Calculate Compound Interest: Rs. 100 Loan Over 1 Year

Have you ever wondered how compound interest really works, especially when the interest rate keeps changing? Let's dive into a fascinating problem involving a loan of Rs. 100. Rocky took out this loan, and the interest rates are anything but constant. They increase progressively each month. We'll break down how to calculate the total amount of interest earned at the end of one year. This isn't your typical, straightforward interest calculation; it's a dynamic journey through ever-increasing rates, making it a great exercise in understanding financial mathematics.

Understanding Progressive Simple Interest

In this problem, Rocky's loan starts with a simple interest rate of 6% per annum for the first month. But that's just the beginning! For the second month, the rate jumps to 12% per annum, then to 24% per annum for the third month, and this pattern continues. The crucial point here is that the interest is simple interest each month, but the rate itself is compounding. This means we need to calculate the interest for each month separately and then add it all up. It's important to note that while the rates are compounding, the interest calculation for each month is based on the principal amount for that specific month, and then the accumulated interest from previous months is added to the principal for the subsequent month's calculation, effectively making it a compound interest scenario over the year. The initial principal is Rs. 100. The interest rates are given per annum, but we are calculating interest monthly. So, for monthly calculations, we need to divide the annual rate by 12.

Month-by-Month Interest Calculation

Let's break down the interest calculation month by month for the entire year.

Month 1: Principal (P1) = Rs. 100 Annual Interest Rate (R1) = 6% p.a. Monthly Interest Rate (r1) = 6% / 12 = 0.5% = 0.005 Interest for Month 1 (I1) = P1 * r1 = 100 * 0.005 = Rs. 0.50 Amount at the end of Month 1 (A1) = P1 + I1 = 100 + 0.50 = Rs. 100.50

Month 2: Principal (P2) = A1 = Rs. 100.50 Annual Interest Rate (R2) = 12% p.a. Monthly Interest Rate (r2) = 12% / 12 = 1% = 0.01 Interest for Month 2 (I2) = P2 * r2 = 100.50 * 0.01 = Rs. 1.005 Amount at the end of Month 2 (A2) = P2 + I2 = 100.50 + 1.005 = Rs. 101.505

Month 3: Principal (P3) = A2 = Rs. 101.505 Annual Interest Rate (R3) = 24% p.a. Monthly Interest Rate (r3) = 24% / 12 = 2% = 0.02 Interest for Month 3 (I3) = P3 * r3 = 101.505 * 0.02 = Rs. 2.0301 Amount at the end of Month 3 (A3) = P3 + I3 = 101.505 + 2.0301 = Rs. 103.5351

This pattern continues for all 12 months. The annual interest rate doubles each month. So, the annual rates will be: 6%, 12%, 24%, 48%, 96%, 192%, 384%, 768%, 1536%, 3072%, 6144%, 12288%.

As you can see, the monthly interest rates will also double each month: 0.5%, 1%, 2%, 4%, 8%, 16%, 32%, 64%, 128%, 256%, 512%, 1024%.

Let's continue the calculation for a few more months to see the trend:

Month 4: Principal (P4) = A3 = Rs. 103.5351 Annual Interest Rate (R4) = 48% p.a. Monthly Interest Rate (r4) = 48% / 12 = 4% = 0.04 Interest for Month 4 (I4) = P4 * r4 = 103.5351 * 0.04 = Rs. 4.141404 Amount at the end of Month 4 (A4) = P4 + I4 = 103.5351 + 4.141404 = Rs. 107.676504

Month 5: Principal (P5) = A4 = Rs. 107.676504 Annual Interest Rate (R5) = 96% p.a. Monthly Interest Rate (r5) = 96% / 12 = 8% = 0.08 Interest for Month 5 (I5) = P5 * r5 = 107.676504 * 0.08 = Rs. 8.61412032 Amount at the end of Month 5 (A5) = P5 + I5 = 107.676504 + 8.61412032 = Rs. 116.29062432

Month 6: Principal (P6) = A5 = Rs. 116.29062432 Annual Interest Rate (R6) = 192% p.a. Monthly Interest Rate (r6) = 192% / 12 = 16% = 0.16 Interest for Month 6 (I6) = P6 * r6 = 116.29062432 * 0.16 = Rs. 18.606500 Amount at the end of Month 6 (A6) = P6 + I6 = 116.29062432 + 18.606500 = Rs. 134.897124

Continuing this process for all 12 months:

Note: Values are rounded to two decimal places for simplicity in the table. The exact calculations will yield slightly different, but much larger, final amounts due to the exponential growth.

