Determining the Rate of Compound Interest for Quadrupling a Sum in Two Years

When seeking to understand the precise rate of compound interest needed to quadruple a sum of money in two years, one can use mathematical formulas and calculations to arrive at the solution. This article explores the process with a step-by-step explanation, along with practical examples and the relevance of the calculation in financial contexts.

Understanding the Formula and Variables

In compound interest calculations, the formula used is:

A P(1 r/n)^(nt)

Where:

A is the amount of money accumulated after n years, including interest. P is the principal amount (the initial sum of money). r is the annual interest rate (in decimal notation). n is the number of times that interest is compounded per year. t is the number of years the money is invested or borrowed for.

Calculating the Required Rate of Compound Interest

To find the rate of compound interest required for a sum of money to become four times itself in 2 years, we use the simplified compound interest formula:

A P(1 r)^n

In this case, we are looking for A 4P and n 2 years.

Substituting these values into the formula gives:

4P P(1 r)^2

Dividing both sides by P (assuming P ≠ 0) and simplifying, we get:

4 (1 r)^2

Next, we take the square root of both sides:

sqrt{4} 1 r

This simplifies to:

2 1 r

Now, solving for r:

r 1

To express r as a percentage, we multiply by 100:

r 1 times; 100 200%

Therefore, the rate of compound interest required for a sum of money to become four times itself in two years is 200% per annum.

Practical Examples and Relevance

Consider a scenario where the sum of money needs to quadruple in two years. Using the example where the principal (P) becomes 2P in the first year and then 4P in the second year:

At 100% interest rate, the sum of money becomes four times itself in two years.

If we let x be the principal sum and r be the rate of interest, with period 2 years, then:

x(1 r/100)^2 4x

(1 r/100)^2 4

Hence, r 100.

Another example where compounding occurs over four years:

P(1 r/100)^4 4P

(1 r/100)^4 4

Hence, r 31.6% approximately.

Conclusion

The rate of compound interest needed for a sum of money to quadruple in two years is 200%. This calculation is crucial for understanding investment growth, loan payments, and interest rates in financial planning. By leveraging the compound interest formula and step-by-step calculations, individuals can make informed decisions about their financial future.