Avoiding calculations in a CI question

Well there is no suitable hint that can be given, since it is more of a intelligent number-play to guess the multiplying factor corresponding to interest rate, so at best, let us see some examples when interest rates are known.

If the interest rate is 10%, the ratio of CI in any two consecutive years will be 10 : 11. And the CI in 1st, 2nd and 3rd year will be in the ratio 100 : 110 : 121.

If the interest rate is 12.5%, the ratio of CI in any two consecutive years will be 8 : 9. And the CI in 1st, 2nd and 3rd year will be in the ratio 64 : 72 : 81.

Knowing that the ratio of CI in 1st, 2nd and 3rd year will be in ratio a² : ab : b², now try to fit such a ratio to the given data.

To appreciate the explanation, you first need to have tried to solve the question using conceptual methods. Only when you experience that whatever approach you try, you cannot avoid a factorising a very tedious quadratic equation, only then will you appreciate the importance of just trying to fit in the given data to a framework, while resorting to some intelligent guess-work.

CI in 3rd year = 2382 – 1518 = 864.

Finding 864 is a multiple of which perfect square …. 2 × 432; 3 × 288 i.e. 144 × 6, multiple of 12².

Hoping that the CI in 1st, 2nd and 3rd year are in the ratio, a² : 12a : 144, it is easier to factorise the quadratic a² + 12a – 1518 = 0 (as compared to the ones you will need to factorise using conceptual methods).

And better still, just try to fit in the value of a.

Which one of the following do you think is the ratio of the CI in the 3 years?
11 : 12 ⇒ 121 : 132 : 144.
Multiplying by 6 interests are 726, 792 and 864. Bingo! 726 + 792 = 1518. Hence this is the answer.
10 : 12 ⇒ 100 : 120 : 144. Is 1518 same as (100 + 120) × 6. No.
9 : 12 ⇒ 81 : 108 : 144. Is 1518 same as (81 + 108) × 6. No, the unit’s digit does not match.

Once we know the multiplying factor is 12/11 i.e. rate of interest is 1/11, finding P is as easy as 726 × 11 = 7986.