Convert 76125 and 55 to single precision floating point With

Convert 76.125 and -5.5 to single precision floating point. Without converting back to decimal, add these two numbers together. Show your steps, and clearly label what different binary numbers represent (exponent, mantissa, sign, etc).

Solution

76.125
We need to find largest power of 2 which is smaller than the number
as here we have 76 then 2^6 = 64 so exponent is 6 and as the exponent is biased by 127 so this means the exponent will be represented as 127+6 = 133.
The mantissa is 76.125/6= 12.6875 here we will take 68 and binary represenation of 68 is 00000000000000001100100 (23 bits)
And signed bit for positive numbers is 0

signed exponent[133] Mantissa[68]
0 10000101 00000000000000001100100

-5.5
The largerst power of 2 which is smaller than the number 2^2 =4 so exponent is 2 , as it is biased by 127 so exponent will be 127+2 = 129
Mantissa = 5.5/2 = 2.75, we will take 75 and binary of 75 is 00000000000000001001111
And signed bit for negative numbers is 1
Signed Exponent[129] Mantissa[75]
1 10000001 00000000000000001001111

Now the two numbers are

01000010100000000000000001100100

11000000100000000000000001001111

sum of these would be 00000011000000000000000010110011