Question

A renewable energy facility stores electricity in three separate battery modules: Module $X$, Module $Y$, and Module $Z$. The ratio of the initial energy stored in Module $X$ to Module $Y$ is $2 : 3$, and the ratio of the initial energy stored in Module $Y$ to Module $Z$ is $5 : 8$. Each module discharges its stored energy at a constant individual rate. Operating alone, Module $Y$ can completely discharge its initial stored energy in $5$ hours. Module $X$ discharges energy at a rate $33\frac{1}{3}\%$ greater than Module $Y$, and Module $Z$ discharges energy at a rate $25\%$ less than the combined discharge rate of Modules $X$ and $Y$. If all three modules begin discharging simultaneously, how many hours will it take to completely discharge the total initial energy stored across all three modules?