While measuring diameter of wire using screw gauge the following readings were noted. Main scale reading is $$1 \text{ mm}$$ and circular scale reading is equal to 42 divisions. Pitch of screw gauge is $$1 \text{ mm}$$ and it has 100 divisions on circular scale. The diameter of the wire is $$\frac{x}{50} \text{ mm}$$. The value of $$x$$ is :

We need to find the diameter of a wire measured using a screw gauge. Recall that the screw gauge reading is given by: $$ \text{Reading} = \text{MSR} + \text{CSR} \times \text{Least Count} $$ where MSR = Main Scale Reading, CSR = Circular Scale Reading, and Least Count = Pitch / Number of divisions on circular scale.

Since the pitch is 1 mm and the circular scale has 100 divisions, the least count is: $$ LC = \frac{\text{Pitch}}{\text{Number of divisions}} = \frac{1 \text{ mm}}{100} = 0.01 \text{ mm} $$. Substituting in the formula gives the diameter: $$ d = 1 + 42 \times 0.01 = 1 + 0.42 = 1.42 \text{ mm} $$. Expressing 1.42 mm in fractional form: $$ d = 1.42 = \frac{142}{100} = \frac{71}{50} \text{ mm} $$. Therefore x = 71. The correct answer is Option (3): 71.