What should be added to `((5)/(6) - (7)/(8))` to get `(1)/(2)`?

Answer

To solve the problem of what should be added to \(\frac{5}{6} - \frac{7}{8}\) to get \(\frac{1}{2}\), we can follow these steps:

### Step 1: Set up the equation
We need to find \(x\) such that:
\[ \frac{5}{6} - \frac{7}{8} + x = \frac{1}{2} \]
This can be rearranged to:
\[ x = \frac{1}{2} - \left(\frac{5}{6} - \frac{7}{8}\right) \]

### Step 2: Calculate \(\frac{5}{6} - \frac{7}{8}\)
To perform the subtraction, we need a common denominator. The least common multiple (LCM) of 6 and 8 is 24.

Convert \(\frac{5}{6}\) and \(\frac{7}{8}\) to have a denominator of 24:
\[ \frac{5}{6} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24} \]
\[ \frac{7}{8} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24} \]

Now, subtract:
\[ \frac{20}{24} - \frac{21}{24} = \frac{20 - 21}{24} = \frac{-1}{24} \]

### Step 3: Substitute back into the equation
Now substitute \(\frac{-1}{24}\) back into the equation for \(x\):
\[ x = \frac{1}{2} - \left(\frac{-1}{24}\right) \]
This simplifies to:
\[ x = \frac{1}{2} + \frac{1}{24} \]

### Step 4: Find a common denominator to add \(\frac{1}{2}\) and \(\frac{1}{24}\)
The LCM of 2 and 24 is 24. Convert \(\frac{1}{2}\) to have a denominator of 24:
\[ \frac{1}{2} = \frac{1 \times 12}{2 \times 12} = \frac{12}{24} \]

Now add:
\[ x = \frac{12}{24} + \frac{1}{24} = \frac{12 + 1}{24} = \frac{13}{24} \]

### Final Answer
Thus, the value that should be added is:
\[ \boxed{\frac{13}{24}} \]