Question

The decimal number \[{{\left( 127.25 \right)}_{10}}\], when converted to binary number, takes the form.
(a) \[{{\left( 1111111.11 \right)}_{2}}\]
(b) \[{{\left( 1111110.01 \right)}_{2}}\]
(c) \[{{\left( 1110111.11 \right)}_{2}}\]
(d) \[{{\left( 1111111.01 \right)}_{2}}\]

Answer 1

Hint: For solving this question you should know about converting a decimal value to a Binary number. In this problem we will change a decimal number into a binary number and it is a fixed format for this.

Complete step by step answer:
According to our question it is asked to convert a decimal number to a binary number.
AS there it is a fixed procedure to change any decimal number from decimal to binary number and here we will use that. The process is that we divide to the decimal number after converting it into a whole number and start to divide it and then we write all the remainders as a digit form for every single step and these remainders written at last as last remainder at left side and first remainder at right most side. And this we get our answer.
For our question the decimal value is 127.25.
We will first consider 127 and convert it into binary and then we will consider 0.25 and convert it into binary and then we will combine those values to get our answer.
So, let us first start with 127, we get
\[\begin{align}
  & 2\left| \!{\underline {\,
  127 \,}} \right. 1 \\
 & 2\left| \!{\underline {\,
  63 \,}} \right. 1 \\
 & 2\left| \!{\underline {\,
  31 \,}} \right. 1 \\
 & 2\left| \!{\underline {\,
  15 \,}} \right. 1 \\
 & 2\left| \!{\underline {\,
  7 \,}} \right. 1 \\
 & 2\left| \!{\underline {\,
  3 \,}} \right. 1 \\
 & 2\left| \!{\underline {\,
  1 \,}} \right. 1 \\
 & \left| \!{\underline {\,
  0 \,}} \right. \\
\end{align}\]
So, we get the binary form of 127 as 1111111.
Now, we will convert 0.25 into binary. To convert 0.25 into binary we will multiply it by 2 and will keep multiplying it by 2 till we get resulting fractional part equal to 0. Therefore, we get
0.25$\times $2 = 0 + 0.5
0.5$\times $2=1 + 0
Therefore, we get the binary form of 0.25 as 0.01.
Therefore, we can combinely write binary form of \[{{\left( 127.25 \right)}_{10}}={{\left( 1111111.01 \right)}_{2}}\]

So, the correct answer is “Option d”.

Note: While solving this type of question you have to ensure that the quotient is in a whole number from and if this is not in that form then convert it in that form or we can also use only the number present before decimal and rest part will be multiplied with remainder for get remainder as a digit.
Another method
If we multiply the decimal number with the base raised to the power of decimals in result:
\[127.25\times {{2}^{2}}=509\]
Divided by the base 2 to get the digits from the remainders.
                Remainder (digit)
\[\begin{align}
  & 2\left| \!{\underline {\,
  509 \,}} \right. 1 \\
 & 2\left| \!{\underline {\,
  254 \,}} \right. 0 \\
 & 2\left| \!{\underline {\,
  127 \,}} \right. 1 \\
 & 2\left| \!{\underline {\,
  63 \,}} \right. 1 \\
 & 2\left| \!{\underline {\,
  31 \,}} \right. 1 \\
 & 2\left| \!{\underline {\,
  15 \,}} \right. 1 \\
 & 2\left| \!{\underline {\,
  7 \,}} \right. 1 \\
 & 2\left| \!{\underline {\,
  3 \,}} \right. 1 \\
 & 2\left| \!{\underline {\,
  1 \,}} \right. 1 \\
 & \left| \!{\underline {\,
  0 \,}} \right. \\
\end{align}\]
We can write it as: \[={{\left( 111111101 \right)}_{2}}>>2\]
                                  \[={{\left( 1111111.01 \right)}_{2}}\]
So, the correct option is ‘(d)’.