Understanding Time to Double a Sum at Simple Interest

The concept of simple interest often confounds many, particularly when trying to determine how long it will take for a sum to double. This article will explore the problem of a sum of money doubling over a certain period of time at a given rate of interest. We'll solve a few examples to illustrate the process.

Example 1: Doubling in 20 Years at 11.11% Rate

Let the initial sum be Rs. 100. At a simple interest rate, the amount becomes 4 times in 20 years. We want to find out the time taken for the same sum to double at the same rate of interest.

Given that the amount A is 400 (original sum 100 times; 4) after 20 years, the formula for simple interest is:

(A P frac{PRT}{100})

Substituting the given values:

(400 100 frac{100 times; 100 times; R times; 20}{100})

Simplifying the equation:

(300 frac{20000R}{100}) (Rightarrow 30000 20000R) (Rightarrow R frac{30000}{20000}) 11.11%

To find the time taken for the sum to double at 11.11%, we use the same rate of interest:

(200 100 frac{100 times; 100 times; 11.11 times; T}{100})

Solving for (T):

(T frac{100}{11.11}) 9 years

Example 2: Doubling in 5 Years at 20% Rate

In another scenario, let the sum be Rs. 100 and the amount quadruples over 20 years. We aim to determine the time taken for the same sum to double at the same rate of interest.

Given that the amount A is 500 (original sum 100 times; 5) after 20 years:

(500 100 frac{100 times; 100 times; R times; 20}{100})

Simplifying the equation to find (R 20%):

(400 20000R) (Rightarrow R 20%)

Using the same rate of interest to find the time taken for the sum to double:

(200 100 frac{100 times; 100 times; 20 times; T}{100})

Solving for (T):

(T frac{100}{20}) 5 years

General Formula and Conclusion

From the solved examples, we can generalize the process. For a principal amount (P), if it takes (T) years to become 4 times at a rate of interest (R), the time taken to double is given by:

1. Determine the rate of interest using the formula:

(R frac{100 times; (A - P)}{P times; T})

2. Use the rate of interest to find the time taken to double the amount:

(T frac{100 times; (A - P)}{P times; R})

For example, if a sum becomes 5 times in 20 years at a rate of interest, the time taken to double is 5 years. Conversely, if a sum quadruples in 18 years at a rate of 100/9%, it will take 9 years to double at the same rate. The key takeaway is that the rate of interest doesn't change the time taken to double under simple interest conditions.

Understanding simple interest and its implications on the time taken to double a sum can be crucial for managing personal and business finances. It also helps in comparing different financial instruments and investment opportunities.