Avni and Beena started a business by investing Rs. P and Rs.1.2P respectively. After ‘x’ months, Avni withdrew her entire amount and Chetna entered into business. After end of 9 months, Beena increased her initial investment by 25%. On completion of one year, share of Avni and Beena in entire profit was Rs.14200. If Beena would have increased her investment after ‘x’ months, then ratio of share of Avni and Beena in entire profit would have been 10: 27 respectively.

If ratio of their profit shares was 20: 51, then find the difference between profit shares of Avni and Beena,

The correct answer is Rs.6200.

Let the initial investment of Avni be Rs. P and that of Beena be Rs. 1.2P.

According to the first scenario:
Avni invests Rs. P for x months.
Beena invests Rs. 1.2P for the first 9 months, and then increases her investment by 25% for the remaining 3 months.
Beena's increased investment = $1.2P\times (1+0.25)=1.5P$.
Total equivalent investment-months for Beena = $(1.2P\times 9)+(1.5P\times 3)=10.8P+4.5P=15.3P$.

According to the hypothetical scenario:
If Beena had increased her investment after x months instead of 9 months:
Beena invests Rs. 1.2P for x months and Rs. 1.5P for the remaining $(12-x)$ months.
Total equivalent investment-months for Beena in this case = $(1.2P\times x)+1.5P\times (12-x)=1.2Px+18P-1.5Px=18P-0.3Px$.

The ratio of the profit shares of Avni and Beena in this hypothetical case is given as 10:27.
$\frac{P\times x}{18P-0.3Px}=\frac{10}{27}$
Canceling P from the numerator and denominator:
$\frac{x}{18-0.3x}=\frac{10}{27}$
Cross-multiplying gives:
$27x=10\times (18-0.3x)$
$27x=180-3x$
$30x=180$
$x=6$.

Using $x=6$ in the actual scenario:
Ratio of profit shares of Avni and Beena = $\frac{6P}{15.3P}=\frac{6}{15.3}=\frac{60}{153}=\frac{20}{51}$.

This matches the given ratio of 20:51.
The sum of their profit shares is Rs. 14200.
Let Avni's profit share be 20y and Beena's profit share be 51y.
$20y+51y=14200$
$71y=14200$
$y=200$.

The difference between the profit shares of Avni and Beena is:
$51y-20y=31y$
Difference = $31\times 200=6200$.
Therefore, the difference between the profit shares of Avni and Beena is Rs. 6200.