To solve the equation
\[ \frac{h}{3} + \frac{h}{4} + \frac{h}{4} = 1, \]
we first need to find a common denominator for the fractions. The denominators are 3 and 4, and the least common multiple of these two numbers is 12.
Next, we will rewrite each fraction with the common denominator of 12:
\[ \frac{h}{3} = \frac{4h}{12}, \quad \frac{h}{4} = \frac{3h}{12}. \]
Substituting these into the equation gives:
\[ \frac{4h}{12} + \frac{3h}{12} + \frac{3h}{12} = 1. \]
Now, combine the fractions on the left side:
\[ \frac{4h + 3h + 3h}{12} = 1 \ \frac{10h}{12} = 1. \]
To simplify \(\frac{10h}{12}\), divide the numerator and the denominator by their greatest common divisor, which is 2:
\[ \frac{5h}{6} = 1. \]
Now, multiply both sides of the equation by 6 to eliminate the fraction:
\[ 5h = 6. \]
Next, divide both sides by 5 to solve for \(h\):
\[ h = \frac{6}{5}. \]
To express this as a mixed number, we can convert \(\frac{6}{5}\) to 1 and \(\frac{1}{5}\). Thus,
\[ h = 1 \frac{1}{5}. \]
So, the time it will take Jessica, Neil, and Ryan to decorate the gym is
\[ \frac{6}{5} \text{ hours} \quad \text{or} \quad 1 \frac{1}{5} \text{ hours}. \]
In its simplest form, the solution is
\(\frac{6}{5}\) hours.