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 sells $\frac{1}{3}$ of her stock, so $\frac{2}{3}x$ remain.
2. **Red house:** She sells 2 more, leaving $\frac{2}{3}x - 2$.
3. **Orange house:** She sells half of what was left, meaning half remains. The amount left is $\frac{1}{2}\left(\frac{2}{3}x - 2\right)$.
4. **Final amount:** We are told she has 5 vacuum cleaners left, so we set up the equation:
   $$\frac{1}{2}\left(\frac{2}{3}x - 2\right) = 5$$

Now, 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 $\frac{3}{2}$:
  $$x = 12 \times \frac{3}{2} = 18$$

**Verification:**
- Start with 18.
- Sell $\frac{1}{3}$ (6) at green house $\rightarrow$ 12 left.
- Sell 2 at red house $\rightarrow$ 10 left.
- Sell half of 10 (5) at orange house $\rightarrow$ 5 left.
This matches the problem statement exactly.

\boxed{18}