Question

Carrie spent 
1
4
 of her allowance on a shirt, 
1
3
 of her allowance on a skirt, and $8 on a belt. If she spent $22 in all, how much was Carrie’s allowance?

Equation: 
1
4
a +
1
3
a + 8 = 22

Carrie’s allowance was $

Answer 1

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 \( \boxed{24} \).