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 $x$ be the number of vacuum cleaners Melanie started with.

1.  **First sale:** She sold a third of her vacuum cleaners at the green house.
    $$ \text{Sold at green house} = \frac{1}{3}x $$
    $$ \text{Remaining} = x - \frac{1}{3}x = \frac{2}{3}x $$

2.  **Second sale:** She sold 2 more to the red house.
    $$ \text{Sold at red house} = 2 $$
    $$ \text{Remaining} = \frac{2}{3}x - 2 $$

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

4.  **Final count:** We are given that she has 5 vacuum cleaners left.
    $$ \frac{1}{3}x - 1 = 5 $$

5.  **Solve for $x$:**
    $$ \frac{1}{3}x = 6 $$
    $$ x = 6 \times 3 $$
    $$ x = 18 $$

\boxed{18}