-     Interest on Rs 1440 = Rs 216 for the third year        

Compound Interest: Compound interest is the interest that is earned on the initial principal, plus all previously accumulated interest. It is calculated using the formula: A = P (1 + r/n) ^nt, where A is the total amount, P is the principal amount, r is the rate of interest per annum, t is the time in years and n is the number of times the interest is compounded in a year.

Given:
Manish earns an interest of Rs 1656 for the third year and Rs 1440 for the second year on the same sum.

We need to find the rate of interest if it is lent at compound interest.

To solve this problem, we can use the above formula for compound interest.

Let P = Principal Amount
Let r = Rate of interest
Let t1 = Time for which interest of Rs 1656 was earned
Let t2 = Time for which interest of Rs 1440 was earned

Now, we can write the equation as follows:

1656 = P (1 + r/n) ^n(t1)
1440 = P (1 + r/n) ^n(t2)

Subtracting the two equations, we have:

216 = P (1 + r/n) ^n(t1) - P (1 + r/n) ^n(t2)

Simplifying, we have:

216 = P (1 + r/n)^n(t1 - t2)

Since t1 - t2 = 1

216 = P (1 + r/n)^n

Dividing both sides by P, we have:

(1 + r/n)^n = 216/P

Taking log on both sides, we have:

n log (1 + r/n) = log (216/P)

Rearranging, we have:

r/n = (log (216/P))/n

Now, substituting the given values, we have:

r/n = (log (216/P))/n = (log (216/P))/3

Therefore, the rate of interest (r) = 3 (log (216/P))

Now, substituting the given values, we have:

r = 3 (log (216/P)) = 3 (log (216/1000)) = 3 (log (0.216))

Therefore, the rate of interest (r) = 3 (log (0.216)) = 3 (-1.67) = -5.01

Therefore, the rate of interest (r) = -5.01%

Therefore, the correct answer is Option C - 15.

If you think the solution is wrong then please provide your own solution below in the comments section .