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.  **Green House:** She sold a third of her vacuum cleaners.
    *   Sold: $\frac{1}{3}x$
    *   Remaining: $x - \frac{1}{3}x = \frac{2}{3}x$

2.  **Red House:** She sold 2 more vacuum cleaners.
    *   Sold: $2$
    *   Remaining: $\frac{2}{3}x - 2$

3.  **Orange House:** She sold half of what was left.
    *   Sold: $\frac{1}{2} \left( \frac{2}{3}x - 2 \right)$
    *   Remaining: $\frac{1}{2} \left( \frac{2}{3}x - 2 \right)$

4.  **Final Count:** She has 5 vacuum cleaners left.
    *   Equation: $\frac{1}{2} \left( \frac{2}{3}x - 2 \right) = 5$

Now, we solve for $x$:
*   Multiply both sides by 2:
    $\frac{2}{3}x - 2 = 10$
*   Add 2 to both sides:
    $\frac{2}{3}x = 12$
*   Multiply both sides by 3:
    $2x = 36$
*   Divide by 2:
    $x = 18$

Let's verify:
*   Start with 18.
*   Sell 1/3 (6) at Green House. Remaining: 12.
*   Sell 2 at Red House. Remaining: 10.
*   Sell half of remaining (5) at Orange House. Remaining: 5.
*   Matches the problem statement.

\boxed{18}