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.

Find the value of ‘x’.

To find the value of $x$, let's analyze the investment details of Avni, Beena, and Chetna step-by-step.

Let Avni's initial investment be Rs. $P$ and Beena's initial investment be Rs. $1.2P$.

First Scenario (Actual Case):
Avni invests Rs. $P$ for $x$ months before withdrawing her entire amount.
So, Avni's total investment-month product is:
$I_{\text{Avni}}=P\times x$

Beena invests Rs. $1.2P$ initially. After 9 months, she increases her investment by 25%.
25% of $1.2P$ is:
$0.25\times 1.2P=0.3P$
So, her new investment for the remaining 3 months (since the total period is 1 year or 12 months) is:
$1.2P+0.3P=1.5P$
Thus, Beena's total investment-month product is:
$I_{\text{Beena}}=(1.2P\times 9)+(1.5P\times 3)$
$I_{\text{Beena}}=10.8P+4.5P=15.3P$

Second Scenario (Hypothetical Case):
If Beena would have increased her investment by 25% after $x$ months instead of 9 months:
In this case, Beena's investment is Rs. $1.2P$ for $x$ months, and Rs. $1.5P$ for the remaining $(12-x)$ months.
So, Beena's hypothetical investment-month product is:
$I_{\text{Beena}}^{\text{'}}=1.2P\times x+1.5P\times (12-x)$
$I_{\text{Beena}}^{\text{'}}=1.2Px+18P-1.5Px$
$I_{\text{Beena}}^{\text{'}}=P(18-0.3x)$

We are given that in this hypothetical scenario, the ratio of the profit shares of Avni and Beena is 10:27.
Since profit shares are proportional to the investment-month products:
$\frac{I_{\text{Avni}}}{I_{\text{Beena}}^{\text{'}}}=\frac{10}{27}$
Substitute the expressions for $I_{\text{Avni}}$ and $I_{\text{Beena}}^{\text{'}}$:
$\frac{Px}{P(18-0.3x)}=\frac{10}{27}$

Cancel $P$ from the numerator and denominator:
$\frac{x}{18-0.3x}=\frac{10}{27}$

Cross-multiplying to solve for $x$:
$27x=10(18-0.3x)$
$27x=180-3x$
$27x+3x=180$
$30x=180$
$x=\frac{180}{30}$
$x=6$

Thus, the value of $x$ is 6.