Shopkeeper False Balance Problem: Actual Weight Calculation

This solution explains how to find the actual weight used by a shopkeeper who claims a loss but makes a profit using a false balance.

Understanding the Problem

A shopkeeper claims to sell goods at a 60% loss. However, by using a false balance, he actually gains 70%. We need to determine the actual weight used instead of 1 kg.

Key Concepts

• Marked Price (MP): The price the shopkeeper declares per unit weight.
• Cost Price (CP): The actual cost price per unit weight for the shopkeeper.
• False Balance: Used to provide less weight than stated.
• Gain/Loss Calculation: Gain/Loss % is calculated on the actual cost price.

Step-by-Step Solution

1. Define Variables and Relationships:

   • Let the cost price (CP) of 1 kg of goods be $C$.
   • Let the marked price (MP) declared by the shopkeeper be $M$.
   • The shopkeeper claims a 60% loss. Therefore, the marked price is:

     $M = C \times (1 - \frac{60}{100}) = C \times (1 - 0.60) = 0.40C$

   • Let $w$ be the actual weight (in kg) used instead of 1 kg.

2. Calculate Actual Cost Price and Selling Price:

   • The actual cost price for the goods supplied is the cost of $w$ kg:

     Actual CP = $C \times w$

   • The selling price (SP) is the amount received for $w$ kg. Since he uses a false balance, he charges the price equivalent to 1 kg at his marked rate ($M$).

     Selling Price (SP) = $M \times 1 = M$

3. Apply the Actual Gain Percentage:

   • The shopkeeper gains 70% overall. The formula for gain percentage is:

     Gain % = $ \frac{SP - \text{Actual CP}}{\text{Actual CP}} \times 100 $

   • Substitute the values:

     $70 = \frac{M - (C \times w)}{C \times w} \times 100$

   • Simplify the equation:

     $\frac{70}{100} = \frac{M - Cw}{Cw}$

     $0.70 = \frac{M}{Cw} - 1$

     $1.70 = \frac{M}{Cw}$

     $1.70 Cw = M$

4. Solve for the Actual Weight ($w$):

   • Substitute $M = 0.40C$ into the equation $1.70 Cw = M$:

     $1.70 Cw = 0.40 C$

   • Divide both sides by $C$ (since $C \neq 0$):

     $1.70 w = 0.40$

   • Solve for $w$:

     $w = \frac{0.40}{1.70} = \frac{4}{17}$ kg

5. Convert Weight to Grams and Round:

   • To convert kg to grams, multiply by 1000:

     Actual Weight (grams) = $ \frac{4}{17} \times 1000 = \frac{4000}{17} $ grams

   • Calculate the value and round to one decimal place:

     $ \frac{4000}{17} \approx 235.294... $ grams

     Rounded weight = 235.3 gm