(12) Annie had a sum of money at first. She spent \( \$ 182 \) on a bag and \( \frac{3}{8} \) of the remainder on a blouse. She had \( \frac{1}{3} \) of her money left. (a) What fraction of her money did she spend on the blouse? (b) How much money did she have at first?

Let the amount of money Annie had initially be \( M \). 1. After spending \(\$182\) on the bag, she has \[ M - 182 \] remaining. 2. She then spent \(\frac{3}{8}\) of the remainder on the blouse. The cost of the blouse is \[ \frac{3}{8}(M-182). \] 3. It is given that after both purchases, she had \(\frac{1}{3}\) of her initial money left, so \[ \frac{1}{3}M. \] Notice that the remaining money after the blouse purchase is also obtained by subtracting the blouse cost from the remaining money after the bag purchase: \[ (M-182) - \frac{3}{8}(M-182) = \frac{5}{8}(M-182). \] 4. Set the amount left equal to \(\frac{1}{3}M\): \[ \frac{5}{8}(M-182) = \frac{1}{3}M. \] 5. Multiply both sides by \(24\) (the least common multiple of \(8\) and \(3\)) to eliminate the fractions: \[ 24 \cdot \frac{5}{8}(M-182) = 24 \cdot \frac{1}{3}M. \] Simplify each side: \[ 3 \cdot 5 (M-182) = 8M \quad \Rightarrow \quad 15(M-182) = 8M. \] 6. Distribute and solve for \( M \): \[ 15M - 15 \cdot 182 = 8M. \] Calculate \(15 \cdot 182\): \[ 15 \times 182 = 2730. \] Thus: \[ 15M - 2730 = 8M. \] Subtract \(8M\) from both sides: \[ 7M = 2730, \] so: \[ M = 390. \] 7. Now, find the amount spent on the blouse: \[ \text{Blouse cost} = \frac{3}{8}(M-182) = \frac{3}{8}(390-182) = \frac{3}{8}(208). \] Since \(208 \div 8 = 26\), we have: \[ \text{Blouse cost} = 3 \times 26 = 78. \] 8. (a) To find the fraction of her money that she spent on the blouse, divide the blouse cost by the initial amount: \[ \frac{78}{390} = \frac{1}{5}. \] 9. (b) The amount of money she had at first is \(\$390\). \[ \boxed{\text{(a) } \frac{1}{5}, \quad \text{(b) } \$390.} \]