Question

Which pair of fractions are equivalent to 2/9
6/27, 12/45
6/27, 12/54
20/27, 12/19
5/27, 12/54

Answer 1

To determine which pairs of fractions are equivalent to \( \frac{2}{9} \), we'll compare each fraction in the pairs to \( \frac{2}{9} \).

A fraction \( \frac{a}{b} \) is equivalent to \( \frac{2}{9} \) if:

\[
a \cdot 9 = b \cdot 2
\]

Let's check each pair:

1. **Pair: \( \frac{6}{27} \) and \( \frac{12}{45} \)**

- For \( \frac{6}{27} \):
\[
6 \cdot 9 = 54 \quad \text{and} \quad 27 \cdot 2 = 54 \quad \Rightarrow \quad \frac{6}{27} \text{ is equivalent to } \frac{2}{9}.
\]

- For \( \frac{12}{45} \):
\[
12 \cdot 9 = 108 \quad \text{and} \quad 45 \cdot 2 = 90 \quad \Rightarrow \quad \frac{12}{45} \text{ is NOT equivalent to } \frac{2}{9}.
\]

2. **Pair: \( \frac{6}{27} \) and \( \frac{12}{54} \)**

- We already determined \( \frac{6}{27} \) is equivalent to \( \frac{2}{9} \).
- For \( \frac{12}{54} \):
\[
12 \cdot 9 = 108 \quad \text{and} \quad 54 \cdot 2 = 108 \quad \Rightarrow \quad \frac{12}{54} \text{ is equivalent to } \frac{2}{9}.
\]

3. **Pair: \( \frac{20}{27} \) and \( \frac{12}{19} \)**

- For \( \frac{20}{27} \):
\[
20 \cdot 9 = 180 \quad \text{and} \quad 27 \cdot 2 = 54 \quad \Rightarrow \quad \frac{20}{27} \text{ is NOT equivalent to } \frac{2}{9}.
\]

- For \( \frac{12}{19} \):
\[
12 \cdot 9 = 108 \quad \text{and} \quad 19 \cdot 2 = 38 \quad \Rightarrow \quad \frac{12}{19} \text{ is NOT equivalent to } \frac{2}{9}.
\]

4. **Pair: \( \frac{5}{27} \) and \( \frac{12}{54} \)**

- For \( \frac{5}{27} \):
\[
5 \cdot 9 = 45 \quad \text{and} \quad 27 \cdot 2 = 54 \quad \Rightarrow \quad \frac{5}{27} \text{ is NOT equivalent to } \frac{2}{9}.
\]

- For \( \frac{12}{54} \):
- We already established that \( \frac{12}{54} \text{ is equivalent to } \frac{2}{9} \).

In conclusion, the only pair of fractions that are equivalent to \( \frac{2}{9} \) is:

**\( \frac{6}{27} \) and \( \frac{12}{54} \)**.