I was struggling for a long time with the following question, eventually unable to find it. I'm really frustrated with it because the answer is not difficult per se but to me requires a complete shift in approaching the problem, and I am just unable to make such a leap by myself. So, is it a hard question or am I just stupid?
Here goes: Two pipes together can fill a reservoir in 6 hr 40 min. Find the time each alone will take to fill the reservoir if one of the pipes can fill it in 3 hr less time than the other.
Edit: The solution is in my textbook so not quite asking for that. Once you formulate the quadratic equation it's straightforward but for me this step really takes a leap of the imagination to find it and I wonder if I'm alone in this. Can't really post the answer because then it'll be obvious.
So basically I'm asking if anyone can formulate the equation and tell me if they found it difficult.
Thanks for the willingness to answer already!
Best Answer
Suppose that one pipe takes $t$ hours to fill the reservoir, and the other pipe takes $t-3$ hours.
Then in $t(t-3)$ hours, pipe 1 has filled $t-3$ reservoirs, and pipe 2 has filled $t$ reservoirs. Together, they fill $2t - 3$ reservoirs in $t(t-3)$ hours, or $1$ reservoir in $\frac{t(t-3)}{2t-3}$ hours.
Converting $6$ hours $40$ minutes into hours, we get:
$$\frac{t(t-3)}{2t-3} = \frac{20}{3}$$ $$\Rightarrow 3t(t-3) = 20(2t-3)$$ $$\Rightarrow 3t^2 - 9t = 40t - 60 \implies 3t^2 - 49t + 60 = 0$$ $$\Rightarrow (3t - 4)(t - 15) = 0 \implies t = 15$$
so working alone, one pipe fills it in $15$ hours, and the other fills it in $12$.
The smaller solution is discarded because firstly, it would make $t-3$ negative (which does not make sense), and working alone should result in a longer time than working together.
As a sanity check, you can repeat this process with $t = 15$ (or using any other method), and doing this gives you $\frac{20}{3}$ hours to fill up the reservoir.