Question
Are the ratios 6/14 and 10/23 equivalent? Why or Why not?
Answers
GPT-4o mini
To determine if the ratios \( \frac{6}{14} \) and \( \frac{10}{23} \) are equivalent, we can simplify \( \frac{6}{14} \) and compare it to \( \frac{10}{23} \).
1. **Simplifying \( \frac{6}{14} \)**:
- The greatest common divisor (GCD) of 6 and 14 is 2.
- Dividing both the numerator and the denominator by 2:
\[
\frac{6 \div 2}{14 \div 2} = \frac{3}{7}
\]
2. **Comparing \( \frac{3}{7} \) and \( \frac{10}{23} \)**:
- To compare \( \frac{3}{7} \) and \( \frac{10}{23} \), we can cross-multiply:
\[
3 \times 23 = 69
\]
\[
10 \times 7 = 70
\]
- Since \( 69 \neq 70 \), the ratios are not equivalent.
Therefore, the ratios \( \frac{6}{14} \) and \( \frac{10}{23} \) are not equivalent.
1. **Simplifying \( \frac{6}{14} \)**:
- The greatest common divisor (GCD) of 6 and 14 is 2.
- Dividing both the numerator and the denominator by 2:
\[
\frac{6 \div 2}{14 \div 2} = \frac{3}{7}
\]
2. **Comparing \( \frac{3}{7} \) and \( \frac{10}{23} \)**:
- To compare \( \frac{3}{7} \) and \( \frac{10}{23} \), we can cross-multiply:
\[
3 \times 23 = 69
\]
\[
10 \times 7 = 70
\]
- Since \( 69 \neq 70 \), the ratios are not equivalent.
Therefore, the ratios \( \frac{6}{14} \) and \( \frac{10}{23} \) are not equivalent.