A waiter's total income consists of his base salary and tips. During a specific week, his tips amounted to $\frac{5}{4}$ of his base salary. What fraction of his total weekly income came from his tips?

Let the waiter's base salary be $S$.
According to the problem, the tips received are:

$$\text{Tips}=\frac{5}{4}S$$

The total weekly income is the sum of his base salary and tips:

$$\text{Total Income}=\text{Salary}+\text{Tips}$$

$$\text{Total Income}=S+\frac{5}{4}S=\frac{9}{4}S$$

Now, we calculate the required fraction of income that came from tips:

$$\text{Required Fraction}=\frac{\text{Tips}}{\text{Total Income}}=\frac{\frac{5}{4}S}{\frac{9}{4}S}=\frac{5}{9}$$