Given that the sum of two - negative quanties is 200 , the probability that their product is not less than `3/4` times their greatest product value is :

Options

A. a. `99/200`
B. b. `100/200`
C. c. `87/100`
D. d. none of these

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Problem**: We are given two negative quantities \( A \) and \( B \) such that their sum is 200. We need to find the probability that their product \( AB \) is not less than \( \frac{3}{4} \) times their greatest product value.

2. **Expressing the Relationship**: Since \( A + B = 200 \), we can express \( B \) in terms of \( A \):
\[ B = 200 - A \]

3. **Finding the Greatest Product**: The product \( AB \) is maximized when \( A \) and \( B \) are equal. Therefore, we set:
\[ A = B \]
From \( A + B = 200 \), we have:
\[ 2A = 200 \implies A = 100 \quad \text{and} \quad B = 100 \]
The maximum product \( AB \) when both are equal is:
\[ AB = 100 \times 100 = 10,000 \]

4. **Finding the Threshold Product**: We need to find the condition where the product \( AB \) is not less than \( \frac{3}{4} \) of the maximum product:
\[ AB \geq \frac{3}{4} \times 10,000 = 7,500 \]

5. **Setting Up the Inequality**: We substitute \( B \) into the product:
\[ A(200 - A) \geq 7,500 \]
This simplifies to:
\[ 200A - A^2 \geq 7,500 \]
Rearranging gives us:
\[ A^2 - 200A + 7,500 \leq 0 \]

6. **Finding the Roots of the Quadratic**: We can solve the quadratic equation \( A^2 - 200A + 7,500 = 0 \) using the quadratic formula:
\[ A = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{200 \pm \sqrt{200^2 - 4 \cdot 1 \cdot 7,500}}{2 \cdot 1} \]
\[ = \frac{200 \pm \sqrt{40,000 - 30,000}}{2} = \frac{200 \pm \sqrt{10,000}}{2} = \frac{200 \pm 100}{2} \]
This gives us:
\[ A = \frac{300}{2} = 150 \quad \text{and} \quad A = \frac{100}{2} = 50 \]

7. **Identifying the Range for A**: The product \( AB \geq 7,500 \) holds for:
\[ 50 \leq A \leq 150 \]

8. **Calculating the Favorable Outcomes**: The range of values for \( A \) that satisfy the condition is from 50 to 150, which is a total of:
\[ 150 - 50 = 100 \text{ favorable outcomes} \]

9. **Calculating the Total Outcomes**: The total possible values for \( A \) range from 0 to 200, which gives us:
\[ 200 \text{ total outcomes} \]

10. **Finding the Probability**: The probability that \( AB \) is not less than \( \frac{3}{4} \) times the maximum product is given by:
\[ P = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{100}{200} = \frac{1}{2} \]

### Final Answer:
The probability that their product is not less than \( \frac{3}{4} \) times their greatest product value is \( \frac{1}{2} \).