Find if the following fractions are equivalent:
`6/(28) and 54/(254)`

Answer

To determine if the fractions \( \frac{6}{28} \) and \( \frac{54}{254} \) are equivalent, we will simplify both fractions and compare them.

### Step 1: Simplify \( \frac{6}{28} \)

1. **Find the greatest common divisor (GCD)** of the numerator (6) and the denominator (28).
- The factors of 6 are: 1, 2, 3, 6
- The factors of 28 are: 1, 2, 4, 7, 14, 28
- The common factors are: 1, 2
- The greatest common factor is 2.

2. **Divide both the numerator and denominator by the GCD** (which is 2):
\[ \frac{6 \div 2}{28 \div 2} = \frac{3}{14} \]

### Step 2: Simplify \( \frac{54}{254} \)

1. **Find the GCD** of the numerator (54) and the denominator (254).
- The factors of 54 are: 1, 2, 3, 6, 9, 18, 27, 54
- The factors of 254 are: 1, 2, 127, 254
- The common factors are: 1, 2
- The greatest common factor is 2.

2. **Divide both the numerator and denominator by the GCD** (which is 2):
\[ \frac{54 \div 2}{254 \div 2} = \frac{27}{127} \]

### Step 3: Compare the simplified fractions

Now we have:
- \( \frac{6}{28} \) simplifies to \( \frac{3}{14} \)
- \( \frac{54}{254} \) simplifies to \( \frac{27}{127} \)

To check if \( \frac{3}{14} \) is equal to \( \frac{27}{127} \), we can cross-multiply:
\[ 3 \times 127 \quad \text{and} \quad 14 \times 27 \]
Calculating both:
- \( 3 \times 127 = 381 \)
- \( 14 \times 27 = 378 \)

Since \( 381 \neq 378 \), the fractions are not equivalent.

### Conclusion

Thus, \( \frac{6}{28} \) is not equivalent to \( \frac{54}{254} \).

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