A sum of money amounts to ₹13,380 after 3 years and to ₹20,070 after 6 years at compound interest compounded annually. Find the sum.

Correct option is A

Given:

Amount after 3 years = ₹13,380
Amount after 6 years = ₹20,070
The interest is compounded annually.

Let the principal be P and the rate of interest be R%.

Formula Used:

The formula for compound interest is:
​$$A = P (1 + \frac{R}{100})^n$$​​
Where:
- A is the amount after n years,
- P is the principal,
- R is the rate of interest,
- n is the number of years.

Solution:
Calculate the interest for 3 years and 6 years:
​$$A_3 = P (1 + \frac{R}{100})^3 = 13,380\\\ \\A_6 = P (1 + \frac{R}{100})^6 = 20,070$$​​
Divide the two equations to eliminate P:
​$$\frac{A_6 }{ A_3} = \frac{(P (1 + \frac{R}{100})^6) }{ (P (1 + \frac{R}{100})^3)} \\\ \\\frac{20,070 }{13,380} = (1 + \frac{R}{100})^3 \\\ \\1.5 = (1 + \frac{R}{100})^3$$​​
Taking the cube root of both sides:
​$$1 + \frac{R}{100} = 1.1447\\\ \\\frac{R}{100} = 0.1447 \\\ \\R = 14.47 \%$$​​
Substitute R = 14.47 into the equation for 3 years:
​$$A_3 = P (1 + \frac{14.47}{100})^3 = 13,380 \\\ \\13,380 = P (1.1447)^3 \\\ \\13,380 = P \times1.5P =\frac{ 13,380 }{ 1.5} = ₹8,920$$ 

Alternative method: 

Since the money becomes Rs. 13380 after 3 years and Rs. 20070 after 6 years

=> $$\frac{20070}{13380} = 1.5$$​​

That means the money becomes 1.5 times after 3 more years.

∴ Principal = $$\frac{13380}{1.5 }$$​= Rs. 8920