Question

Perform the following calculations using binary numbers and convert the result back to decimal.

(a)8+5

(b)339+25

(c)47+16

(d)$2\times 9$

Short answer

The decimals are as follows

(a) $8_{10}+5_{10}={\left(1000\right)}_2+{\left(101\right)}_2={\left(1101\right)}_2={13}_{10}$

(b) 364

(c) 63

(d) 18

Step-by-step solution

Part (a)

We will convert the decimal to binary as follows:

Divide 8 by 2, we will get 4, remainder will be 0(LSB).

Now divide 4 by 2, we will get remainder will be .

Now divide 2 by 2 , we will get 1(MSB) remainder will be 0.

Hence the binary equivalent of 8 will be 1000, it can be written as follows:

Similarly, we can find the binary equivalent of as,

$5_{10}={\left(101\right)}_2$

The sum is,

1000

+ 101

----------

1101

-----------

The decimal equivalent of sum is${\left(1101\right)}_2={13}_{10}$

Part (b)

we will convert the decimal to binary as follows:

Divide 339 by 2 , we will get 169 ,remainder will be 1 (LSB).

Now divide 169 by 2, we will get 84 remainder will be 1.

Now divide 84 by 2, we will get 42 remainder will be 0.

………………………………………………………………

………………………………………………………………

Now divide 2 by 2 , we will get 1 (MSB) remainder will be 0.

Hence the binary equivalent of 339 will be 101010011, it can be written as follows:

Similarly we can find the binary equivalent of .

${25}_{10}={\left(11001\right)}_2$

101010011

+11001

----------------

101101100

-----------------

${\left(\text{101101100}\right)}_2={364}_{10}$

Part (c)

we will convert the decimal to binary as follows:

Divide 47 by 2, we will get ,remainder will be 1(MSB)

Now divide 23 by 2 , we will get remainder will be 1.

………………………………………………………………

………………………………………………………………

Now divide 2 by 2 , we will get 1(MSB) remainder will be 0.

Hence the binary equivalent of 47 will be 101111, it can be written as follows:

Similarly, we can find the binary equivalent of

${16}_{10}={\left(10000\right)}_2$

101111

+1000

------------

111111

-------------

${\left(111111\right)}_2={63}_{10}$

Part (d)

we will convert the decimal to binary as follows:

Divide 9 by 2 , we will get ,remainder will be 1 (LSB) .

Now divide 4 by 2, we will get 2 remainder will be 0 .

………………………………………………………………

………………………………………………………………

Now divide 2 by 2 , we will get 1 (MSB) remainder will be 0.

Hence the binary equivalent of 9 will be 1001, it can be written as follows:

$9_{10}={\left(1001\right)}_2$

Similarly, we can find the binary equivalent of 2.

$2_{10}={\left(10\right)}_2$

1001

X10

-----------

0000

1001

--------------

10010

--------------

${\left(10010\right)}_2={18}_{10}$