Are the following numbers in continued proportion? a. 9, 12, 16 b. 4, 11, 44

Answer

To determine if the given numbers are in continued proportion, we will follow these steps for both parts of the question.

### Part a: 9, 12, 16

1. **Identify the numbers**: The numbers we are examining are 9, 12, and 16.

2. **Set up the ratios**: For continued proportion, we need to check if the ratio of the first two numbers is equal to the ratio of the last two numbers. We can express this as:
\[ \frac{9}{12} \quad \text{and} \quad \frac{12}{16} \]

3. **Cross-multiply the ratios**: For the numbers to be in continued proportion, the following condition must hold:
\[ 9 \times 16 \quad \text{should equal} \quad 12 \times 12 \]
Let's calculate:
\[ 9 \times 16 = 144 \]
\[ 12 \times 12 = 144 \]

4. **Compare the results**: Since both products are equal:
\[ 144 = 144 \]
This means that the numbers 9, 12, and 16 are in continued proportion.

### Part b: 4, 11, 44

1. **Identify the numbers**: The numbers we are examining are 4, 11, and 44.

2. **Set up the ratios**: Again, we will check if the ratio of the first two numbers is equal to the ratio of the last two numbers:
\[ \frac{4}{11} \quad \text{and} \quad \frac{11}{44} \]

3. **Cross-multiply the ratios**: We will check the condition:
\[ 4 \times 44 \quad \text{should equal} \quad 11 \times 11 \]
Let's calculate:
\[ 4 \times 44 = 176 \]
\[ 11 \times 11 = 121 \]

4. **Compare the results**: Since the products are not equal:
\[ 176 \neq 121 \]
This means that the numbers 4, 11, and 44 are not in continued proportion.

### Final Conclusion:
- Part a: 9, 12, and 16 are in continued proportion.
- Part b: 4, 11, and 44 are not in continued proportion.