Step 1: Understanding the Question:
The problem asks for the total time required to fill a tank if all three taps (A, B, and C) are opened simultaneously.
The provided text contains a repetitive typo from the OCR, stating "A and C together can fill it in 10 h" and then "A and C together can fill it in 7(1/2)".
Standard logical structure for such problems implies the three unique pairings are A+B, B+C, and A+C.
Thus, the correct intended values are A and B in 6 hours, B and C in 10 hours, and A and C in 7.5 hours ($15/2$ hours).
Step 2: Key Formula or Approach:
To solve this, we will determine the portion of the tank filled by each pair in one hour.
Let the rates of the taps be denoted by $\frac{1}{A}$, $\frac{1}{B}$, and $\frac{1}{C}$.
Adding the three combined equations yields twice the combined rate of all three taps together: $2\left(\frac{1}{A} + \frac{1}{B} + \frac{1}{C}\right)$.
Step 3: Detailed Explanation:
First, let us write down the 1-hour work rates for each pair of taps based on the corrected problem statement.
The part filled by A and B in 1 hour is $\frac{1}{A} + \frac{1}{B} = \frac{1}{6}$.
The part filled by B and C in 1 hour is $\frac{1}{B} + \frac{1}{C} = \frac{1}{10}$.
The part filled by A and C in 1 hour is $\frac{1}{A} + \frac{1}{C} = \frac{1}{7.5} = \frac{2}{15}$.
We will now add these three equations together.
\[ \left(\frac{1}{A} + \frac{1}{B}\right) + \left(\frac{1}{B} + \frac{1}{C}\right) + \left(\frac{1}{A} + \frac{1}{C}\right) = \frac{1}{6} + \frac{1}{10} + \frac{2}{15} \]
This simplification results in:
\[ 2\left(\frac{1}{A} + \frac{1}{B} + \frac{1}{C}\right) = \frac{1}{6} + \frac{1}{10} + \frac{2}{15} \]
Next, we find a common denominator for the fractions on the right side, which is 30.
\[ \frac{1}{6} = \frac{5}{30} \]
\[ \frac{1}{10} = \frac{3}{30} \]
\[ \frac{2}{15} = \frac{4}{30} \]
Now, substitute these back into the sum.
\[ 2\left(\frac{1}{A} + \frac{1}{B} + \frac{1}{C}\right) = \frac{5 + 3 + 4}{30} = \frac{12}{30} \]
We can reduce the fraction $\frac{12}{30}$ to $\frac{2}{5}$.
\[ 2\left(\frac{1}{A} + \frac{1}{B} + \frac{1}{C}\right) = \frac{2}{5} \]
Dividing both sides by 2 gives the combined 1-hour work rate of all three taps working simultaneously.
\[ \frac{1}{A} + \frac{1}{B} + \frac{1}{C} = \frac{1}{5} \]
Since the combined rate is $\frac{1}{5}$ of the tank per hour, all three taps will take exactly 5 hours to fill the tank completely.
Step 4: Final Answer:
The time taken by all three taps to fill the tank is 5 h.