Understanding Profit Calculation with Faulty Weights

This problem involves a businessman who uses faulty weights to gain extra profit during both the buying and selling processes. Cheating with weights affects the actual quantity of material received or given, impacting the cost price and selling price per unit of actual weight. We need to calculate the percentage increase in his profit solely due to this dishonest practice.

Analyzing the Cheating Mechanism

The businessman cheats by 12% each time. This means:

• When Buying: He pays for a certain weight but receives 12% more actual weight. If he intends to buy 100 units of weight (and pays for 100 units), he actually gets $100 \times (1 + 0.12) = 112$ units of actual weight.
• When Selling: He charges for a certain weight but delivers 12% less actual weight. If he intends to sell 100 units of weight (and charges for 100 units), he actually gives $100 \times (1 - 0.12) = 88$ units of actual weight.

Calculating the Profit from Faulty Weights

Let's assume the true price (Cost Price or Selling Price if there were no cheating) of one unit of weight is $P$.

Consider the quantity the businessman obtains when he pays for 100 units using the faulty weight during buying:

• He pays for 100 units, so his cost is $100 \times P = 100P$.
• Due to cheating, he receives 112 units of actual weight for this cost.
• So, the actual cost price for 112 units is $100P$.

Now, consider selling these 112 units of actual weight using the faulty weight during selling:

• When selling, 88 units of actual weight are charged as 100 units using the faulty weight.
• This means 1 unit of actual weight is charged as $100 / 88$ units using the faulty weight.
• So, 112 units of actual weight will be charged as $112 \times (100/88)$ units using the faulty weight.
• The total revenue for selling 112 actual units will be the price of $112 \times (100/88)$ units at the true price $P$ per unit.
• Total Revenue $= \left(112 \times \frac{100}{88}\right) \times P$
• Total Revenue $= \left(\frac{11200}{88}\right) \times P = \left(\frac{1400}{11}\right) \times P$

Now we can calculate the profit made on the 112 units of actual weight:

• Total Cost for 112 units = $100P$
• Total Revenue from selling 112 units = $\frac{1400}{11}P$
• Profit = Total Revenue - Total Cost $= \frac{1400}{11}P - 100P$
• Profit $= \left(\frac{1400 - 1100}{11}\right)P = \frac{300}{11}P$

The profit percentage is calculated on the cost price. The cost price for the 112 units was $100P$.

Profit Percentage $= \frac{\text{Profit}}{\text{Cost}} \times 100\%$

Profit Percentage $= \frac{\frac{300}{11}P}{100P} \times 100\%$

Profit Percentage $= \frac{300}{1100} \times 100\%$

Profit Percentage $= \frac{3}{11} \times 100\%$

To convert the fraction $\frac{3}{11}$ to a percentage:

$\frac{3}{11} \times 100 = \frac{300}{11}$

Now, perform the division:

So, the rise in his profit percentage using faulty weights only is approximately 27.27%.

Summary of Calculation

Let's summarize the steps in a table:

Effective Cost Price per actual unit $= \frac{100P}{112}$

Effective Selling Price per actual unit $= \frac{100P}{88}$

Profit Percentage $= \left(\frac{\text{Effective SP per unit} - \text{Effective CP per unit}}{\text{Effective CP per unit}}\right) \times 100\%$

Profit Percentage $= \left(\frac{\frac{100P}{88} - \frac{100P}{112}}{\frac{100P}{112}}\right) \times 100\%$

Profit Percentage $= \left(\frac{\frac{100}{88} - \frac{100}{112}}{\frac{100}{112}}\right) \times 100\%$

Profit Percentage $= \left(\frac{\frac{1}{0.88} - \frac{1}{1.12}}{\frac{1}{1.12}}\right) \times 100\%$

Profit Percentage $= \left(\frac{1.12 - 0.88}{0.88 \times 1.12} \times 1.12\right) \times 100\%$

Profit Percentage $= \left(\frac{0.24}{0.88}\right) \times 100\%$

Profit Percentage $= \frac{0.24}{0.88} \times 100 = \frac{24}{88} \times 100 = \frac{3}{11} \times 100 = \frac{300}{11} \approx 27.27\%$

The calculated profit percentage is approximately 27.27%.

Revision Table: Faulty Weights & Profit

Additional Information: Dishonest Dealer Problems

Problems involving dishonest dealers and faulty weights are common in competitive exams focusing on quantitative aptitude. The key is to always think about the actual quantity of the item being traded versus the declared or weighed quantity.

Common scenarios include:

• Dealer gives less weight than marked (profit on selling).
• Dealer takes more weight than marked (profit on buying).
• Dealer marks up price AND uses faulty weights.
• Dealer claims to sell at cost price but uses faulty weights.

The percentage profit or loss is always calculated based on the actual cost price incurred for the actual quantity traded.