ratios. Are they equivalent? Explain or show \[ 3: 11 \text { and 15:55 } \]

To determine whether the ratios \( 3:11 \) and \( 15:55 \) are equivalent, we can simplify both ratios and compare them. ### Step 1: Understand What It Means for Ratios to Be Equivalent Two ratios \( a:b \) and \( c:d \) are **equivalent** if the fractions \( \frac{a}{b} \) and \( \frac{c}{d} \) are equal. In other words: \[ \frac{a}{b} = \frac{c}{d} \] ### Step 2: Simplify the Second Ratio \( 15:55 \) 1. **Find the Greatest Common Divisor (GCD)** of the numerator and the denominator. For \( 15 \) and \( 55 \), the GCD is \( 5 \). 2. **Divide both terms of the ratio by the GCD** to simplify: \[ \frac{15}{55} = \frac{15 \div 5}{55 \div 5} = \frac{3}{11} \] So, \( 15:55 \) simplifies to \( 3:11 \). ### Step 3: Compare the Simplified Ratios Now, both ratios are: \[ 3:11 \quad \text{and} \quad 3:11 \] Since they are identical after simplification, the ratios are **equivalent**. ### Conclusion Yes, the ratios \( 3:11 \) and \( 15:55 \) are equivalent because both simplify to \( 3:11 \).