In a group of 50 people, two tests were conducted, one for diabetes and one for blood pressure. 30 people were diagnosed with diabetes and 40 people were diagnosed with high blood pressure. What is the minimum number of people who were having diabetes and high blood pressure?

Options

A. 0
B. 10
C. 20
D. 30

Answer

To solve the problem, we need to find the minimum number of people who have both diabetes and high blood pressure. We can use the principle of inclusion-exclusion in set theory.

### Step-by-Step Solution:

1. **Define the Sets**:
- Let \( A \) be the set of people diagnosed with diabetes.
- Let \( B \) be the set of people diagnosed with high blood pressure.

2. **Identify the Sizes of the Sets**:
- The number of people in set \( A \) (diabetes) is \( |A| = 30 \).
- The number of people in set \( B \) (high blood pressure) is \( |B| = 40 \).

3. **Total Number of People**:
- The total number of people in the group is \( |A \cup B| = 50 \).

4. **Use the Inclusion-Exclusion Principle**:
- According to the principle, the total number of people in either set \( A \) or set \( B \) can be expressed as:
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
- Where \( |A \cap B| \) is the number of people who have both conditions (diabetes and high blood pressure).

5. **Substitute the Known Values**:
- We know:
\[ 50 = 30 + 40 - |A \cap B| \]

6. **Simplify the Equation**:
- Rearranging the equation gives:
\[ 50 = 70 - |A \cap B| \]
- Therefore:
\[ |A \cap B| = 70 - 50 = 20 \]

7. **Conclusion**:
- The minimum number of people who have both diabetes and high blood pressure is \( |A \cap B| = 20 \).

### Final Answer:
The minimum number of people who were having both diabetes and high blood pressure is **20**.