It takes 12 minutes to fill an entire bathtub using both the cold and hot water. If just the cold water is used, it takes 18 minutes to fill the bathtub. How long would it take to fill the bathtub if just the hot water were used? The equation can be used to solve for the rate, r, for the hot water alone to fill the bathtub. The hot water can fill of the tub in 1 minute. It would take minutes for the hot water to fill the bathtub.

We are gonna name V the complete volume of the bathtub, which is filled a certain amount of minutes. Each filling rate or speed is gonna be expressed as: [tex]\frac{V}{t}[/tex].

So, if we apply this consideration to each case we have:

Using cold and hot water: [tex]\frac{V}{12}[/tex]

Using only cold water: [tex]\frac{V}{18}[/tex]

Using only hot water: [tex]\frac{V}{x}[/tex]; because we don't knot the time it takes to fill the bathtub with hot water.

Now, as you can see, Cold and Hot water is a sum of cold water only and hot water only:

[tex]\frac{V}{12}=\frac{V}{18}+\frac{V}{x}[/tex]

Solving the equation for x:

[tex]\frac{V}{12}=\frac{xV+18V}{18x} \\\frac{V}{12}=\frac{(x+18)V}{18x}\\\\\frac{18xV}{V}=12(x+18)\\18x=12x+216\\18x-12x=216\\6x=216\\x=\frac{216}{6}=36[/tex]

Therefore, it takes 36 minutes to fill the bathtub using just hot water.