Let P is the sum and rate of interest per annum is r%.

Given S₁ = 13230

and S₂ = 13891.5

Rate of interest half-yearly = r/2 %

Now, time n₁ = 2 and n₂ = 3          {since rate of interest is calculated as half yearly}

Now A = P(1 + r/100)ⁿ

So, 13230 = P{1 + (r/2)/100}²

=> 13230 = P{1 + r/200}² .................1

=> 13891.5 = P{1 + (r/2)/100}³

=> 13891.5 = P{1 + r/200}³   .............2

Now, divide equation 2 by equation 1, we get

=>  13891.5/13230 = [P{1 + r/200}³ ]/[P{1 + r/200}² ]

=> 1 + r/200 = 13891.5/13230

=> 1 + r/200 = 1.05

=> r/200 = 1.05 - 1

=> r/200 = .05

=> r/200 = 5/100

=> r/2 = 5

=> r = 2*5

=> r = 10

So, rate of interest is 10 %

Now, from equation 1, we get

      13230 = P{1 + 10/200}² 

=> 13230 = P{1 + 1/20}² 

 => 13230 = P{21/20}²

=> 13230 = P{(21/20)*(21/20)}

=> 13230 = 441P/400

=> P = (13230 * 400)/441

=> P = 30 * 400

=> P = 12000

So, the sum is Rs 12000