To solve the equation \( \frac{1}{4} a + \frac{1}{3} a + 8 = 22 \) and find the value of \( a \) (which represents Carrie's allowance), follow these steps:
Step 1: Combine Like Terms
First, we need to combine the terms involving \( a \). To do this, we will find a common denominator for the fractions \( \frac{1}{4} \) and \( \frac{1}{3} \).
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:
\[ \frac{3}{12} a + \frac{4}{12} a + 8 = 22 \]
Step 2: Combine the Fractions
Now, combine the fractions:
\[ \left( \frac{3}{12} a + \frac{4}{12} a \right) + 8 = 22 \] \[ \frac{7}{12} a + 8 = 22 \]
Step 3: Isolate the Variable
Next, we want to isolate the term with \( a \). Subtract 8 from both sides:
\[ \frac{7}{12} a = 22 - 8 \] \[ \frac{7}{12} a = 14 \]
Step 4: Solve for \( a \)
To solve for \( a \), multiply both sides of the equation by the reciprocal of \( \frac{7}{12} \), which is \( \frac{12}{7} \):
\[ a = 14 \times \frac{12}{7} \]
Step 5: Simplify
Calculate \( 14 \times \frac{12}{7} \):
\[ a = \frac{14 \times 12}{7} \] \[ a = \frac{168}{7} \] \[ a = 24 \]
Conclusion
Carrie's allowance is \( \boxed{24} \).