The simple interest on a certain sum for 3 years in Rs. 225 and the compound interest on the same sum for 2 years is Rs. 165. Find the rate percent per annum. 

The Simple interest on a certain sum for 3 years in Rs. 225 and the compound interest on the same sum for 2 years is Rs. 165. Find the rate percent per annum.

The rate of interest per annum is 20%.

Given:

• Simple Interest (SI) for 3 years = Rs. 225

We find that the SI for 1 year is 225/3 = Rs. 75.

Since the compound interest for the second year is Rs. 90, and the simple interest for the second year is Rs. 75, the difference between compound interest and simple interest for the second year is 90 - 75 = Rs. 15.

Now, to find the rate of interest:

Rate of interest = (Difference / Simple interest for 1 year) * 100

= (15 / 75) * 100

= 0.20 * 100

= 20%

So, the rate of interest per annum is 20%.

Simple vs. Compound Interest: Understanding the Difference

Both simple interest and compound interest are ways to calculate the interest earned on money over time, but they differ in how they account for that growth:

Simple Interest:

• Calculated only on the initial principal amount (the original amount borrowed or invested).
• Interest earned in each period remains constant throughout the investment or loan term.
• Often used for short-term loans or investments, where the compounding effect is minimal.

Formula:

Simple Interest (SI) = Principal (P) x Rate (R) x Time (T) / 100

Compound Interest:

• Takes into account both the principal amount and the accumulated interest from previous periods. This is often referred to as "interest on interest."
• Interest earned grows exponentially over time, leading to a faster growth in the overall amount compared to simple interest.
• Used for long-term investments and loans, where the compounding effect can significantly increase the final amount.

Formula:

Amount = Principal (P) x (1 + Rate (R) / Number of compounding periods (n))^ (Time (T) x n)

Here's an example to illustrate the difference:

Let's say you invest $1000 for 5 years at a 10% annual interest rate.

• Simple Interest:
  • SI = 1000 x 10 x 5 / 100 = $500
  • Total amount after 5 years: $1000 (principal) + $500 (interest) = $1500
• Compound Interest:
  • Assuming annual compounding:
    • Amount = 1000 (1 + 0.1)^5 = $1638.62
  • The difference highlights the power of compound interest over time.