From what should `((3)/(7) + (4)/(21))` be subtracted to get `(5)/(6)` ?

Answer

To solve the problem, we need to find the value of \( x \) such that when we subtract \( \left(\frac{3}{7} + \frac{4}{21}\right) \) from \( x \), we get \( \frac{5}{6} \).

### Step-by-step Solution:

1. **Set up the equation:**
\[ x - \left(\frac{3}{7} + \frac{4}{21}\right) = \frac{5}{6} \]

2. **Calculate \( \frac{3}{7} + \frac{4}{21} \):**
- First, find a common denominator for \( \frac{3}{7} \) and \( \frac{4}{21} \). The least common multiple (LCM) of 7 and 21 is 21.
- Convert \( \frac{3}{7} \) to have a denominator of 21:
\[ \frac{3}{7} = \frac{3 \times 3}{7 \times 3} = \frac{9}{21} \]
- Now, add \( \frac{9}{21} + \frac{4}{21} \):
\[ \frac{9}{21} + \frac{4}{21} = \frac{9 + 4}{21} = \frac{13}{21} \]

3. **Substitute back into the equation:**
\[ x - \frac{13}{21} = \frac{5}{6} \]

4. **Isolate \( x \):**
- Add \( \frac{13}{21} \) to both sides:
\[ x = \frac{5}{6} + \frac{13}{21} \]

5. **Find a common denominator for \( \frac{5}{6} \) and \( \frac{13}{21} \):**
- The LCM of 6 and 21 is 42.
- Convert \( \frac{5}{6} \):
\[ \frac{5}{6} = \frac{5 \times 7}{6 \times 7} = \frac{35}{42} \]
- Convert \( \frac{13}{21} \):
\[ \frac{13}{21} = \frac{13 \times 2}{21 \times 2} = \frac{26}{42} \]

6. **Add the two fractions:**
\[ x = \frac{35}{42} + \frac{26}{42} = \frac{35 + 26}{42} = \frac{61}{42} \]

### Final Answer:
\[ x = \frac{61}{42} \]