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The problem in the image asks to find the original price of each share at the beginning of 2023, given the following information:

- The price of Biotech Company shares at the end of 2023 was **$41.60**.
- The price fell by **13.68% in 2024**.
- The overall fall over the two-year period (2023 and 2024) was **16.62%**.

### Let's solve it step-by-step:

#### 1. Calculate the price at the end of 2024.
Since the price fell by 13.68% in 2024, the price at the end of 2024 is:
\[
\text{Price at the end of 2024} = 41.60 \times (1 - 0.1368)
\]
\[
\text{Price at the end of 2024} = 41.60 \times 0.8632 = 35.90
\]

So, the price at the end of 2024 is **$35.90**.

#### 2. Use the overall fall to find the price at the beginning of 2023.
The overall fall over the two years is 16.62%. This means the price at the end of 2024 (i.e., $35.90) is 83.38% (or 100% - 16.62%) of the price at the beginning of 2023.

Let the price at the beginning of 2023 be $ P_{\text{beginning\ of\ 2023}} $. We can write the equation:
\[
35.90 = P_{\text{beginning\ of\ 2023}} \times (1 - 0.1662)
\]
\[
35.90 = P_{\text{beginning\ of\ 2023}} \times 0.8338
\]
Solving for $ P_{\text{beginning\ of\ 2023}} $:
\[
P_{\text{beginning\ of\ 2023}} = \frac{35.90}{0.8338} \approx 43.05
\]

So, the original price at the beginning of 2023 was approximately **$43.05**.

Would you like further clarification or details on this process?

---

Here are five related questions you might explore:

1. How would the solution change if the price fell by 15% in 2024 instead of 13.68%?
2. Can you find the percentage increase needed in 2025 to return to the original price of 2023?
3. How would the overall percentage decrease change if the price at the end of 2023 was $45 instead of $41.60?
4. What is the difference in percentage if the price fell by 16.62% in just one year instead of over two years?
5. What would be the price at the beginning of 2023 if the price at the end of 2024 was $30?

### Tip:
When calculating percentage decreases over multiple years, always apply the decrease year by year rather than summing the percentages directly.

The price of the common shares of Biotech Company was valued at $41.60 at the end of 2023, fell by 13.68% in 2024, and overall fell by 16.62% over the two-year period 2023/2024. What was the original price of each share at the beginning of 2023?

Find out how to calculate the original share price at the beginning of 2023 for Biotech Company, using percentage decrease rates over two years. Given a 13.68% drop in 2024 and an overall 16.62% fall from 2023 to 2024, solve step-by-step with detailed formulas and explanations. Suitable for high school students in Grades 10-12.

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