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 a man invested Rs. 1.2P for x years on simple interest at the rate of 2.5x % p.a. and received total interest of Rs. 6480, then find P?

The correct answer is 6000.

Step 1: Understand the investment of Avni and Beena and find the value of $x$.

Let Avni's initial investment be $P$ and Beena's initial investment be $1.2P$.
Avni withdrew her entire amount after $x$ months. Thus, Avni's investment duration is $x$ months.
Equivalent investment of Avni = $P\times x$

If Beena had increased her investment by 25% after $x$ months instead of 9 months, then:
For the first $x$ months, Beena's investment is $1.2P$.
For the remaining $(12-x)$ months, her investment is increased by 25%, which becomes:
$1.2P\times 1.25=1.5P$

Therefore, Beena's total equivalent investment in this hypothetical case is:
$(1.2P\times x)+[1.5P\times (12-x\left)\right]$
$=1.2Px+18P-1.5Px$
$=P(18-0.3x)$

The hypothetical ratio of the profit share of Avni and Beena is given as 10 : 27:
$\frac{Px}{P(18-0.3x)}=\frac{10}{27}$
$\frac{x}{18-0.3x}=\frac{10}{27}$
$27x=10\times (18-0.3x)$
$27x=180-3x$
$30x=180$
$x=6$

Step 2: Solve the second part of the question to find $P$.

A man invested Rs. $1.2P$ for $x$ years on simple interest at the rate of $2.5x\%$ p.a.
Using $x=6$:
Principal ($\mathrm{PR}$) = $1.2P$
Time ($T$) = $6$ years
Rate ($R$) = $2.5\times 6=15\%$ p.a.

The simple interest formula is:
$\mathrm{SI}=\frac{\text{Principal}\times \text{Rate}\times \text{Time}}{100}$

Given the received interest is Rs. 6480:
$6480=\frac{1.2P\times 15\times 6}{100}$
$6480=\frac{1.2P\times 90}{100}$
$6480=1.2P\times 0.9$
$6480=1.08P$
$P=\frac{6480}{1.08}$
$P=6000$

Thus, the value of $P$ is Rs. 6000.