Equalizing Water Levels in Two Cups with Differing Filling Rates

Imagine you have two cups—one is full and the other is empty. You also have two taps: one tap fills the empty cup three times faster than the other tap fills the full cup. The question is: at what percentage do both cups have equal levels of water?

Mathematical Explanation

We start with the initial conditions: the empty cup tap fills a full cup in $n$ minutes, and the half-full cup tap fills a full cup in $3n$ minutes. This means:

Step 1: Determining the Filling Rate

The rate at which the empty cup tap fills the cup is $1/n$ cups per minute. The rate at which the half-full cup tap fills the cup is $1/(3n)$ cups per minute.

Step 2: Calculating the Time to Equalize Water Levels

Let's denote the time in minutes the taps are on as $t$. The amount of water filled by the empty cup tap is $t cdot frac{1}{n} frac{t}{n}$, and for the half-full cup tap, it is $t cdot frac{1}{3n} frac{t}{3n}$. For the cups to be equal, the equation becomes:

$$frac{t}{n} frac{t}{3n} frac{1}{2}$$

Simplifying, we get:

Step 3: Solving for Time

Subtract $frac{t}{3n}$ from both sides: $$frac{t}{3n} frac{1}{2}$$ Multiplying both sides by $3n$: $$t frac{3n}{2}$$ Since $3n 4$ minutes (from the initial condition), $t 2$ minutes.

Both cups will have equal levels of water after 2 minutes.

Step 4: Calculating the Water Level

After 2 minutes, the amount of water added by the empty cup tap is:

$$2 cdot frac{1}{n} frac{2}{n}$$

And for the half-full cup tap:

$$2 cdot frac{1}{3n} frac{2}{3n}$$

The total amount of water in the full cup:

$$frac{2}{3n} frac{1}{2}$$

Since $3n 4$, the total amount of water in the full cup after 2 minutes is:

$$frac{2}{4} frac{1}{2} frac{1}{2} frac{1}{2} 1$$

However, we need to convert this to a percentage. The calculation is:

$$frac{1}{2} 0.5$$

$$0.5 times 100 50%$$

50.00000%

Therefore, both cups will have equal levels of water at 50% when they both contain half a cup each.

Conclusion

Both cups will have equal levels of water at 75% full after 1 minute when the empty cup tap fills 3/4 of the cup per unit time and the half-full cup tap fills 1/4 of the cup per unit time.

These calculations and assumptions ensure that the problem is fully understood and the correct solution is derived using mathematical reasoning and algebraic manipulation.