Question

If diameter reduces by 50% during wire drawing, area reduces by

• A. 25%
• B. 75%
• C. 50%
• D. 10%

Hint:

For any circular or square section undergoing uniform scaling of its linear dimension \( L \) by a factor \( k \) (where \( L_{\text{new}} = k L_{\text{old}} \)):
- New Area \( A_{\text{new}} = k^2 A_{\text{old}} \).
- In this case, \( k = 0.5 \implies k^2 = 0.25 \).
- Area Reduction = \( 1 - k^2 = 1 - 0.25 = 0.75 \rightarrow 75\% \).
This quick scaling trick avoids tedious calculations during exams.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question is a quantitative problem asking for the percentage reduction in the cross-sectional area of a wire when its diameter is reduced by exactly 50% during a wire drawing operation.
This is a standard geometric calculation applied in bulk metal forming processes.

Step 2: Key Formula or Approach:
Let the initial diameter of the wire be \( d_0 \) and the final diameter be \( d_1 \).
The cross-sectional area of a circular wire is given by:
\[ A = \frac{\pi}{4} d^2 \]
The initial area is:
\[ A_0 = \frac{\pi}{4} d_0^2 \]
The percentage reduction in area (\( r \)) is defined as:
\[ r = \frac{A_0 - A_1}{A_0} \times 100\% \]

Step 3: Detailed Explanation:

• Relationship Between Diameters: Since the diameter is reduced by 50%, the final diameter \( d_1 \) is half of the initial diameter:
\[ d_1 = d_0 - (0.50 \cdot d_0) = 0.50 \cdot d_0 \]

• Final Area Calculation:
Substitute \( d_1 \) into the area equation:
\[ A_1 = \frac{\pi}{4} (0.50 \cdot d_0)^2 = 0.25 \cdot \left(\frac{\pi}{4} d_0^2\right) = 0.25 \cdot A_0 \]
- This shows that reducing the diameter by half reduces the cross-sectional area to one-fourth of its original value.

• Percentage Reduction in Area:
Substitute \( A_1 = 0.25 \cdot A_0 \) into the percentage reduction formula:
\[ r = \frac{A_0 - 0.25 \cdot A_0}{A_0} \times 100\% \]
\[ r = \frac{0.75 \cdot A_0}{A_0} \times 100\% = 75\% \]
- Thus, a 50% reduction in diameter corresponds to a 75% reduction in cross-sectional area.


Step 4: Final Answer:
The calculated area reduction is exactly 75%.
Hence, option (B) is the correct answer.