Explanation:

Correct option 2 : 0.5P = 15Q 

The marked price of the article is ((100+ Q)/100) x P
The discount given is 25%,
So the selling price of the article is ((100-25)/100) x Marked price = 0.75 x ((100 + Q)/100) x P
The profit earned is Rs. (Q + 20),
So,
Selling price - Cost price = Q + 20
0.75 x ((100+Q)/100) x P - P = Q + 20
(3/4) x (P + PQ/100) PQ+ 20 -------------------------(1)
Case II:
The marked price of the article is ((100+ Q+5)/100) x P = ((105 + Q)/100) x P
The discount given is 25%,
So the selling price of the article is ((100-25)/100) x Marked price = 0.75 x ((105+Q)/100) x P
The profit earned is Rs. (Q+65),
So,
Selling price - Cost price = Q + 65
0.75 x ((105 + Q)/100) x P - P = Q +65
3/4 x (P + Px (Q+5)/100) - P=Q+ 65----------------(2)
Subtract equation (1) from equation (2), (3/4) x (5P/100) = 45
P = Rs.1200.
Put this in equation (1),
(3/4) x (1200+ 12Q) - 1200 = Q+ 20
900 +9Q-1200 = Q + 20
8Q=320
Q = 40.
Checking the options:
0.25P = 1.2Q-82
LHS = 0.25 x 1200 = 300.
RHS = 1.2 x 40 - 64 = -16.
LHS is not equal to RHS.
0.5P = 15Q
LHS = 0.5 x 1200 = 600.
RHS = 15 x 40 = 600.
LHS = RHS.
P = 30 + 100
LHS = 1200,
RHS = 30+ 10 x 40 = 430
LHS is not equal to RHS.
P = 20Q
LHS = 1200,
RHS = 20 x 40 = 800.
LHS is not equal to RHS.
Hence, the correct answer is 2.