A sum of money at compound interest amounts to fifth-hold itself in 7 years. In how many years will it be 25 times itself?

Let’s denote the principal amount as \(P\), the rate of compound interest as \(r\), and the number of years as \(n\).

According to the given information, the amount becomes fivefold in 7 years. Therefore, we can write the compound interest formula as follows:

\[ A = P(1 + r)^n \]

For the amount to be fivefold:

\[ 5P = P(1 + r)^7 \]

Simplifying:

\[ (1 + r)^7 = 5 \]

Now, we need to find in how many years the amount will be 25 times itself. We can set up a similar equation:

\[ 25P = P(1 + r)^n \]

\[ (1 + r)^n = 25 \]

We know that \((1 + r)^7 = 5\), so we can express 25 in terms of 5:

\[ (1 + r)^n = 5^2 \]

\[ (1 + r)^n = ((1 + r)^7)^2 \]

\[ (1 + r)^n = (1 + r)^{14} \]

Thus, we can see that \(n = 14\).

Therefore, it will take 14 years for the sum of money to become 25 times itself at the same rate of compound interest.