Determining the Time for Simple Interest to be 75% of the Principal at Specific Interest Rates

In financial mathematics, understanding the relationship between the principal, the rate of interest, and the time is crucial for various applications, including loans, investments, and government bonds. A common question in this domain is to determine the time required for the simple interest on a certain sum to be 75% of the principal at a given annual interest rate. Let's explore how to solve such a problem through various examples and approaches.

Example 1

Let x be the principal sum. The rate of interest is 15% per annum. We need to find the required period n in years such that the simple interest is 0.75 times the principal.

Mathematically, we can express this problem as:

x * (10/100) * n 0.75x

Now, let's solve for n:

x * 10 * n / 100 0.75x

10n / 100 0.75

n (0.75 * 100) / 10

n 7.5 years

Example 2

Consider an initial amount of P. The interest is 10%, and the simple interest is 0.75 times the principal. The formula for simple interest is given by:

Simple Interest (P * N * R) / 100, where N is the number of years and R is the rate in percentage.

Given:

(0.75P) (P * N * 10) / 100

Solving for N:

75P / 100 (P * N * 10) / 100

75 N * 10

N 75 / 10

N 7.5 years

Example 3

Assume the principal is Rs. 100. The rate of interest is 15%. The simple interest would be Rs. 100 * 0.75 Rs. 75.

The formula for time is:

Time (N) (100 * Interest) / (P * R)

Time (N) (100 * 75) / (100 * 15)

Time (N) 75 / 15

N 5 years

Example 4

Let P be the principal. The simple interest is 0.75 times the principal.

Mathematically, we write:

(0.75P) (P * N * 15) / 100

Solving for N:

75P / 100 (P * 15 * N) / 100

75 15N

N 75 / 15

N 5 years

Example 5

Let P be the principal. The interest rate is 15%. The simple interest is 0.75 times the principal.

Mathematically, we write:

(0.75P) (P * N * 15) / 100

Solving for N:

0.75P (0.15P * N)

N 0.75 / 0.15

N 5 years

Therefore, the time required for the simple interest to be 75% of the principal at 15% per annum is 5 years.