King Midas spent $\frac{100}{x}\%$ of his gold deposit yesterday. He is set to earn gold today. What percentage
of the amount of gold King Midas currently has would he need to earn today to end up with as much
gold as he started?
1. Let the initial amount of gold be \( G \).
2. King Midas spent \(\frac{100}{x}\%\) of his gold deposit yesterday. This means he spent:
   \[
   \frac{100}{x} \times \frac{G}{100} = \frac{G}{x}
   \]
3. Therefore, the amount of gold left after spending is:
   \[
   G - \frac{G}{x} = G \left(1 - \frac{1}{x}\right) = G \left(\frac{x-1}{x}\right)
   \]
4. Let \( a\% \) be the percentage of the remaining gold that King Midas needs to earn today to end up with the original amount \( G \).
5. The amount of gold he needs to earn is:
   \[
   \frac{a}{100} \times G \left(\frac{x-1}{x}\right)
   \]
6. To end up with the original amount \( G \), the total amount of gold after earning should be \( G \):
   \[
   G \left(\frac{x-1}{x}\right) + \frac{a}{100} \times G \left(\frac{x-1}{x}\right) = G
   \]
7. Factor out \( G \left(\frac{x-1}{x}\right) \):
   \[
   G \left(\frac{x-1}{x}\right) \left(1 + \frac{a}{100}\right) = G
   \]
8. Divide both sides by \( G \left(\frac{x-1}{x}\right) \):
   \[
   1 + \frac{a}{100} = \frac{x}{x-1}
   \]
9. Subtract 1 from both sides:
   \[
   \frac{a}{100} = \frac{x}{x-1} - 1
   \]
10. Simplify the right-hand side:
    \[
    \frac{a}{100} = \frac{x - (x-1)}{x-1} = \frac{1}{x-1}
    \]
11. Multiply both sides by 100 to solve for \( a \):
    \[
    a = 100 \times \frac{1}{x-1} = \frac{100}{x-1}
    \]

The final answer is \(\boxed{\frac{100}{x-1}\%}\)