Step 1: Set up the equations

Let the two numbers be $x$ and $y$. According to the problem, their sum is $36$ and $5$ times one number is equal to $4$ times the other.

$x + y = 36$ and $5x = 4y$

Step 2: Express one variable in terms of the other

From the ratio equation $5x = 4y$, we can express $y$ in terms of $x$: $y = \frac{5}{4}x$

$y = \frac{5}{4}x$

Step 3: Substitute into the sum equation

Substitute $y = \frac{5}{4}x$ into the sum equation: $x + \frac{5}{4}x = 36x \left(1 + \frac{5}{4}\right) = 36\frac{9}{4}x = 36$

$\frac{9}{4}x = 36$

Step 4: Solve for $x$ and $y$

Solve for $x$: $x = 36 \times \frac{4}{9} = 16$ Now, find $y$: $y = 36 - 16 = 20$

$x = 16, \quad y = 20$

Step 5: Identify the greater number

Comparing the two numbers ($16$ and $20$), the greater of the two numbers is $20$.

$20$