NIMCET 2007 — Computer PYQ

To convert a decimal number with a fractional part into its binary equivalent, we must separate the conversion into two parts: the integer part ($35$) and the fractional part ($0.75$).

Step 1: Convert the Integer Part (35)

To convert the integer part from base-10 to base-2, we repeatedly divide by $2$ and record the remainders from bottom to top (Least Significant Bit to Most Significant Bit):

• $35÷2=17$ with a remainder of $1$ (LSB)

• $17÷2=8$ with a remainder of $1$

• $8÷2=4$ with a remainder of $0$

• $4÷2=2$ with a remainder of $0$

• $2÷2=1$ with a remainder of $0$

• $1÷2=0$ with a remainder of $1$ (MSB)

Reading the remainders from bottom to top gives:

$$(35{)}_{10}=(100011{)}_2$$

Step 2: Convert the Fractional Part (0.75)

To convert the fractional part, we repeatedly multiply the fraction by $2$ and record the integer part of the result from top to bottom:

• $0.75\times 2=1.50⟹$ Record $1$

• Take the remaining fractional part ($0.50$):

  $0.50\times 2=1.00⟹$ Record $1$

Since the fractional part has become $.00$, the conversion process ends. Reading the recorded integers from top to bottom gives:

$$(0.75{)}_{10}=(0.11{)}_2$$

Step 3: Combine Both Parts

Now, combine the binary results obtained from the integer and fractional parts together:

$$(35.75{)}_{10}=(100011{)}_2+(0.11{)}_2=(100011.11{)}_2$$

Conclusion

The decimal number $(35.75{)}_{10}$ is equivalent to $(100011.11{)}_2$ in binary.

Correct Answer: (a) $(100011.11{)}_2$