Answer: Option D. -> 21%
$$\eqalign{
& \text{Required %} \cr
& = \left(\frac{35\times1000}{167.5\times1000}\times100\right)\% \cr
& = \frac{350\times100}{1675}\% \cr
& = \frac{1400}{67}\% \cr
& = 20.89\% \cr
& \approx 21\% \cr} $$

Answer: Option B. -> 13 : 16
Ratio of number of students from Town B to that from Town A
$$\eqalign{
& = \frac{32.5\times1000}{40\times1000} \cr
& = \frac{325}{400} \cr
& = \frac{13}{16} \cr
& = 13:16 \cr} $$

Answer: Option A. -> 33500
Average number of students appearing in Aptitude Test from all towns
  $$ = \frac{\left(40+32.5+17.5+42.5+35\right)\times1000}{5}$$
$$\eqalign{
& = \frac{167.5\times1000}{5} \cr
& = 33.5\times1000 \cr
& = 33500 \cr} $$

Answer: Option A. -> 243%
$$\eqalign{
& \text{Required percentage} \cr
& = \left(\frac{42.5\times1000}{17.5\times1000}\times100\right)\% \cr
& = \left(\frac{425}{175}\times100\right)\% \cr
& = \frac{425\times4}{7}\% \cr
& = \frac{1700}{7}\% \cr
& = 242.85\% \cr
& \approx 243\% \cr} $$

Answer: Option A. -> Rs. 2.25 lakh
Let the investment of A in 2008 be Rs. $$x$$. Then,
$$\eqalign{
& \left(\frac{160}{100}\times x\right) = 24\text{ lakh} \cr
& \Rightarrow x = \left(24\times\frac{5}{8}\right) \text{lakh} \cr
& \Rightarrow x = 15 \text{ lakh} \cr} $$
∴ Investment of A in 2008 = 15 lakh
Profit of A in 2008
= Rs. (24 - 15) lakh
= Rs. 9 lakh
Investment of A in 2007
= Investment of A in 2008
= Rs. 15 lakh
Profit of A in 2007
$$\eqalign{
& =\text{Rs.} \left(\frac{45}{100}\times15\text{ lakh}\right) \cr
& =\text{Rs.} \frac{27}{4}\text{ lakh} \cr
& =\text{Rs. 6.75 lakh} \cr} $$
Difference between the profits of A in 2007 and 2008
= Rs. (9 - 6.75) lakh
= Rs. 2.25 lakh

Answer: Option A. -> Rs. 13.8 lakh
Let the investment of A in 2006 be Rs. $$x$$
Then,
$$\eqalign{
& \text{Profit = 55% of Rs. }x \cr
& =\text{Rs.}\left(x\times\frac{55}{100}\right) \cr
& =\text{Rs.}\left(\frac{11x}{20}\right) \cr
& \therefore \frac{11x}{20} = 10.12\text{ lakh} \cr
& \Rightarrow \frac{11x}{20} = 10.12\times100000 \cr
& \Rightarrow \frac{11x}{20} = 1012000 \cr
& \Rightarrow x = 1012000\times\frac{20}{11} \cr
& \Rightarrow x = 1840000 \cr
& \Rightarrow x = \text{18.4 lakh} \cr} $$
  $$\Rightarrow$$ Investment of A in 2006 = 18.4 lakh

Answer: Option D. -> Rs. 20 lakh
Average profit of A and B in 2010
$$\eqalign{
& = \frac{1}{2}\times\left[\left(\frac{90}{100}\times25\text{ lakh}\right)+\left(\frac{70}{100}\times25\text{ lakh}\right)\right] \cr
& = \frac{1}{2}\times\left(\frac{45}{2}+\frac{35}{2}\right)\text{lakh} \cr
& = \frac{80}{4}\text{ lakh} \cr
& = \text{20 lakh} \cr} $$

Answer: Option C. -> 34 : 31
Let the investment of each in 2005 be Rs. $$x$$
Then,
(Income of A) : (Income of B)
  = Rs. ($$x$$ + 70% of $$x$$) : Rs. ($$x$$ + 55% of $$x$$) [∵ Income = (Investment) + (Profit)]
$$\eqalign{
& = \left(x+\frac{70x}{100}\right) : \left(x+\frac{55x}{100}\right)\cr
& = \left(x+\frac{7x}{10}\right) : \left(x+\frac{11x}{20}\right) \cr
& = \frac{17x}{10} : \frac{31x}{20} \cr
& = 34 : 31 \cr} $$

Answer: Option B. -> Rs. 20.70 lakh
Let the investment of A in 2009 be Rs. $$x$$
Then,
Investment of B
$$\eqalign{
& = \text{50% of Rs. }x\cr
& =\text{Rs.} \left(x\times\frac{50}{100}\right) \cr
& =\text{Rs. } \frac{x}{2} \cr
& \therefore \left(x+\frac{x}{2}\right) = 27 \text{ lakh} \cr
& \Rightarrow \frac{3x}{2} = 27 \text{ lakh} \cr
& \Rightarrow x = \left(27\times\frac{2}{3}\right) \text{lakh} \cr
& \Rightarrow x = \text{18 lakh} \cr} $$
  ∴ Investment of A in 2009 = Rs. 18 lakh and investment of B = Rs. 9 lakh.
$$\eqalign{
& \text{Profit of A} \cr
& =\text{Rs.} \left(18\text{ lakh}\times \frac{75}{100}\right) \cr
& = \text{Rs. 13.50 lakh} \cr
& \text{Profit of B} \cr
& =\text{Rs.} \left(9\text{ lakh}\times \frac{80}{100}\right) \cr
& = \text{Rs. 7.20 lakh} \cr
& \therefore \text{Total profit of A and B} \cr
& = \text{Rs.}\left(13.50+7.20\right)\text{lakh} \cr
& = \text{Rs. 20.70 lakh} \cr} $$

Answer: Option C. -> Rs. 10.23 lakh
$$\eqalign{
& \text{Investment of B in 2004 = 12 lakh} \cr
& \text{Income of B in 2004} \cr
& =\text{Rs.}\left(\frac{155}{100}\times12\text{ lakh}\right) \cr
& = \text{Rs. 18.6 lakh} \cr
& \text{Profit earned in 2005} \cr
& =\text{Rs.}\left(\frac{55}{100}\times18.6\text{ lakh}\right) \cr
& =\text{Rs.}\left(\frac{11}{10}\times9.3 \text{ lakh}\right) \cr
& =\text{Rs. 10.23 lakh} \cr} $$