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.
    *   Remaining: $\frac{2}{3}x - 2$

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

4.  **Final Count:** We are told 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 $\frac{3}{2}$:
    $$ x = 12 \times \frac{3}{2} $$
    $$ x = \frac{36}{2} $$
    $$ x = 18 $$

**Verification (Working Backwards):**
*   She has 5 left.
*   Before the orange house, she had double that amount (since she sold half): $5 \times 2 = 10$.
*   Before the red house, she had 2 more than that (since she sold 2): $10 + 2 = 12$.
*   Before the green house, the 12 represents the remaining two-thirds (since she sold one-third). To find the total, we divide by $\frac{2}{3}$ (or multiply by $\frac{3}{2}$): $12 \times \frac{3}{2} = 18$.

Melanie started with 18 vacuum cleaners.

\boxed{18}