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 solve this step by step.

Let $ x $ be the number of vacuum cleaners Melanie started with.

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**Step 1: Sold a third at the green house.**

She sold $ \frac{1}{3}x $ vacuum cleaners at the green house.

Remaining after green house:
$$
x - \frac{1}{3}x = \frac{2}{3}x
$$

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**Step 2: Sold 2 more to the red house.**

She sold 2 vacuum cleaners at the red house.

Remaining after red house:
$$
\frac{2}{3}x - 2
$$

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**Step 3: Sold half of what was left at the orange house.**

She sold half of $ \frac{2}{3}x - 2 $, so remaining after orange house is half of that:

Remaining after orange house:
$$
\frac{1}{2} \left( \frac{2}{3}x - 2 \right)
$$

We are told she has **5 vacuum cleaners left**, so:

$$
\frac{1}{2} \left( \frac{2}{3}x - 2 \right) = 5
$$

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} $:

$$
x = 12 \cdot \frac{3}{2} = 18
$$

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✅ So, Melanie started with **18** vacuum cleaners.

$$
\boxed{18}
$$