The Problem:
The problem states that one-fifth of a number is increased by 5, and the result is equal to 4 less than one-fourth of the same number. We need to find the number.

Step-by-Step Solution:

Step 1: Understand the problem:
Let's assume the number as 'x'.
According to the problem, one-fifth of 'x' increased by 5 is equal to 4 less than one-fourth of 'x'.

Step 2: Translate the problem into an equation:
We can translate the problem statement into the following equation:
1/5 * x + 5 = 1/4 * x - 4

Step 3: Solve the equation:
To solve the equation, we need to isolate the variable 'x' on one side of the equation.

Step 4: Simplify the equation:
Multiplying both sides of the equation by 20 to eliminate the fractions, we get:
4 * (1/5 * x + 5) = 4 * (1/4 * x - 4)
4 * (1/5 * x) + 4 * 5 = 4 * (1/4 * x) - 4 * 4
4/5 * x + 20 = 1/4 * x - 16

Step 5: Combine like terms:
Let's combine the terms with 'x' on one side and the constants on the other side of the equation.
4/5 * x - 1/4 * x = -16 - 20
(4/5 - 1/4) * x = -36

Step 6: Simplify further:
To simplify the equation, we need to find a common denominator for 5 and 4, which is 20.
(16/20 - 5/20) * x = -36
11/20 * x = -36

Step 7: Solve for 'x':
To solve for 'x', we need to isolate it on one side of the equation. We can do this by multiplying both sides of the equation by the reciprocal of 11/20, which is 20/11.

(11/20 * x) * (20/11) = -36 * (20/11)
x = -720/11

Step 8: Simplify the result:
The number is -720/11.

Therefore, the solution to the problem is x = -720/11.