Why not the other way around?

[aaronhma]

Prof. Loh says that since we're trying to drain the pool and the filling the pool is working against it, the equation is:

$$\frac{1}{185} + \frac{1}{74} - \frac{1}{x} = \frac{1}{111}$$

By similar logic, we're trying to fill the pool, and draining the pool is working against it, so the equation is:

$$\frac{1}{x} - \frac{1}{74} - \frac{1}{185} = \frac{1}{111}$$

Are these two equivalent to each other? Thanks!

[quacker88]

Really good idea you have there, @aaronhma !

One thing though (soo close!)-- since you're kinda looking at the problem from the perspective of the filling pipe, we actually have to flip the sign of the $\frac{1}{111}$. Here's why:

Let's look at the first equation:

$$\frac{1}{185} + \frac{1}{74} - \frac{1}{x} = \frac{1}{111}$$

What this equation literally means is that, in $1$ minute, the large pipe drains $\frac{1}{185}$ of the pool, the small pipe drains $\frac{1}{74}$ of the pool, and the filling pipe fills or "un-drains" $\frac{1}{x}$ of the pool, which is why we subtract the $\frac{1}{x}$. This altogether is equal to draining $\frac{1}{111}$ of the pool.

But looking at the second equation, which is from the perspective of the filling pipe, we have:

$$\frac{1}{x} - \frac{1}{74} - \frac{1}{185} = \frac{1}{111}$$

This equation says that, in $1$ minute, the filling pipe fills $\frac{1}{x}$ of the pool, the small pipe drains or "un-fills" $\frac{1}{74}$ of the pool, which is why we subtract $\frac{1}{74}$, the large pipe drains or "un-fills" $\frac{1}{185}$ of the pool, which is why we subtract$\frac{1}{185}$.

But here's the catch: this altogether is equal to draining $\frac{1}{111}$ of the pool. We have "un-filled" the pool by that much. This means we have less water than when we started.

So, the correct equation would be actually equal to $-\frac{1}{111}$:

$$\frac{1}{x} - \frac{1}{74} - \frac{1}{185} = -\frac{1}{111}$$

Notice how this is exactly the same as Prof. Loh's equation but multiplied all the way on both sides by $-1$.
Really great thinking you did there, and it really goes to show how many different ways there are to solve the same problem! Well done

[aaronhma]

This post is deleted!