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's denote the total number of vacuum cleaners Melanie started with as X.

At the green house, she sold a third of them, so she had X - X/3 left.
At the red house, she sold 2 more, so she had X - X/3 - 2 left.
At the orange house, she sold half of what was left, so she had (X - X/3 - 2)/2 left.

We know that after the orange house, she has 5 vacuum cleaners left, so we can set up the equation:

(X - X/3 - 2)/2 = 5

Now, let's solve for X step by step:

1. Multiply both sides by 2 to get rid of the fraction:
X - X/3 - 2 = 10

2. Multiply everything by 3 to get rid of the fraction:
3X - X - 6 = 30

3. Combine like terms:
2X - 6 = 30

4. Add 6 to both sides:
2X = 36

5. Divide both sides by 2:
X = 18

Melanie started with 18 vacuum cleaners.

\boxed{18}