To solve the equation \(\frac{1}{4} a + \frac{1}{3} a + 8 = 22\) and find the amount of Carrie's allowance, we can follow these steps:
1. Combine the terms with \(a\) on the left side of the equation:
[tex]\[
\frac{1}{4} a + \frac{1}{3} a + 8 = 22
\][/tex]
2. Find a common denominator for the fractions \(\frac{1}{4}a\) and \(\frac{1}{3}a\).
The least common denominator (LCD) for 4 and 3 is 12. Rewrite the fractions with this common denominator:
[tex]\[
\frac{1}{4} a = \frac{3}{12} a
\][/tex]
[tex]\[
\frac{1}{3} a = \frac{4}{12} a
\][/tex]
3. Add the fractions together:
[tex]\[
\frac{3}{12} a + \frac{4}{12} a = \frac{7}{12} a
\][/tex]
4. Substitute this back into the original equation:
[tex]\[
\frac{7}{12} a + 8 = 22
\][/tex]
5. Isolate the \(a\) term by moving the constant term (8) to the right side of the equation:
[tex]\[
\frac{7}{12} a = 22 - 8
\][/tex]
Simplify the right side:
[tex]\[
\frac{7}{12} a = 14
\][/tex]
6. Solve for \(a\) by multiplying both sides by the reciprocal of \(\frac{7}{12}\), which is \(\frac{12}{7}\):
[tex]\[
a = 14 \times \frac{12}{7}
\][/tex]
Simplify the multiplication:
[tex]\[
a = 14 \times \frac{12}{7} = 14 \times 1.71428571 \approx 24
\][/tex]
Therefore, Carrie's allowance [tex]\(a\)[/tex] is 24 dollars.
