Understanding Compound Interest Growth

This question involves compound interest, where the interest earned also earns interest over time. This leads to exponential growth. Let the principal sum be '$P$'. Let the annual rate of compound interest be '$r$' (expressed as a decimal), and let the time be in years.

The formula for the amount '$A$' after '$t$' years is given by:

Analyzing the Given Information

We are given two key pieces of information:

• The sum of money triples itself in 17 years. This means that when $t = 17$ years, the amount $A = 3P$.
• We need to find the time ($t$) when the amount becomes 81 times the original sum ($A = 81P$).

Applying the Compound Interest Formula

Step 1: Use the first condition to find a relationship.

Using the formula $A = P(1+r)^t$, we substitute the first condition:

Divide both sides by $P$ (assuming $P \neq 0$):

This equation tells us that the growth factor over 17 years is 3.

Step 2: Set up the equation for the second condition.

We want to find the time '$t$' (let's call it $Y$ for years) when the amount is $81P$.

Divide both sides by $P$:

Step 3: Relate the two conditions.

Notice that 81 can be expressed as a power of 3:

Now substitute the value of 3 from Step 1 into this equation:

Using the rule of exponents $(a^m)^n = a^{m \times n}$, we get:

Step 4: Solve for the time '$Y$'.

We have two expressions for 81:

By comparing the exponents, we can see that:

Conclusion

Therefore, it will take 68 years for the sum of money to amount to 81 times itself at the given rate of compound interest.