What number is represented by the byte 00000111?

Here's one way to decode binary numbers. Make a column for each digit, and starting from the right, put a "1" and then as you move left put double the value of the column next to it. Remember, right-to-left ( THIS WAY ). You'll get something like this, which is a horizontal version of the helper table shown in the quz:

128  64  32  16   8   4   2   1   <-- 8 columns for 8 digits

Then, put your binary number on the next line, spread into the columns. Now below that, multiply each top number by the binary digit and put the result below. It's easy, since binary digits are either 1 or 0, so you either put the number on top again or zero.

Then finally, add up all the numbers on the bottom row and that's your answer. Like this:

128  64  32  16   8   4   2   1   <-- starting columns
  0   0   0   0   0   1   1   1   <-- multiply by your binary digits
--- --- --- --- --- --- --- ---
  0   0   0   0   0   4   2   1   <-- add these up:  4 + 2 + 1  =  7

Also, since zeros in front don't count you can skip them. So in this case we really only needed 3 columns.