Every day a mango seller sells half his stock, 10% of the stock overnight gets spoiled. If 1983 mangoes rotted over 3 nights then what did he start with on the first day?

Let's solve this step-by-step:

1. **Define Variables and Equations**:
   - Let \( x \) be the initial number of mangoes.
   - Every night, 10% of the remaining stock gets spoiled. This means 90% of the remaining mangoes are left unspoiled.

2. **Daily and Overnight Calculations**:
   - At the end of the first day, the seller sells half of his remaining mangoes. This leaves him with \( 0.5x \) mangoes.
   - Overnight, 10% of these mangoes spoil, meaning he is left with \( 0.9 \times 0.5x = 0.45x \) mangoes the next morning.
   - This process repeats each day.

3. **Set Up for Three Days**:
   - After Day 1: The seller has \( 0.45x \) mangoes left.
   - After Day 2: He sells half again, so he has \( 0.5 \times 0.45x = 0.225x \) left. Then, 10% spoil overnight, leaving \( 0.9 \times 0.225x = 0.2025x \) for the next morning.
   - After Day 3: He sells half, resulting in \( 0.5 \times 0.2025x = 0.10125x \), and after 10% spoilage, he is left with \( 0.9 \times 0.10125x = 0.091125x \) the next day.

4. **Total Spoiled Mangoes Calculation**:
   - Spoiled mangoes each night are as follows:
     - First night: \( 0.05x \)
     - Second night: \( 0.0225x \)
     - Third night: \( 0.010125x \)
   - Sum of spoiled mangoes: 
     \[
     0.05x + 0.0225x + 0.010125x = 0.082625x
     \]
5. **Equate Spoiled Mangoes to 1983**:
   - Given that 1983 mangoes rotted over the three nights, we can set up the equation:
     \[
     0.082625x = 1983
     \]
   - Solving for \( x \):
     \[
     x = \frac{1983}{0.082625} \approx 24000
     \]

Therefore, the seller started with **24,000 mangoes** on the first day.

Would you like additional details on any part of the calculation?

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Here are 5 related questions to explore this topic further:

1. How would the initial number change if the spoilage rate was 15% instead of 10%?
2. What would the remaining stock be after each day without spoilage?
3. How many mangoes would be left if he continued for a fourth day under the same conditions?
4. If he sold only 40% of his stock each day, what initial stock would match the given spoilage?
5. How does compounding spoilage overnight impact the overall loss?

**Tip**: Compounding percentages over multiple periods can be calculated using the formula for exponential decay.

Solve for the initial stock of mangoes given daily sales of half the stock and 10% spoilage overnight. Learn to set up decay formulas and use algebra to find that 1983 mangoes rotted over three nights. Perfect for Grades 10-12 students exploring exponential decay and percentage calculations.

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