Understanding the Problem: Dishonest Dealer and Faulty Weights

This problem involves a dealer who uses faulty weights to cheat customers during both the buying and selling process. The cheating percentage is given as \(x\%\) in both transactions. We need to find the overall gain percentage the dealer makes on their initial investment (outlay).

Impact of Faulty Weight in Buying

When the dealer buys goods, they use a faulty weight to get more quantity than they pay for. If they defraud by \(x\%\) in buying, it means for every 100 units of weight they are supposed to get according to the payment, they actually receive \(100 + x\) units of weight. In other words, they pay the price of 100 units but receive \(100+x\) actual units.

Let's assume the true cost price per unit of weight is \(C\). When the dealer buys, they pay for 100 units of weight, costing \(100C\). However, they receive \(100+x\) actual units of weight. Therefore, the effective cost price (CP) for the dealer for \(100+x\) actual units is \(100C\).

The effective CP per actual unit of weight is: \( \text{Effective CP} = \frac{\text{Total Cost}}{\text{Actual Quantity Received}} = \frac{100C}{100+x} \) We can assume the true cost price \(C=1\) for simplicity, as the gain is solely due to faulty weights. \( \text{Effective CP per actual unit} = \frac{100}{100+x} \)

Impact of Faulty Weight in Selling

When the dealer sells goods, they use a faulty weight to give less quantity than they charge for. If they defraud by \(x\%\) in selling, it means for every 100 units of weight they are supposed to sell according to the charge, they actually give \(100 - x\) units of weight. In other words, they charge the price of 100 units but give \(100-x\) actual units.

Let's assume the true selling price per unit of weight is \(S\). When the dealer sells, they give \(100-x\) actual units but charge the price of 100 units, which is \(100S\). Therefore, the effective selling price (SP) for the dealer for \(100-x\) actual units is \(100S\).

The effective SP per actual unit of weight is: \( \text{Effective SP} = \frac{\text{Total Revenue}}{\text{Actual Quantity Sold}} = \frac{100S}{100-x} \) Again, assume the true selling price \(S=1\) for simplicity (or assume \(S=C=1\)). \( \text{Effective SP per actual unit} = \frac{100}{100-x} \)

Calculating the Total Gain Percentage

The dealer's gain comes from the difference between the effective selling price and the effective cost price per unit of actual weight. The gain percentage is calculated on the effective cost price (outlay).

Gain per actual unit = Effective SP - Effective CP \( \text{Gain} = \frac{100}{100-x} - \frac{100}{100+x} \) \( \text{Gain} = 100 \left( \frac{1}{100-x} - \frac{1}{100+x} \right) \) \( \text{Gain} = 100 \left( \frac{(100+x) - (100-x)}{(100-x)(100+x)} \right) \) \( \text{Gain} = 100 \left( \frac{100+x - 100+x}{10000 - x^2} \right) \) \( \text{Gain} = 100 \left( \frac{2x}{10000 - x^2} \right) = \frac{200x}{10000 - x^2} \)

Gain Percentage = \( \frac{\text{Gain}}{\text{Effective CP}} \times 100 \) \( \text{Gain } \% = \frac{\frac{200x}{10000 - x^2}}{\frac{100}{100+x}} \times 100 \) \( \text{Gain } \% = \frac{200x}{(100-x)(100+x)} \times \frac{100+x}{100} \times 100 \) \( \text{Gain } \% = \frac{200x}{100-x} \times \frac{100}{100} \) \( \text{Gain } \% = \frac{200x}{100-x} \)

So, the gain percent on his outlay is \( \left( \frac{200x}{100-x} \right)\%\). Note: Some interpretations of "defrauds to the extent of x%" lead to \( \left( 2x + \frac{x^2}{100} \right)\% \), but the phrasing with faulty weights typically implies the error is in the weight measurement itself, leading to the calculation above.

Comparing with Options

Let's compare our derived gain percentage formula \( \left( \frac{200x}{100-x} \right)\% \) with the given options:

1. \(2x\%\)
2. \( \left( {\frac{{10}}{x} + {x^2}} \right)\% \)
3. None of these
4. \( \left( {x + \frac{{{x^2}}}{{100}}} \right)\% \)

Our calculated gain percentage \(\left( \frac{200x}{100-x} \right)\%\) does not match options 1, 2, or 4. Therefore, based on the standard interpretation of cheating using faulty weights, the correct answer is "None of these".

Conclusion

By defrauding \(x\%\) in both buying and selling using faulty weights, the dealer achieves a total gain percentage on outlay of \( \left( \frac{200x}{100-x} \right)\%\). As this result is not provided in options 1, 2, or 4, the correct choice is "None of these".

Revision Table: Dishonest Dealer Calculation

Additional Information: Faulty Weight Problems

Faulty weight problems are common in profit and loss calculations. They usually involve the dealer cheating either the seller (while buying) or the buyer (while selling), or both.

• Cheating while buying: The dealer gets more quantity than intended for the price paid. This reduces the effective cost price per unit for the dealer. If the scale shows 1000g but gives \(1000(1+x/100)\)g, the dealer pays for 1000g but gets \(1000(1+x/100)\)g.
• Cheating while selling: The dealer gives less quantity than intended for the price charged. This increases the effective selling price per unit for the dealer. If the scale shows 1000g but gives \(1000(1-x/100)\)g, the dealer charges for 1000g but gives \(1000(1-x/100)\)g.
• The percentage \(x\%\) can be applied to the weight itself (e.g., the weight measures \(100 \pm x\) instead of 100) or to the value obtained/given relative to the price paid/charged. The wording "defrauds to the extent of x% by using faulty weight" most commonly implies the percentage error is in the weight delivered/received relative to the stated weight.
• The total gain percentage calculation combines these two effects. The gain is always calculated on the effective cost price (the actual cost incurred to acquire the goods).