Which of the following fractions are equivalent:
`8/(12), 16/(22)`

Answer

To determine whether the fractions \( \frac{8}{12} \) and \( \frac{16}{22} \) are equivalent, we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM) of the Denominators
The denominators are 12 and 22. We need to find the LCM of these two numbers.

- The prime factorization of 12 is \( 2^2 \times 3 \).
- The prime factorization of 22 is \( 2 \times 11 \).

To find the LCM, we take the highest power of each prime factor:
- For 2, the highest power is \( 2^2 \).
- For 3, the highest power is \( 3^1 \).
- For 11, the highest power is \( 11^1 \).

Thus, the LCM is:
\[ LCM = 2^2 \times 3^1 \times 11^1 = 4 \times 3 \times 11 = 132 \]

### Step 2: Convert Each Fraction to Have the Same Denominator
Now, we will convert both fractions to have the denominator of 132.

**For \( \frac{8}{12} \):**
To convert \( \frac{8}{12} \) to a denominator of 132:
- We find what we need to multiply 12 by to get 132:
\[ 132 \div 12 = 11 \]
- We multiply both the numerator and the denominator by 11:
\[ \frac{8 \times 11}{12 \times 11} = \frac{88}{132} \]

**For \( \frac{16}{22} \):**
To convert \( \frac{16}{22} \) to a denominator of 132:
- We find what we need to multiply 22 by to get 132:
\[ 132 \div 22 = 6 \]
- We multiply both the numerator and the denominator by 6:
\[ \frac{16 \times 6}{22 \times 6} = \frac{96}{132} \]

### Step 3: Compare the Two Converted Fractions
Now we compare the two fractions:
- \( \frac{88}{132} \) and \( \frac{96}{132} \)

Since \( 88 \neq 96 \), we conclude that:
\[ \frac{8}{12} \neq \frac{16}{22} \]

### Conclusion
Therefore, the fractions \( \frac{8}{12} \) and \( \frac{16}{22} \) are not equivalent.

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