Calculating Gain Percentage for a Dishonest Dealer

This problem involves a common scenario where a shopkeeper tries to make extra profit by using tampered weights. We need to find the actual gain percentage when a dishonest dealer claims to sell goods at cost price but uses a faulty weight.

Understanding the Dealer's Trick with Faulty Weight

The core of the problem lies in the discrepancy between the weight the dealer claims to sell and the weight they actually provide. Here's the breakdown:

• The dealer professes to sell grains at the cost price.
• However, instead of using a standard 1 kg (which is 1000 g), the dealer uses a weight of only 925 g. This means for every 1 kg the customer thinks they are receiving, they are actually getting only 925 g.
• The dealer effectively buys goods at the price of 925 g but sells them at the price of 1000 g.

Step-by-Step Gain Percentage Calculation

To find the gain percentage, we compare the profit made against the actual cost incurred by the dealer.

Step 1: Determine the Cost Price (CP) Perspective

The dealer incurs cost based on the actual amount of grain given to the customer. Since the dealer uses a 925 g weight instead of 1 kg (1000 g), the cost price is effectively for 925 g of grain.

Let the cost price of 1 gram of grain be '$c$'.

Actual Cost Price (CP) = Cost of 925 g = $925 \times c$

Step 2: Determine the Selling Price (SP) Perspective

The dealer charges the customer the price equivalent to 1 kg (1000 g) of grain, even though they only give 925 g.

Apparent Selling Price (SP) = Price for 1000 g = $1000 \times c$

Step 3: Calculate the Gain Amount

The gain is the difference between the selling price and the cost price.

Gain = SP - CP

Gain = $(1000 \times c) - (925 \times c)$

Gain = $75 \times c$

Step 4: Calculate the Gain Percentage

The formula for gain percentage is:

Gain Percentage = $ (\frac{\text{Gain}}{\text{CP}}) \times 100 $

Substitute the values we found:

Gain Percentage = $ (\frac{75 \times c}{925 \times c}) \times 100 $

The '$c$' (cost price per gram) cancels out:

Gain Percentage = $ (\frac{75}{925}) \times 100 $

Step 5: Simplify and Find the Numerical Value

First, simplify the fraction $\frac{75}{925}$. Both numbers are divisible by 25:

$ 75 = 3 \times 25 $

$ 925 = 37 \times 25 $

So, the fraction becomes:

$ \frac{75}{925} = \frac{3}{37} $

Now, calculate the gain percentage:

Gain Percentage = $ \frac{3}{37} \times 100 = \frac{300}{37} $

Performing the division:

$ \frac{300}{37} \approx 8.108108... $

This is approximately $8.11\%$.

Interpreting the Options

The options provided are in decimal format (e.g., 0.0811). This suggests that the question might be asking for the gain as a decimal fraction of the cost price (Gain / CP) rather than the percentage value.

Let's calculate the gain fraction directly:

Gain Fraction = $ \frac{\text{Gain}}{\text{CP}} = \frac{75 \times c}{925 \times c} = \frac{75}{925} $

As calculated before:

$ \frac{75}{925} = \frac{3}{37} $

Converting this fraction to a decimal:

$ \frac{3}{37} \approx 0.08108108... $

Rounding this value to four decimal places, we get 0.0811.

This matches the correct answer option. Therefore, the dealer's gain, expressed as a decimal fraction and rounded, is approximately 0.0811.