A sum of money triples itself in 7 years. In how many years it amounts to 9 times of itself, if the interest is compounded annually?

Let’s denote the principal amount as \(P\) and the interest rate as \(r\) (expressed as a decimal).

Given:
– The money triples itself in 7 years. This means that the amount \(A\) after 7 years is \(3P\).
– The interest is compounded annually.

The formula for compound interest is:
\[ A = P(1 + r)^t \]
where \(A\) is the amount after \(t\) years, \(P\) is the principal, \(r\) is the interest rate, and \(t\) is the time in years.

From the given information, we have:
\[ 3P = P(1 + r)^7 \]
\[ 3 = (1 + r)^7 \]
\[ (1 + r) = \sqrt[7]{3} \]

Now, we need to find the time \(t\) when the amount becomes 9 times of itself:
\[ 9P = P(1 + r)^t \]
\[ 9 = (1 + r)^t \]
\[ 9 = (\sqrt[7]{3})^t \]
\[ 9 = 3^{\frac{t}{7}} \]
\[ 3^2 = 3^{\frac{t}{7}} \]
\[ 2 = \frac{t}{7} \]
\[ t = 14 \]

Conclusion

The money will amount to 9 times of itself in 14 years if the interest is compounded annually.