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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I need to work backwards from the final number of vacuum cleaners to find the initial amount.

Let me define variables and track the sales step by step.

Let x = initial number of vacuum cleaners Melanie had

Step 1: Sales at the green house
- She sold 1/3 of her vacuum cleaners at the green house
- Sold: x/3 vacuum cleaners
- Remaining: x - x/3 = 2x/3 vacuum cleaners

Step 2: Sales at the red house
- She sold 2 more vacuum cleaners at the red house
- Sold: 2 vacuum cleaners
- Remaining: 2x/3 - 2 vacuum cleaners

Step 3: Sales at the orange house
- She sold half of what was left at the orange house
- Sold: (1/2)(2x/3 - 2) = (2x/3 - 2)/2 = x/3 - 1 vacuum cleaners
- Remaining: (2x/3 - 2) - (x/3 - 1) = 2x/3 - 2 - x/3 + 1 = x/3 - 1 vacuum cleaners

Step 4: Final condition
- Melanie has 5 vacuum cleaners left
- So: x/3 - 1 = 5
- Therefore: x/3 = 6
- Therefore: x = 18

Let me verify this answer:
- Started with: 18 vacuum cleaners
- Green house: sold 18/3 = 6, remaining: 18 - 6 = 12
- Red house: sold 2, remaining: 12 - 2 = 10
- Orange house: sold 10/2 = 5, remaining: 10 - 5 = 5 ✓

The answer checks out.

$\boxed{18}$