A trader sells one variety of sugar at ₹36 per kg, incurring a 10% loss, and another variety at ₹48 per kg, incurring a 20% loss. If he mixes the two varieties in equal quantities and sells the mixture at ₹42 per kg, what is his overall profit or loss percentage?

Correct Answer: Option 1 (Loss of 16%)

Step-by-step Explanation:

Let us analyze the cost price and selling price for each variety of sugar step-by-step.

Step 1: Find the Cost Price (CP) of the First Variety

The trader sells the first variety at a selling price (SP₁) of ₹36 per kg, incurring a 10% loss.

The relation between Selling Price and Cost Price during a loss is given by:

${\text{SP}}_1={\text{CP}}_1\times (1-\frac{\text{Loss \%}}{100}$

Substituting the given values:

$36={\text{CP}}_1\times (1-\frac{10}{100})$

$36={\text{CP}}_1\times \frac{90}{100}$

${\text{CP}}_1=\frac{36\times 100}{90}=\text{₹}40\text{per kg}$

Step 2: Find the Cost Price (CP) of the Second Variety

The trader sells the second variety at a selling price (SP₂) of ₹48 per kg, incurring a 20% loss.

Using the same cost price formula:

${\text{SP}}_2={\text{CP}}_2\times (1-\frac{\text{Loss \%}}{100}$

$48={\text{CP}}_2\times (1-\frac{20}{100})$

$48={\text{CP}}_2\times \frac{80}{100}$

${\text{CP}}_2=\frac{48\times 100}{80}=\text{₹}60\text{per kg}$

Step 3: Calculate the Total Cost Price and Selling Price of the Mixture

Since the two varieties are mixed in equal quantities, let us assume he mixes 1 kg of the first variety and 1 kg of the second variety (total quantity = 2 kg).

Total Cost Price of 2 kg of mixture:

$\text{Total CP}={\text{CP}}_1+{\text{CP}}_2=40+60=\text{₹}100$

The mixture is sold at ₹42 per kg.

Total Selling Price of 2 kg of mixture:

$\text{Total SP}=2\times 42=\text{₹}84$

Step 4: Calculate the Overall Profit or Loss Percentage

Since Total SP (₹84) is less than Total CP (₹100), there is an overall loss.

$\text{Total Loss}=\text{Total CP}-\text{Total SP}=100-84=\text{₹}16$

Overall Loss Percentage is calculated as:

$\text{Loss \%}=\left(\frac{\text{Total Loss}}{\text{Total CP}}\right)\times 100$

$\text{Loss \%}=\left(\frac{16}{100}\right)\times 100=16\text{\%}$

Thus, the trader incurs an overall Loss of 16%.