Compound Interest Time Calculation

This problem involves calculating the time required for an investment to grow based on compound interest principles.

Let the principal sum be P. The formula for the amount (A) accumulated after time (t) years at a certain compound interest rate is:

$ A = P(1 + r)^t $

where r is the annual interest rate.

Condition 1: Amount Triples in 4 Years

We are given that the amount becomes 3 times the principal (3P) in 4 years. Using the formula:

$ 3P = P(1 + r)^4 $

Divide both sides by P:

$ 3 = (1 + r)^4 $

This equation establishes the growth factor over 4 years.

Condition 2: Amount Becomes 9 Times

We need to find the time (let's denote it as T) for the amount to become 9 times the principal (9P).

$ 9P = P(1 + r)^T $

Divide both sides by P:

$ 9 = (1 + r)^T $

Relating the Conditions for Time Calculation

Notice that $9$ is the square of $3$ (i.e., $9 = 3^2$). We can substitute the expression for $3$ from the first condition into the equation for the second condition:

$ 9 = 3^2 = \left((1 + r)^4\right)^2 $

Using the property of exponents $ (x^a)^b = x^{a \times b} $:

$ 9 = (1 + r)^{4 \times 2} $

$ 9 = (1 + r)^8 $

Final Time Determination

Now we have two expressions equal to 9:

• From Condition 2: $ 9 = (1 + r)^T $
• Derived relation: $ 9 = (1 + r)^8 $

By comparing these two equations, we can determine the value of T:

$ T = 8 $

Thus, Suresh should invest his money for 8 years so that it amounts to 9 times the sum he initially lent.