Calculating the Total Interest Earned

To find the total interest earned at the end of one year, we need to sum up the interest earned in each of the 12 months. However, a more direct way to calculate the total amount is to take the final amount at the end of Month 12 and subtract the initial principal amount.

Let's re-calculate more precisely without rounding intermediate steps for better accuracy:

A_final = P_initial * (1 + r1) * (1 + r2) * ... * (1 + r12)

Where r_n is the monthly interest rate for month n.

• r1 = 0.005
• r2 = 0.01
• r3 = 0.02
• r4 = 0.04
• r5 = 0.08
• r6 = 0.16
• r7 = 0.32
• r8 = 0.64
• r9 = 1.28
• r10 = 2.56
• r11 = 5.12
• r12 = 10.24

Let's perform the cumulative multiplication:

Amount after Month 1 = 100 * (1 + 0.005) = 100.5 Amount after Month 2 = 100.5 * (1 + 0.01) = 101.505 Amount after Month 3 = 101.505 * (1 + 0.02) = 103.5351 Amount after Month 4 = 103.5351 * (1 + 0.04) = 107.676504 Amount after Month 5 = 107.676504 * (1 + 0.08) = 116.29062432 Amount after Month 6 = 116.29062432 * (1 + 0.16) = 134.8971243112 Amount after Month 7 = 134.8971243112 * (1 + 0.32) = 178.0641907574 Amount after Month 8 = 178.0641907574 * (1 + 0.64) = 292.0253336422 Amount after Month 9 = 292.0253336422 * (1 + 1.28) = 665.8377497042 Amount after Month 10 = 665.8377497042 * (1 + 2.56) = 2370.364398924 Amount after Month 11 = 2370.364398924 * (1 + 5.12) = 14506.37728245 Amount after Month 12 = 14506.37728245 * (1 + 10.24) = 163049.4603027

The total amount at the end of one year is approximately Rs. 163049.46.

To find the total interest earned, we subtract the initial principal from the final amount:

Total Interest = Final Amount - Initial Principal Total Interest = Rs. 163049.46 - Rs. 100 Total Interest = Rs. 162949.46

This demonstrates the power of compounding, especially with rapidly increasing interest rates. Even a small initial amount can grow exponentially under such conditions. The options provided in the question (Rs. 2145, Rs. 2130.5) seem to be based on a different interpretation or perhaps a misunderstanding of how the interest compounds over the year with these escalating rates. The calculated amount clearly shows a much higher figure due to the geometric progression of the interest rates.

Key Takeaways

This problem highlights several important financial concepts:

• Compounding: The interest earned in one period is added to the principal, and then the next period's interest is calculated on this new, larger principal. This is why the amount grows so rapidly.
• Exponential Growth: When the rate of growth itself is growing (as the interest rate doubles each month), the overall growth becomes exponential.
• Importance of Interest Rates: Even small differences in interest rates can lead to significant differences in the final amount over time, especially in a compounding scenario.
• Calculation Accuracy: For problems involving rapid growth or many periods, precise calculations are essential. Rounding too early can lead to substantial errors in the final result.

It's fascinating to see how a seemingly simple loan with initially low rates can balloon into such a large sum due to the aggressive increase in interest rates. This serves as a powerful illustration of financial mathematics in action.

If you're interested in exploring more about compound interest and its effects, you might find resources on Investopedia very helpful. They offer detailed explanations and tools to understand various financial concepts, including the long-term impact of compounding.