Question

A binary number is represented by ${{\left( cdccddcccddd \right)}_{2}}$, where $c>d$. What is its decimal equivalent?
A. 1848
B. 2048
C. 2842
D. 2872

Answer 1

Hint: To solve this question first we convert the given number into the 0’s and 1’s form. Then, to change a binary number to a decimal number we had to multiply a digit starting from the right with ${{2}^{n}}$ where $n$ is the position of that digit starting from 0 and then add the expansion of digits to get the equivalent decimal number.

Complete step-by-step solution
We have given that a binary number is represented by ${{\left( cdccddcccddd \right)}_{2}}$.
We know that a subscript $2$ is added in the base of binary numbers to distinguish them from other numbers and a subscript $10$ is added in the base of decimal numbers.
As we know that the binary numbers are in the form of 0’s and 1’s. So, first, we convert the given number into the actual binary form.
Also, we have given a condition $c>d$, so we replace $c$ with $1$ and $d$ with $0$,
We get ${{\left( cdccddcccddd \right)}_{2}}={{\left( 101100111000 \right)}_{2}}$
To convert a binary number to a decimal number we multiply the digits with ${{2}^{n}}$ starting from the rightmost digits, power of $2$ is increased and will be equal to the position of that digit.
So, the conversion of ${{\left( 101100111000 \right)}_{2}}$ to decimal will be,
$\begin{align}
  & {{\left( 101100111000 \right)}_{2}}=\left( 1\times {{2}^{11}} \right)+\left( 0\times {{2}^{10}} \right)+\left( 1\times {{2}^{9}} \right)+\left( 1\times {{2}^{8}} \right)+\left( 0\times {{2}^{7}} \right)+\left( 0\times {{2}^{6}} \right)+\left( 1\times {{2}^{5}} \right)+\left( 1\times {{2}^{4}} \right)+\left( 1\times {{2}^{3}} \right)+\left( 0\times {{2}^{2}} \right) \\
 & +\left( 0\times {{2}^{1}} \right)+\left( 0\times {{2}^{0}} \right) \\
\end{align}$ When we solve this equation we get,
${{\left( 101100111000 \right)}_{2}}=2048+0+512+256+0+0+32+16+8+0+0+0$
\[{{\left( 101100111000 \right)}_{2}}={{\left( 2872 \right)}_{10}}\]
The required decimal number is \[{{\left( 2872 \right)}_{10}}\].
Hence, Option D is the correct answer.

Note: The point to note in this question is that first, we have to convert the given number in the form of $0$ and $1$ because binary numbers can represent in the form of only 0’s and 1’s. For example, the number 1010 is a binary number. Also, remember that multiplying the binary digits with ${{2}^{n}}$ will be done from right to left only.