How many numbers can you make between 0-99 using exactly four 4s and the operations $$ +,−,×,÷?

  37

Step-by-step explanation:

I find 37 numbers in that range:

  {0, 1/16, 1/8, 1/5, 1/3, 1/2, 3/4, 4/5, 1, 5/4, 4/3, 2, 3, 7/2, 15/4, 4, 17/4, 9/2, 5, 6, 7, 8, 9, 12, 15, 16, 17, 20, 24, 28, 32, 36, 48, 60, 64, 68, 80}

For these numbers we have used only the indicated operations. No minus signs were used.

Additional comment

Of the 320 ways the numbers can be combined, 15 result in division by zero. Many other results are outside the 0-99 range of interest. In total, there are 56 unique rational results.