Question

2/3,2/7,5/6
Arrange in decreasing order
Find common denominator

Answer 1

To arrange the fractions \( \frac{2}{3} \), \( \frac{2}{7} \), and \( \frac{5}{6} \) in decreasing order, we'll first find a common denominator.

The denominators are 3, 7, and 6. The least common multiple (LCM) of these numbers will be the common denominator.

1. **Find the LCM**:
- The prime factorization of the denominators:
- \( 3 = 3^1 \)
- \( 7 = 7^1 \)
- \( 6 = 2^1 \times 3^1 \)

To find the LCM, take the highest power of each prime factor:
- \( 2^1 \) from \( 6 \)
- \( 3^1 \) from \( 3 \) and \( 6 \)
- \( 7^1 \) from \( 7 \)

So, the LCM is:
\[
2^1 \times 3^1 \times 7^1 = 2 \times 3 \times 7 = 42
\]

2. **Convert each fraction to have a denominator of 42**:
- For \( \frac{2}{3} \):
\[
\frac{2}{3} = \frac{2 \times 14}{3 \times 14} = \frac{28}{42}
\]

- For \( \frac{2}{7} \):
\[
\frac{2}{7} = \frac{2 \times 6}{7 \times 6} = \frac{12}{42}
\]

- For \( \frac{5}{6} \):
\[
\frac{5}{6} = \frac{5 \times 7}{6 \times 7} = \frac{35}{42}
\]

3. **List the fractions with the common denominator**:
- \( \frac{28}{42} \)
- \( \frac{12}{42} \)
- \( \frac{35}{42} \)

4. **Arrange in decreasing order**:
- Compare the numerators:
- \( 35 > 28 > 12 \)

Thus, the order from largest to smallest is:
\[
\frac{5}{6}, \frac{2}{3}, \frac{2}{7}
\]

So, the final arrangement in decreasing order is:
\[
\frac{5}{6}, \frac{2}{3}, \frac{2}{7}
\]