$A$ sells an article to $B$ at a gain of $20 \%$,$B$ sells it to $C$ at a gain of $10 \%$,and $C$ sells it to $D$ at a gain of $12 \frac{1}{2} \%$. If $D$ pays $₹ 29.70$,then $A$ purchased the article for (in $₹$):

$A$ fruit seller bought $240$ bananas at the rate of $Rs. 48$ per dozen. He sells half of them at the rate of $Rs. 5$ per banana. $\frac{1}{6}$ of the remaining bananas are found to be rotten. At what price per banana must he sell the remaining bananas to get a profit of $25\%$ on his entire investment?

Raman sells a machine for $Rs. 39$ lakh at a loss. Had he sold it for $Rs. 49$ lakh,his gain would have been three times the loss. What is the cost price of the machine? (in $Rs. \text{ lakh}$)

$A$ trader buys $500 \, kg$ of cotton for $Rs. 9,000$. $10 \%$ of this cotton is spoiled due to rain. At what rate $(Rs./ kg)$ should he sell the rest to earn $10 \%$ profit?

In these questions,two equations numbered $I$ and $II$ are given. You have to solve both the equations and mark the appropriate option.
$I.$ $2x^2 + 19x + 45 = 0$
$II.$ $2y^2 + 11y + 12 = 0$

$A$ sells his house worth $Rs. 10\, lakh$ to $B$ at a loss of $10\%$. Later,$B$ sells it back to $A$ at $10\%$ profit. The result of the two transactions is: