Question

A digital voltmeter (DVM) with a \( 3 \frac{1}{2} \) digit display can show a maximum count of:

• A. 999
• B. 1999
• C. 3999
• D. 9999

Hint:

For a \( N \frac{1}{2} \) digit display:
The number of full digits is \( N \), and the maximum count is always \( 1\underbrace{99\dots9}_{N \text{ times}} \).
For \( 3 \frac{1}{2} \) digits, \( N = 3 \), giving 1 followed by three 9s, which is 1999.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the maximum display count of a digital voltmeter (DVM) characterized by a \( 3 \frac{1}{2} \) digit display.
The fractional designation in digital displays indicates the capability of the most significant digit (MSD) compared to the other full digits.

Step 2: Key Formula or Approach:
A \( 3 \frac{1}{2} \) digit display consists of 3 full digits and 1 half-digit:
- A "full digit" can display any integer value from 0 to 9 (10 distinct states).
- A "half-digit" (the fractional part \( \frac{1}{2} \)) is the Most Significant Digit (MSD) and can only display a value of 0 or 1 (2 distinct states).

Step 3: Detailed Explanation:
Let us break down the range of the display:

• Structure of the Display: Let the digits be represented as \( [D_{\text{half}}][D_1][D_2][D_3] \).
- The three full digits \( D_1, D_2, D_3 \) can each take any value from the set \( \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \).
- The half-digit \( D_{\text{half}} \) can only take values from the set \( \{0, 1\} \).

• Maximum Count Formulation:
- The highest value for the full digits is obtained when each is set to its maximum value, which is 9. Therefore, \( D_1 D_2 D_3 = 999 \).
- The highest value for the half-digit is 1. Therefore, \( D_{\text{half}} = 1 \).
- Combining these gives the maximum possible displayed value as 1999.

• Alternative Configurations:
- If the MSD could display up to 3, it would be called a \( 3 \frac{3}{4} \) digit display, which could show up to 3999.
- Since this is a \( 3 \frac{1}{2} \) digit display, it cannot exceed the 1999 count.

Step 4: Final Answer:
The maximum count that a \( 3 \frac{1}{2} \) digit DVM display can show is 1999.