Step 1: Understand the profit relation in false weight

When a dealer professes to sell at cost price but gains 25% by using a false weight, the profit percentage formula based on weight is given by: $\text{Gain}\% = \frac{\text{True Weight} - \text{False Weight}}{\text{False Weight}} \times 100$

$\text{Gain}\% = \frac{\text{True Weight} - \text{False Weight}}{\text{False Weight}} \times 100$

Step 2: Convert percentage to fraction

Express $25\%$ as a fraction: $25\% = \frac{25}{100} = \frac{1}{4}$ This means for every $4$ units of weight actually given (False Weight), he makes a profit of $1$ unit.

$\text{Gain ratio} = \frac{1}{4}$

Step 3: Set up the ratio of weights

The true weight (what the customer pays for, i.e., $100\text{ gm}$) corresponds to the total parts: $4 \text{ (false weight)} + 1 \text{ (profit)} = 5 \text{ units}$.

$5 \text{ units} = 100\text{ gm}$

Step 4: Calculate the false weight

Since $5 \text{ units} = 100\text{ gm}$, $1 \text{ unit} = \frac{100}{5} = 20\text{ gm}$. The false weight is $4 \text{ units}$: $\text{False Weight} = 4 \times 20\text{ gm} = 80\text{ gm}$

$\text{False Weight} = 80\text{ gm}$