An investor lent-out a certain sum on simple interest and the same sum on compound interest at the same rate of interest per annum. He noticed that the ratio of the difference of the compound interest and the simple interest for 4 years to the difference of the compound interest and the simple interest for 3 years is 20:8. The approximate rate of interest per annum is given by,

Let '$$p$$' be the principal amount and '$$r$$' be the interest rate per annum.

Simple Interest for t years = $$ptr$$

Compound Interest for t years = $$p\left(1+r\right)^t-p$$

From the given information,

$$\frac{\left(p\left(1+r\right)^4-p\right)-4pr}{\left(p\left(1+r\right)^3-p\right)-3pr}=\frac{20}{8}$$

$$\frac{\left(\left(1+r\right)^4-1\right)-4r}{\left(\left(1+r\right)^3-1\right)-3r}=\frac{5}{2}$$

$$2\times\ \left(\left(1+r\right)^4-1-4r\right)=5\times\ \left(\left(1+r\right)^3-1-3r\right)$$

$$2\times\ \left(1+r\right)^4-2-8r=5\times\ \left(1+r\right)^3-5-15r$$

$$2\times\ \left(1+r\right)^4=5\times\ \left(1+r\right)^3-3-7r$$

$$\ 2r^4+8r^3+12r^2+8r+2=\ 5r^3+15r^2+15r+5-3-7r$$

$$\ 2r^4+8r^3+12r^2=\ 5r^3+15r^2$$

$$\ 2r^2+3r-3=0$$

Hence, $$r=\frac{-3\pm\ \sqrt{\ 9+24}}{2\times\ 2}$$

$$r=\frac{-3\pm\ \sqrt{\ 33}}{4}$$

As interest rate cannot be negative,

$$r=\frac{-3+\ \sqrt{\ 33}}{4}$$ 

$$r=0.6861$$

Hence, the approximate value of $$r=69\%$$

Option (A) is correct.