 3.6 years.

Additional Insights

Consider the following additional insights to better understand the loan scenario:

Calculation Breakdown

The breakdown of the calculations shows that over three years, interest accrued at 3% per year would amount to about $5,850. The total balance due in March 2017 would be $70,850. However, an extra $1,150 was paid, indicating an overpayment. This $1,150 can be attributed to the interest accrued on the extra days or months beyond the three years.

Daily Accrual Factor

To further break this down, we can use the daily accrual factor calculated as follows:

1950/365 ≈ 5.34 per day

(1150 / 5.34) ≈ 215 days

Since the first three years do not include a leap year, adding 215 days to March 2017 (end of the third year) lands us in August of the fourth year.

Conclusion

In conclusion, the duration of the loan, given the provided details, is approximately 3.6 years, or 3 years and 7 months. This comprehensive approach to understanding simple interest and loan duration will help borrowers and lenders alike in accurately calculating and managing loan repayments.