gsm8k-0013

# GSM8K Problem 14

Melanie is a door-to-door saleswoman. She sold a third of her vacuum cleaners at the green house, 2 more to the red house, and half of what was left at the orange house. If Melanie has 5 vacuum cleaners left, how many did she start with?

Reason step by step, then give ONLY the final answer on the last line as \boxed{ANSWER}.

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Let's denote the total number of vacuum cleaners Melanie started with as \( x \).

1. She sold a third of her vacuum cleaners at the green house:
   \[
   \text{Sold at green house} = \frac{x}{3}
   \]
   \[
   \text{Remaining after green house} = x - \frac{x}{3} = \frac{2x}{3}
   \]

2. She sold 2 more to the red house:
   \[
   \text{Sold at red house} = 2
   \]
   \[
   \text{Remaining after red house} = \frac{2x}{3} - 2
   \]

3. She sold half of what was left at the orange house:
   \[
   \text{Sold at orange house} = \frac{1}{2} \left( \frac{2x}{3} - 2 \right) = \frac{x}{3} - 1
   \]
   \[
   \text{Remaining after orange house} = \left( \frac{2x}{3} - 2 \right) - \left( \frac{x}{3} - 1 \right) = \frac{x}{3} - 1
   \]

4. According to the problem, Melanie has 5 vacuum cleaners left:
   \[
   \frac{x}{3} - 1 = 5
   \]

5. Solve for \( x \):
   \[
   \frac{x}{3} = 6
   \]
   \[
   x = 18
   \]

Therefore, Melanie started with \(\boxed{18}\) vacuum cleaners.