Convert percentage to fraction

Convert the gain percentage $11 \frac{1}{9}\%$ into a fraction. $11 \frac{1}{9}\% = \frac{100}{9} \times \frac{1}{100} = \frac{1}{9}$.

$\frac{100}{9}\% = \frac{1}{9}$

Set up the gain ratio

The formula for gain when using false weights is $\text{Gain}\% = \frac{\text{Error}}{\text{False Weight}} \times 100$. This implies $\frac{\text{True Weight}}{\text{False Weight}} = 1 + \text{Gain}$. Given the gain is $\frac{1}{9}$, the ratio is $1 + \frac{1}{9} = \frac{10}{9}$.

$\frac{\text{True}}{\text{False}} = \frac{10}{9}$

Calculate false weight

We know the True Weight is $1000$ gm. Setting up the equation $\frac{1000}{x} = \frac{10}{9}$, we solve for $x$.

$x = \frac{1000 \times 9}{10} = 900 \text{ gm}$