To determine if the ratios \( \frac{82}{24} \) and \( \frac{41}{12} \) are equivalent, we can simplify each ratio and compare their values.
First, let's simplify \( \frac{82}{24} \).
We can simplify by finding the greatest common divisor (GCD) of the numerator and the denominator.
The GCD of 82 and 24 can be determined by prime factorization:
- \( 82 = 2 \times 41 \)
- \( 24 = 2^3 \times 3 \)
The only common factor is \( 2 \). Therefore, we divide both by \( 2 \):
\[ \frac{82 \div 2}{24 \div 2} = \frac{41}{12} \]
Next, we look at the second ratio \( \frac{41}{12} \).
Since we have already simplified \( \frac{82}{24} \) to \( \frac{41}{12} \), the ratios clearly match:
\[ \frac{82}{24} = \frac{41}{12} \]
Thus, both ratios are equivalent. Therefore, we conclude:
\[ \text{Yes, the ratios are equivalent.} \]
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