Solution:

Given, the sum of money invested doubles itself in four years at the rate of compound interest.

Let the principal amount be P and the rate of interest be r.

After 4 years, the amount becomes 2P.

Using the compound interest formula, we get:

2P = P(1 + r/100)^4

Simplifying the equation, we get:

(1 + r/100)^4 = 2

Taking the fourth root on both sides, we get:

1 + r/100 = 2^(1/4)

r/100 = 2^(1/4) - 1

r = 100(2^(1/4) - 1)

r ≈ 7.18%

Therefore, the rate of interest is approximately 7.18%.

To find out the time taken for the amount to become 32 times of itself, we can use the following formula:

Amount = Principal(1 + r/100)^n

where n is the number of years.

Let the time taken for the amount to become 32 times of itself be t years.

Then we have:

32P = P(1 + r/100)^t

Simplifying the equation, we get:

(1 + r/100)^t = 32

Taking the logarithm on both sides, we get:

t log(1 + r/100) = log 32

t = log 32 / log(1 + r/100)

Substituting the value of r, we get:

t ≈ 10.04 years

Therefore, it will take approximately 10.04 years for the amount to become 32 times of itself at the same rate of compound interest.

Conclusion:

The sum of money invested doubles itself in four years at the rate of approximately 7.18% compound interest. It will take approximately 10.04 years for the amount to become 32 times of itself at the same rate of compound interest.

20%