Introduction to Simple Interest and Time Calculation

Understanding the principles of simple interest and time calculation is fundamental in financial mathematics. The current problem involves determining how long it will take for a sum of money to become seven times itself under the condition of simple interest. This article will explore the logic and formulas used in solving such problems, providing a clear and detailed process.

Simple Interest Basics

Simple interest is calculated using the formula:

[ text{Simple Interest (SI)} frac{P times R times T}{100} ]

Where:

P is the principal (initial amount of money). R is the rate of interest per year. T is the time in years.

Problem Setup and Solution

Given:

A sum of money becomes five times its principal in 5 years at simple interest. We need to calculate the time required for the same sum to become seven times its principal.

Solution 1: Using Given Information

Let's denote the principal as (P). If the sum of money becomes five times itself in 5 years, the amount is (5P). The simple interest earned over this period is (4P).

Simple Interest (SI) (4P) [ text{Time (T)} 5 text{ years} ]

Using the simple interest formula:

[ text{SI} frac{P times R times T}{100} ]

Solving for (R):

[ 4P frac{P times R times 5}{100} ] [ R frac{4P times 100}{P times 5} 80% ]

Now, for the sum to become seven times the principal, the interest should be (6P).

[ text{Time (T)} frac{6P times 100}{P times 80} frac{600}{80} 7.5 text{ years} ]

Solution 2: Using Rate of Interest Directly

Given that the principle becomes 5 times itself in 5 years, the rate of interest per year can be calculated as 80%. For the sum to become 7 times itself, we can use the following approach:

[ text{Time (T)} frac{6P times 100}{P times 80} frac{600}{80} 7.5 text{ years} ]

Solution 3: Step-by-Step Calculation

Assume the principal (P).

Given (P ) becomes (3P) in 4 years, it becomes (7P) (double of (3P)) in another 4 years, which is a total of 8 years. However, the problem specifically asks for the time to become 7 times, not 8.

The increase in 4 years is 200%, so in 2 years, the increase is 100%. Therefore, for 600% increase, the time required is 12 years.

Conclusion

Through various methods, we have determined that it takes 7.5 years for a sum of money to become 7 times itself under simple interest. This highlights the importance of understanding and applying simple interest formulas effectively.

Keywords

simple interest, compound interest, time calculation, financial mathematics