In that case, can the problems, minus the proven incorrect one, both yield the same answer?
They do yield exactly the same answer. What is confusing you (I think) is that x stands for something different in the two formulations.
Correct Formulation 1
\(\displaystyle x = gallons\ in\ tank.\)
\(\displaystyle input\ rate\ of\ fast\ pipe = x / 12.\)
\(\displaystyle input\ rate\ of\ slow\ pipe = x / 15.\)
\(\displaystyle input\ rate\ of\ both\ pipes\ together = r = \dfrac{x}{12} + \dfrac{x}{15}.\)
\(\displaystyle hours\ to\ fill\ tank\ using\ both\ pipes = h.\)
\(\displaystyle But\ r = \dfrac{x}{h}= \dfrac{x}{12} + \dfrac{x}{15} = \dfrac{9x}{60} \implies h = \dfrac{60}{9}\ hours = 400\ minutes.\)
x is in the numerators because it represents gallons.
Correct Formulation 2
\(\displaystyle g = gallons\ in\ tank.\)
\(\displaystyle input\ rate\ of\ fast\ pipe = g / 12.\)
\(\displaystyle input\ rate\ of\ slow\ pipe = g / 15.\)
\(\displaystyle input\ rate\ of\ both\ pipes\ together = r = \dfrac{g}{12} + \dfrac{g}{15} = g\left(\dfrac{1}{12} + \dfrac{1}{15}\right).\)
\(\displaystyle hours\ to\ fill\ tank\ using\ both\ pipes = x.\)
\(\displaystyle But\ r = \dfrac{g}{x}= g\left(\dfrac{1}{12} + \dfrac{1}{15}\right) \implies \dfrac{1}{x} = \dfrac{1}{12} + \dfrac{1}{15} = \dfrac{9}{60} \implies x = \dfrac{60}{9}\ hours = 400\ minutes.\)
x is in the denominator because it represents hours.