A tiny metallic rectangular sheet has length and breadth of 5 mm and 2.5 mm , respectively. Using a specially designed screw gauge which has pitch of 0.75 mm and 15 divisions in the circular scale, you are asked to find the area of the sheet. In this measurement, the maximum fractional error will be $$\frac{x}{100}$$ where x is __________.

A rectangular metallic sheet has length $$L = 5$$ mm and breadth $$B = 2.5$$ mm, and the screw gauge used has pitch = 0.75 mm with 15 divisions on the circular scale.

We first find the least count (LC) of the screw gauge.

$$ LC = \frac{\text{Pitch}}{\text{Number of divisions}} = \frac{0.75}{15} = 0.05 \text{ mm} $$

Next, we calculate the area of the sheet.

$$ A = L \times B = 5 \times 2.5 = 12.5 \text{ mm}^2 $$

We then determine the maximum fractional error in this area.

$$ \frac{\Delta A}{A} = \frac{\Delta L}{L} + \frac{\Delta B}{B} $$

Here, we take $$\Delta L = \Delta B = LC = 0.05$$ mm (since the maximum absolute error equals the least count).

$$ \frac{\Delta A}{A} = \frac{0.05}{5} + \frac{0.05}{2.5} = 0.01 + 0.02 = 0.03 = \frac{3}{100} $$

So $$\frac{x}{100} = \frac{3}{100}$$, giving $$x = 3$$.

Hence the answer is 3.