Understanding the Shopkeeper's False Scale Problem

This problem involves a shopkeeper who uses a deceptive practice to make a profit despite advertising a loss. The shopkeeper claims to sell cloth at a 7% loss on the cost price. However, they use a false scale, meaning they give the customer less cloth than what is paid for (1 metre). This discrepancy allows them to achieve an actual gain of 24%.

We need to find the actual length of cloth the shopkeeper gives when the customer believes they are receiving 1 metre.

Setting up the False Scale Calculation

Let's assume the cost price of 1 metre of cloth is $C$.

• The shopkeeper advertises selling at a 7% loss. So, the advertised selling price for 1 metre is $C - 0.07C = 0.93C$. This is the price the customer pays for what they *think* is 1 metre.
• Let the actual length of cloth the shopkeeper gives out instead of 1 metre be $L$ metres.
• The actual cost price for the shopkeeper for the cloth they *actually* give out ($L$ metres) is $C \times L$.
• The actual selling price the shopkeeper receives for this $L$ metres of cloth is the price of 1 metre, which is the advertised selling price: $0.93C$.
• The shopkeeper's actual gain is 24%. This gain is calculated on the actual cost price.

Calculating the Actual Length Used

The formula for actual gain percentage is:

$$ \text{Actual Gain \%} = \frac{\text{Actual Selling Price} - \text{Actual Cost Price}}{\text{Actual Cost Price}} \times 100 $$

We know the actual gain is 24%, the actual selling price is $0.93C$, and the actual cost price is $CL$. Substituting these values into the formula:

$$ 24 = \frac{0.93C - CL}{CL} \times 100 $$

Let's solve for $L$:

Divide both sides by 100:

$$ 0.24 = \frac{0.93C - CL}{CL} $$

Multiply both sides by $CL$:

$$ 0.24 \times CL = 0.93C - CL $$

$$ 0.24CL = 0.93C - CL $$

Add $CL$ to both sides:

$$ 0.24CL + CL = 0.93C $$

$$ (0.24 + 1)CL = 0.93C $$

$$ 1.24CL = 0.93C $$

Since $C$ is the cost price per metre (and must be non-zero), we can divide both sides by $C$:

$$ 1.24L = 0.93 $$

Now, solve for $L$:

$$ L = \frac{0.93}{1.24} $$

To simplify the fraction, we can multiply the numerator and denominator by 100:

$$ L = \frac{93}{124} $$

This value of $L$ is in metres. We need to find the length in centimetres. Since 1 metre = 100 centimetres, we multiply $L$ by 100:

$$ \text{Actual length in cm} = \frac{93}{124} \times 100 $$

We can simplify the fraction $\frac{93}{124}$. Both 93 and 124 are divisible by 31 ($93 = 3 \times 31$ and $124 = 4 \times 31$).

$$ \text{Actual length in cm} = \frac{3 \times 31}{4 \times 31} \times 100 $$

$$ \text{Actual length in cm} = \frac{3}{4} \times 100 $$

$$ \text{Actual length in cm} = 3 \times \frac{100}{4} $$

$$ \text{Actual length in cm} = 3 \times 25 $$

$$ \text{Actual length in cm} = 75 \text{ cm} $$

Therefore, the actual length the shopkeeper uses instead of 1 metre (100 cm) is 75 cm.

Summarizing the Problem and Solution

Here's a breakdown of the key information and the result:

The shopkeeper tricks the customer by giving only 75 cm of cloth while charging for 100 cm at a price that appears to be at a loss, but results in a significant actual gain.

Revision Table: Profit and Loss with False Weights

Additional Information on False Scale Problems

Problems involving false weights or scales are common in quantitative aptitude sections of many exams. The core idea is that the seller cheats by giving less quantity than promised, thereby increasing their effective selling price relative to the actual cost of the goods given.

• Identifying the deception: The key is to recognize that the 'selling price' for the actual quantity given is the price charged for the *marked* quantity.
• Calculating Profit/Loss: Always calculate the actual profit or loss based on the cost price of the quantity *actually* sold (the smaller quantity given by the seller) and the price *actually* received (the price charged for the larger, marked quantity).
• Common variations: These problems can involve selling at cost price but using a false weight, selling at a gain/loss percentage while using a false weight, or mixing false weights with other discounts or markups.
• Unit Consistency: Ensure that the units (e.g., metres, grams, kilograms, percentages) are consistent throughout the calculation. Convert everything to a base unit early in the process.

Understanding the difference between the advertised transaction and the actual transaction is crucial for solving these types of problems accurately.