Question

The decimal equivalent of the binary number \[{\left( {11010.101} \right)_2}\] is
A. 9.625
B. 25.265
C. 26.625
D. 26.265

Answer 1

Hint: For converting any binary number into decimal we should first know that binary numbers are 0 and 1. So, to change any binary number to decimal number we have to apply formula that is for each digit starting from right we had to multiply that digit by \[{2^n}\] where n will be the position of that digit starting from 0 (rightmost digit) and then at last add expansion of digits to get the required decimal number.

Complete step-by-step answer:
As we know that decimal numbers are those numbers which can have from 0 to 9 (like 9854) but the binary numbers are in the form of only 0’s and 1’s (like 1010).
Let us first take an example to change a binary number to the decimal number
So, to change 1010 into decimal we should use the formula as
\[\left( {1 \times {2^3}} \right) + \left( {0 \times {2^2}} \right) + \left( {1 \times {2^1}} \right) + \left( {0 \times {2^0}} \right)\] (power of 2 increases and will be equal to the position of its corresponding digit like 0,1,2,3….)
So, on solving above equation we get,
8 + 0 + 2 + 0 = 10
So, the equivalent decimal of binary 1010 will be 10.
But in the given question we had to change the binary number \[{\left( {11010.101} \right)_2}\] to decimal number
As we can see that there are 2 parts of the binary in the number \[{\left( {11010.101} \right)_2}\] i.e. part before decimal and part after decimal.
Step 1 – In part one (i.e. before decimal = 11010)
Multiplying the binary digits by \[{2^n}\] will be done from right to left (where n will be the position of digit).
So, the conversion of 11010 to decimal will be,
\[\left( {1 \times {2^4}} \right) + \left( {1 \times {2^3}} \right) + \left( {0 \times {2^2}} \right) + \left( {1 \times {2^1}} \right) + \left( {0 \times {2^0}} \right)\]
16 + 8 + 0 + 2 + 0 = 26
Step 2 :- In the second part ( i.e. after decimal = .101 )
Now to change the binary part of the number after the decimal point we had to multiply each digit of the binary number by \[{2^{ - n}}\], where n will be the position of digit starting from 1 (leftmost digit after point).
So (.101) binary converts into decimal by
\[\left( {1 \times {2^{ - 1}}} \right) + \left( {0 \times {2^{ - 2}}} \right) + \left( {1 \times {2^{ - 3}}} \right)\]
\[\dfrac{1}{2} + 0 + \dfrac{1}{8} = \dfrac{{4 + 1}}{8} = \dfrac{5}{8} = 0.625\]
Step 3:- Adding up the results we get from step 1 and step 2 will give us the resultant decimal value of binary number \[{\left( {11010.101} \right)_2}\]
Adding both values 26 + 0.625 = 26.625
So, the decimal equivalent of binary number \[{\left( {11010.101} \right)_2}\] will be equal to 26.625
Hence, the correct option will be C.


Note:- Whenever we come up with this type of problem then to change any given binary number to decimal number, we have to just apply the formula. And according to that, to convert the number before the decimal point we had to start from the rightmost digit before the point and multiply each digit by \[{2^n}\] where n will be the position of that digit starting from 0 (rightmost digit) and then add them up. After that to convert the number after point we had to start from the leftmost digit after point and multiply each digit by \[{2^{ - n}}\] where n will be the position of that digit starting from 1 (leftmost digit) and then add them up. Then the resultant equivalent of the binary number in decimal will be the sum of both parts. And this will be the easiest and efficient way to find the solution of the problem.