Question

What is the sum of the two numbers ${\left( {11110} \right)_2}$and ${\left( {1010} \right)_2}$?
A) ${\left( {101000} \right)_2}$
B) ${\left( {110000} \right)_2}$
C) ${\left( {100100} \right)_2}$
D) ${\left( {101100} \right)_2}$

Answer 1

Hint:
When we are converting any binary number into a decimal number, we should first know that binary numbers are 0 and 1. So, for changing the value of 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 solution:
As we know that decimal numbers are those numbers which can have any digits from 0 to 9 (for e.g. 4924) but the binary numbers can have only 0’s and 1’s (like 1010).
Let us first take an to ${\left( {11110} \right)_2}$change a binary number to the decimal number.
So, to change ${\left( {11110} \right)_2}$into decimal we should use the formula as
\[\left( {1 \times {2^4}} \right) + \left( {1 \times {2^3}} \right) + \left( {1 \times {2^2}} \right) + \left( {1 \times {2^1}} \right) + \left( {0 \times {2^0}} \right)\] (Here the power of two is increasing and is equal to the position of its corresponding digit starting with zero)
So, on solving above equation we get,
\[ = 16 + 8 + 4 + 2 + 0 = 30\]
So, the equivalent decimal of binary ${\left( {11110} \right)_2}$ will be 30.
Now we will take the second number ${\left( {1010} \right)_2}$and change a binary number to the decimal number.
So, to change ${\left( {1010} \right)_2}$into decimal we should use the formula as
\[\left( {1 \times {2^3}} \right) + \left( {0 \times {2^2}} \right) + \left( {1 \times {2^1}} \right) + \left( {0 \times {2^0}} \right)\] (Here the power of two is increasing and is equal to the position of its corresponding digit starting with zero)
So, on solving above equation we get,
\[ = 8 + 0 + 2 + 0 = 10\]
So, the equivalent decimal of binary ${\left( {1010} \right)_2}$ will be 10.
Now we will add both numbers
$ \Rightarrow 30 + 10 = 40$
Now we have to convert 40 into the binary number.
For changing a decimal number to a binary number, we can use this method:
We will divide the decimal number by 2 and noting the remainder as 1 or 0.
If the number you are left after a division is greater than 0, go back to last step.
Then write your remainders in reverse order to get your binary number equivalent to the decimal number.
So decimal number 40:
\[ \Rightarrow 40{\text{ }} \div {\text{ }}2{\text{ }} = {\text{ }}20\], here remainder is 0
\[ \Rightarrow 20{\text{ }} \div {\text{ }}2{\text{ }} = {\text{ }}10\], here remainder is 0
\[ \Rightarrow 10{\text{ }} \div {\text{ }}2{\text{ }} = {\text{ }}5\], here remainder is 0
\[ \Rightarrow 5{\text{ }} \div {\text{ }}2{\text{ }} = {\text{ }}2\], here reminder is 1
\[ \Rightarrow 2{\text{ }} \div {\text{ }}2{\text{ }} = {\text{ }}1\], here reminder is 0
\[ \Rightarrow 1{\text{ }} \div {\text{ }}2{\text{ }} = {\text{ }}0\], here reminder is 1
This gives us ${\left( {101000} \right)_2}$.

Hence, the correct option will be A.

Note:
Verification:
We will check the answer by converting the above mentioned binary number into decimal number:
So, to change ${\left( {101000} \right)_2}$into decimal we should use the formula as
\[\left( {1 \times {2^5}} \right) + \left( {0 \times {2^4}} \right) + \left( {1 \times {2^3}} \right) + \left( {0 \times {2^2}} \right) + \left( {0 \times {2^1}} \right) + \left( {0 \times {2^0}} \right)\] (Here the power of two is increasing and is equal to the position of its corresponding digit starting with zero)
So, on solving above equation we get,
\[ = 32 + 0 + 8 + 0 + 0 + 0 = 40\]
So, the equivalent decimal of binary ${\left( {101000} \right)_2}$ will be 40.
Whenever we come up with this type of problem then to change any given binary number to decimal number, or decimal number to binary number we have to just apply the formula.