If a sum of money doubles itself in 10 years at compound interest, then in how many years will it become 16 times of itself at the same rate?

Correct option is C

Given:

After every 10 years the sum of money doubles.

Formula Used:

$$\text{Amount} = P \cdot (1 + r)^t$$​

where, P =  Principle amount ,  r = rate, t = time period

Solution:

​If the money doubles in 10 years, it means:

​$$P \cdot (1 + r)^{10} = 2P$$​​

​$$(1 + r)^{10} = 2$$ 

​$$(1+r) = 2^{\frac{1}{10}}$$​​

Now, for the money to become 16 times the initial amount:

​$$P \cdot (1 + r)^t = 16P$$​​

​$$(1 + r)^t = 16$$

​$$(1 + r)^t = 2^4$$​​

Since $$(1 + r) = 2^{\frac{1}{10}}$$, So;

​$$\left[2^{\frac{1}{10}}\right]^t = 2^4$$​​

​$$2^{\frac{t}{10}} = 2^4$$​

Comparing the exponents, we get:

​$$\frac{t}{10} = 4$$

$$t= 40$$ 

Thus, In 40 years it became 16 time of itself.