A shopkeeper sells his goods at cost price. If by using false weights he gains 3*19/27%, then what weight does he use for 1 kg (in g, rounded off to 2 decimal places)?

The correct answer/option is 964.29.

Let us understand how to arrive at this answer step-by-step.

When a shopkeeper sells goods at cost price but uses a false weight, his gain percentage can be calculated using a standard formula:
Gain % = (Error / (True Value - Error)) * 100
Here, the True Value is 1 kg, which is equal to 1000 g.
Let the false weight used by the shopkeeper be x g.
Therefore, the Error in weight is (1000 - x) g.

Substituting these values into the formula, we get:
Gain % = ((1000 - x) / x) * 100

We are given that the shopkeeper gains 3¹⁹⁄₂₇%. Let us convert this mixed fraction into an improper fraction:
Gain % = (3 * 27 + 19) / 27 = (81 + 19) / 27 = 100 / 27 %

Now, we equate the two expressions for Gain %:

$$\frac{100}{27}=\frac{1000-x}{x}\times 100$$

Dividing both sides by 100, we simplify the equation to:

$$\frac{1}{27}=\frac{1000-x}{x}$$

Next, we perform cross-multiplication to solve for x:
x = 27 * (1000 - x)
x = 27000 - 27x
x + 27x = 27000
28x = 27000

Now, we divide 27000 by 28 to find the value of x:

$$x=\frac{27000}{28}$$

$$x=\frac{6750}{7}$$

Evaluating this division gives:
x ≈ 964.2857...

Rounding to two decimal places, we get:
x = 964.29 g

Thus, the weight he uses for 1 kg is 964.29 g.