Question

For a certain bathtub, it takes the hot water faucet

Answer 1

So, we know both faucets open, can do the whole job in 6minutes

ok, how much has the faster cold water faucet done in 1minute then?

well, let's say the the rate of speed of the cold water faucet is "c", so in 1minute it has done 1/c of the whole work

now, we know the hot water faucet is slower, it takes it 3 times as long, that means is 1/3 the speed of the cold water faucet, so, how much does the hot water faucet do in 1min?   well, we know is 1/3 fast as the cold one, the cold one is 1/c, what's 1/3 of 1/c? well, 1/3 * 1/c

now, we know they both added together, can do the whole job in 6 minutes, so in 1minute, they've done 1/6 of the job then

thus

$\bf \begin{array}{cccccclllllll} \cfrac{1}{c}&+&\left( \cfrac{1}{3}\cdot \cfrac{1}{c} \right)&=&\cfrac{1}{6}\\\\ \cfrac{1}{c}&+&\cfrac{1}{3c}&=&\cfrac{1}{6}\\\\ \uparrow &&\uparrow &&\uparrow \\ \textit{cold water rate}&&\textit{hot water rate}&&\textit{total done by both} \end{array}\\\\ -----------------------------$

$\bf \cfrac{3+1}{3c}=\cfrac{1}{6}\implies 24=3c\implies \cfrac{24}{3}=c\implies \boxed{8=c} \\\\\\ \textit{now, if the cold water faucet can do it in 8 minutes by itself}\\\\ \textit{then the hot water faucet can do it in 3 times as long} \\\\\\ 3\cdot 8\implies \boxed{24}$