The float using 7 bit exponent and 16 bit signficand for 277.431 was shown 
to be :

0 1000111 0001010101101110

Converting back :

Bias 
  Use formula 2^(n-1) - 1 where n is the size of the exponent.
    2^(7-1)-1 = 63 bias

Exponent 10000111

   1   0   0   0   1   1   1
   1   2   4   8  17  35  71
 
  71 - 63 (bias) = 8

 Signifcand  0001010101101110
  
  Remember to put the invisible 1. back on :
    (b at the end of the string of 0s and 1s means binary)
 
   1.0001010101101110b * 2^8

   100010101.01101110b  


   1   0   0   0   1   0   1   0   1
   1   
       2   
           4   
               8  
                  17  
                      34  
                          69 
                             138 
                                 277 


  .01101110

  .
  0  .5
  1  .25                 .25
  1  .125                .125
  0  .0625
  1  .03125              .03125
  1  .015625             .015625
  1  .0078125            .0078125
  0  .00390625
   
                         .4296375
  
         Original        277.431