gsm8k-0012

# GSM8K Problem 13

Carlos is planting a lemon tree. The tree will cost $90 to plant. Each year it will grow 7 lemons, which he can sell for $1.5 each. It costs $3 a year to water and feed the tree. How many years will it take before he starts earning money on the lemon tree?

Reason step by step, then give ONLY the final answer on the last line as \boxed{ANSWER}.

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We are given:

- **Initial cost to plant the tree**: $90 (one-time cost)
- **Each year**, the tree grows **7 lemons**, sold at **$1.5 each** → **Revenue per year** = 7 × $1.5 = **$10.5**
- **Annual cost** for water and feed: **$3**
- So, **net profit per year** = Revenue − Cost = $10.5 − $3 = **$7.5**

We want to know: **How many years until Carlos starts earning money**, i.e., when **total net profit becomes positive**.

Let $ n $ be the number of years.

- Total cost = initial planting cost + annual costs = $90 + 3n$
- Total revenue = $10.5n$
- Net profit = Revenue − Total cost = $10.5n − (90 + 3n) = (10.5 − 3)n − 90 = 7.5n − 90$

We want the smallest integer $ n $ such that net profit > 0:

$$
7.5n - 90 > 0 \\
7.5n > 90 \\
n > \frac{90}{7.5} = 12
$$

So, $ n > 12 $, meaning the smallest integer $ n $ is **13**.

Let’s double-check:

- After 12 years: net profit = $7.5 × 12 − 90 = 90 − 90 = 0$ → break-even (not yet earning).
- After 13 years: $7.5 × 13 − 90 = 97.5 − 90 = 7.5 > 0$ → starts earning.

✅ So, it takes **13 years** before he starts earning money.

$$
\boxed{13}
$$