Renu loses 15% by selling an article A for ₹ 382.50 and gains 12% by selling article B for ₹ 616. If she sells both articles for ₹ 1,180, then her percentage gain is :

The correct option is 18%.

Step-by-Step Solution:

Step 1: Find the Cost Price of Article A

Selling Price of Article A (${\mathrm{SP}}_{\mathrm{A}}$) = ₹ 382.50

Loss percentage on Article A = 15%

The formula relating Cost Price ($\mathrm{CP}$), Selling Price ($\mathrm{SP}$), and Loss percentage is:

$\mathrm{CP}=\frac{100\times \mathrm{SP}}{100-\text{Loss\%}}$

Substituting the given values into the formula:

${\mathrm{CP}}_{\mathrm{A}}=\frac{100\times 382.50}{100-15}$

${\mathrm{CP}}_{\mathrm{A}}=\frac{38250}{85}=\text{₹}450$

Step 2: Find the Cost Price of Article B

Selling Price of Article B (${\mathrm{SP}}_{\mathrm{B}}$) = ₹ 616

Gain percentage on Article B = 12%

The formula relating Cost Price ($\mathrm{CP}$), Selling Price ($\mathrm{SP}$), and Gain percentage is:

$\mathrm{CP}=\frac{100\times \mathrm{SP}}{100+\text{Gain\%}}$

Substituting the given values into the formula:

${\mathrm{CP}}_{\mathrm{B}}=\frac{100\times 616}{100+12}$

${\mathrm{CP}}_{\mathrm{B}}=\frac{61600}{112}=\text{₹}550$

Step 3: Calculate Total Cost Price and Total Selling Price

Total Cost Price of both articles (${\mathrm{CP}}_{\text{total}}$):

${\mathrm{CP}}_{\text{total}}={\mathrm{CP}}_{\mathrm{A}}+{\mathrm{CP}}_{\mathrm{B}}=450+550=\text{₹}1000$

New combined Selling Price of both articles (${\mathrm{SP}}_{\text{total}}$) = ₹ 1,180

Step 4: Calculate Total Profit and Percentage Gain

Total Profit = Total Selling Price - Total Cost Price

$\text{Total Profit}=1180-1000=\text{₹}180$

Now, calculate the percentage gain on the total transaction:

$\text{Percentage Gain}=\left(\frac{\text{Total Profit}}{{\mathrm{CP}}_{\text{total}}}\right)\times 100$

$\text{Percentage Gain}=\left(\frac{180}{1000}\right)\times 100=18\%$

Thus, her percentage gain is 18%.