The float using 8 bit exponent and 15 bit signficand for 277.431 was shown
to be :

0 10000111 000101010110111

Converting back :

Bias
  Use formula 2^(n-1) - 1 where n is the size of the exponent.
    2^(8-1)-1 = 127 bias

Exponent 10000111

   1   0   0   0   0   1   1    1
   1   2   4   8  16  32  66  134 x2
   ------------------------------
   1   2   4   8  16  33  67  135  +

  135 - 127 (bias) = 8

 Signifcand  000101010110111

  Remember to put the hidden 1. back :
    (b at the end of the string of 0s and 1s means binary)

   1.000101010110111b * 2^8

   100010101.0110111b

                    0 (start)
   1    (0x2) + 1   1 (this will always be 1)
   0    (1x2) + 0   2
   0    (2x2) + 0   4
   0    (4x2) + 0   8
   1   (16x2) + 1  17
   0   (17x2) + 0  34
   1   (34x2) + 1  69
   0   (60x2) + 0 138
   1  (138x2) + 1 277

   1   0   0   0   1   0   1   0   1
 256 128  64  32  16   8   4   2   1
 256 + 0 + 0 + 0 +16 + 0 + 4 + 0 + 1 


  .01101110

  .
  0  .5
  1  .25                 .25
  1  .125                .125
  0  .0625
  1  .03125              .03125
  1  .015625             .015625
  1  .0078125            .0078125
  0  .00390625

                         .4296875

         Stored          277.4296875

         Original        277.431