Let \( a \) represent Carrie’s allowance.
The equation given is:
\[ \frac{1}{4}a + \frac{1}{3}a + 8 = 22 \]
To solve for \( a \), we first combine the fractions on the left side of the equation. To do this, we need a common denominator. The least common multiple of 4 and 3 is 12. We can rewrite each fraction:
\[ \frac{1}{4}a = \frac{3}{12}a \] \[ \frac{1}{3}a = \frac{4}{12}a \]
Now, substitute these back into the equation:
\[ \frac{3}{12}a + \frac{4}{12}a + 8 = 22 \]
Combine the fractions:
\[ \frac{3 + 4}{12}a + 8 = 22 \] \[ \frac{7}{12}a + 8 = 22 \]
Next, isolate the term with \( a \) by subtracting 8 from both sides:
\[ \frac{7}{12}a = 22 - 8 \] \[ \frac{7}{12}a = 14 \]
Now, to solve for \( a \), multiply both sides by the reciprocal of \( \frac{7}{12} \), which is \( \frac{12}{7} \):
\[ a = 14 \times \frac{12}{7} \]
Calculating this gives:
\[ a = \frac{168}{7} = 24 \]
Thus, Carrie’s allowance was \( \boxed{24} \).