Solving the False Balance Problem

This solution details the calculation to determine the actual weight used by a shopkeeper who claims a percentage loss but achieves a different percentage gain using a false balance.

Step-by-Step Calculation

• Define Variables and Conditions:

  • Assume the Cost Price (CP) of 1000 gm is $100$.
  • The shopkeeper claims a 20% loss.
  • The actual gain achieved is 40%.
  • Let the actual weight used be $x$ grams.

• Calculate the Selling Price (SP) based on the claimed loss:

  The SP for the intended quantity (1000 gm) is set according to the claimed loss:

  SP = $CP_{1000} \times (1 - \text{Loss Percentage})$

  SP = $100 \times (1 - 0.20) = 100 \times 0.80 = 80$.

  This SP of 80 represents the total revenue received for the transaction.

• Calculate the actual Cost Price (CP) for the weight supplied:

  The CP per gram = $100 / 1000 = 0.1$.

  The actual cost incurred by the shopkeeper for the weight $x$ grams is:

  Actual CP = $x \times CP_{per\_gram} = x \times 0.1 = \frac{x}{10}$.

• Apply the Gain Percentage Formula:

  The shopkeeper gains 40% on the actual cost. The formula for gain percentage is:

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

  Substitute the known values:

  $40 = \frac{80 - (x/10)}{(x/10)} \times 100$

• Solve for the actual weight $x$:

  Convert the percentage to decimal:

  $0.40 = \frac{80 - (x/10)}{(x/10)}$

  Multiply both sides by $(x/10)$:

  $0.40 \times \frac{x}{10} = 80 - \frac{x}{10}$

  $0.04x = 80 - 0.1x$

  Gather terms with $x$ on one side:

  $0.04x + 0.1x = 80$

  $0.14x = 80$

  Isolate $x$:

  $x = \frac{80}{0.14} = \frac{8000}{14} = \frac{4000}{7}$

• Calculate the final value and round:

  $x \approx 571.42857...$ gm

  Rounding the result to one decimal place gives the actual weight used:

  Actual weight = 571.4 gm.

Conclusion

The shopkeeper uses an actual weight of approximately 571.4 gm when intending to sell 1 kg, while claiming a 20% loss, ultimately achieving a 40% gain.