Understanding Compound Interest Growth

This problem deals with the concept of compound interest. In compound interest, the earnings from interest are added to the principal amount, and then the new, larger principal earns interest. We are given that a sum of money triples in 8 years and need to find out how long it takes to become 9 times its original value.

Step 1: Setting up the Growth Scenario

Let the initial principal amount be '$P$'.

We are told that the money triples itself in 8 years. This means the final amount '$A$' becomes $3P$ when the time '$t$' is 8 years.

The formula for compound interest is:

Where:

• $A$ = Final Amount
• $P$ = Principal Amount
• $r$ = Annual interest rate
• $t$ = Time in years

Applying this to our first condition:

To simplify, we divide both sides by $P$ (assuming $P \neq 0$):

This equation shows the growth factor over 8 years.

Step 2: Determining the Time for 9 Times Growth

We want to find the time '$T$' when the amount becomes 9 times the principal, i.e., $A = 9P$.

Using the compound interest formula again:

Dividing both sides by $P$:

Step 3: Relating the Growth Factors

Notice that 9 is the square of 3, i.e., $9 = 3^2$.

We can substitute the expression for $3$ from Step 1 ($3 = (1+r)^8$) into the equation $9 = 3^2$:

Using the exponent rule $(x^a)^b = x^{a \times b}$:

Now we have two expressions equal to 9:

1. $9 = (1+r)^T$ (from Step 2)
2. $9 = (1+r)^{16}$ (derived above)

By comparing these two equations, we can equate the exponents:

Step 4: Simple Logic Explanation

Another way to think about this is that compound interest causes exponential growth. If the money multiplies by 3 in 8 years, to multiply by 9 (which is $3 \times 3$), it needs to undergo this tripling process twice.

Since each tripling takes 8 years, two tripling periods would take:

Final Answer

Therefore, the sum of money will amount to 9 times itself in 16 years.