To find out how much Carrie’s allowance was, we need to solve the equation given in terms of her allowance \( a \).
The equation provided is:
[tex]\[
\frac{1}{4}a + \frac{1}{3}a + 8 = 22
\][/tex]
### Step-by-Step Solution
1. Combine like terms involving \( a \):
First, we combine the fractions that involve \( a \). To do this, we need a common denominator. The least common multiple of 4 and 3 is 12.
Therefore, we convert the fractions:
[tex]\[
\frac{1}{4}a = \frac{1 \times 3}{4 \times 3}a = \frac{3}{12}a
\][/tex]
[tex]\[
\frac{1}{3}a = \frac{1 \times 4}{3 \times 4}a = \frac{4}{12}a
\][/tex]
Adding these fractions together:
[tex]\[
\frac{3}{12}a + \frac{4}{12}a = \frac{3 + 4}{12}a = \frac{7}{12}a
\][/tex]
2. Rewriting the equation:
Substituting back into the original equation, we get:
[tex]\[
\frac{7}{12}a + 8 = 22
\][/tex]
3. Isolate \( a \):
To isolate \( a \), we first move the constant term (8) to the right side of the equation by subtracting 8 from both sides:
[tex]\[
\frac{7}{12}a = 22 - 8
\][/tex]
[tex]\[
\frac{7}{12}a = 14
\][/tex]
4. Solve for \( a \):
To solve for \( a \), multiply both sides of the equation by the reciprocal of \( \frac{7}{12} \), which is \( \frac{12}{7} \):
[tex]\[
a = 14 \times \frac{12}{7}
\][/tex]
[tex]\[
a = 14 \times \frac{12}{7} = 2 \times 12 = 24
\][/tex]
Therefore, Carrie's allowance was [tex]\(\$24\)[/tex].
