Question

moving from 4-bits to 8-bits two schools are merging and to adapt to the increased size of the student body, the school is changing its system for keeping track of all student ids from a system that uses 4-bit binary numbers to one that uses 8-bit binary numbers (aka 1 byte). which of the following best describes what effect this change will have? a. the 8-bit system will be able to represent twice as many values as the 4-bit system. b. the 8-bit system will be able to represent sixteen times as many values as the 4-bit system. c. the 8-bit system will be able to represent eight times as many values as the 4-bit system. d. the 8-bit system will be able to represent the same number of values as the 4-bit system, it will just include more zeros.

Explanation:

Step1: Recall number of values in n-bit system

The number of distinct values an \( n \)-bit binary system can represent is \( 2^n \). For a 4 - bit system, \( n = 4 \), so the number of values is \( 2^4=16 \). For an 8 - bit system, \( n = 8 \), so the number of values is \( 2^8 = 256 \).

Step2: Find the ratio of values

To find how many times more values the 8 - bit system can represent than the 4 - bit system, we calculate the ratio of the number of values of the 8 - bit system to the 4 - bit system. The ratio is \( \frac{2^8}{2^4}=\frac{256}{16} = 16 \). So the 8 - bit system can represent 16 times as many values as the 4 - bit system.

Answer:

B. The 8 - bit system will be able to represent sixteen times as many values as the 4 - bit system.