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: Calculate number of values for 4 - bit system

The number of values that an $n$-bit binary system can represent is given by $2^n$. For a 4 - bit system, $n = 4$, so the number of values is $2^4=16$.

Step2: Calculate number of values for 8 - bit system

For an 8 - bit system, $n = 8$, so the number of values is $2^8 = 256$.

Step3: Find the ratio

To find out how many times more values the 8 - bit system can represent compared to the 4 - bit system, we calculate the ratio $\frac{2^8}{2^4}$. Using the rule of exponents $\frac{a^m}{a^n}=a^{m - n}$, we have $\frac{2^8}{2^4}=2^{8 - 4}=2^4 = 16$.

Answer:

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