P sells an item to Q at 50% profit. Q sells the same item to R at 60% loss. If R paid ₹ 3234 for the item, then at what price (in ₹) is the item purchased by P?

The correct option is 5390 (or ₹ 5390).

Let's solve this problem step-by-step to find the cost price of the item for P.

Step 1: Define the variables
Let the cost price of the item for P be $x$ rupees.

Step 2: Calculate the price at which Q buys the item
P sells the item to Q at a profit of 50%.
Selling Price for P (which is Cost Price for Q) = $x+0.50\times x=1.50x$

Step 3: Calculate the price at which R buys the item
Q sells the item to R at a loss of 60%.
Selling Price for Q (which is Cost Price for R) = $1.50x\times (1-0.60)$

$\text{Selling Price for Q}=1.50x\times 0.40=0.60x$

Step 4: Equate to the price paid by R and solve for $x$
We are given that R paid ₹ 3234 for the item.

$0.60x=3234$

$x=\frac{3234}{0.60}$

$x=\frac{32340}{6}=5390$

Therefore, the price at which the item was purchased by P is ₹ 5390.