Each person in Cambridge drinks a (possibly different) $12$ ounce mixture of water and apple juice,
where each drink has a positive amount of both liquids. Marc McGovern, the mayor of Cambridge, drinks $\frac{1}{6}$ of the total amount of water drunk and $\frac{1}{8}$ of the total amount of apple juice drunk. How many people are in Cambridge?


1. Let \( p \) be the number of people in Cambridge.
2. Let \( w \) be the total amount of water consumed by all people in Cambridge.
3. Let \( a \) be the total amount of apple juice consumed by all people in Cambridge.
4. Since each person drinks a 12-ounce mixture of water and apple juice, we have:
   \[
   w + a = 12p
   \]
5. Marc McGovern drinks \(\frac{1}{6}\) of the total amount of water and \(\frac{1}{8}\) of the total amount of apple juice. Therefore, the amount of water Marc drinks is:
   \[
   \frac{w}{6}
   \]
   and the amount of apple juice Marc drinks is:
   \[
   \frac{a}{8}
   \]
6. Since Marc drinks a total of 12 ounces, we have:
   \[
   \frac{w}{6} + \frac{a}{8} = 12
   \]
7. To solve for \( p \), we first express \( a \) in terms of \( w \) using the equation \( w + a = 12p \):
   \[
   a = 12p - w
   \]
8. Substitute \( a = 12p - w \) into the equation \(\frac{w}{6} + \frac{a}{8} = 12\):
   \[
   \frac{w}{6} + \frac{12p - w}{8} = 12
   \]
9. Clear the fractions by multiplying through by the least common multiple of 6 and 8, which is 24:
   \[
   24 \left( \frac{w}{6} + \frac{12p - w}{8} \right) = 24 \cdot 12
   \]
   \[
   4w + 3(12p - w) = 288
   \]
10. Simplify the equation:
    \[
    4w + 36p - 3w = 288
    \]
    \[
    w + 36p = 288
    \]
11. Substitute \( w = 288 - 36p \) back into the equation \( w + a = 12p \):
    \[
    288 - 36p + a = 12p
    \]
    \[
    a = 12p - (288 - 36p)
    \]
    \[
    a = 48p - 288
    \]
12. Since \( a \) must be positive, we have:
    \[
    48p - 288 > 0
    \]
    \[
    48p > 288
    \]
    \[
    p > 6
    \]
13. Similarly, since \( w \) must be positive, we have:
    \[
    288 - 36p > 0
    \]
    \[
    288 > 36p
    \]
    \[
    p < 8
    \]
14. Combining the inequalities, we get:
    \[
    6 < p < 8
    \]
15. Since \( p \) must be an integer, the only possible value is:
    \[
    p = 7
    \]

The final answer is \(\boxed{7}\).