Put these fractions in order from least to greatest : `(2)/(3).(3)/(13),(5)/(7),(2)/(9)`

Answer

To order the fractions \( \frac{2}{3}, \frac{3}{13}, \frac{5}{7}, \frac{2}{9} \) from least to greatest, we will follow these steps:

### Step 1: Identify the fractions
The fractions we need to compare are:
- \( \frac{2}{3} \)
- \( \frac{3}{13} \)
- \( \frac{5}{7} \)
- \( \frac{2}{9} \)

### Step 2: Find the least common multiple (LCM) of the denominators
The denominators are 3, 13, 7, and 9. We need to find the LCM of these numbers.

- The prime factorization of each number is:
- \( 3 = 3^1 \)
- \( 13 = 13^1 \)
- \( 7 = 7^1 \)
- \( 9 = 3^2 \)

To find the LCM, we take the highest power of each prime:
- Highest power of 3: \( 3^2 \)
- Highest power of 7: \( 7^1 \)
- Highest power of 13: \( 13^1 \)

Thus, the LCM is:
\[ LCM = 3^2 \times 7^1 \times 13^1 = 9 \times 7 \times 13 = 819 \]

### Step 3: Convert each fraction to have the same denominator
Now we will convert each fraction to have a denominator of 819.

1. For \( \frac{2}{3} \):
\[ \frac{2}{3} = \frac{2 \times 273}{3 \times 273} = \frac{546}{819} \]

2. For \( \frac{3}{13} \):
\[ \frac{3}{13} = \frac{3 \times 63}{13 \times 63} = \frac{189}{819} \]

3. For \( \frac{5}{7} \):
\[ \frac{5}{7} = \frac{5 \times 117}{7 \times 117} = \frac{585}{819} \]

4. For \( \frac{2}{9} \):
\[ \frac{2}{9} = \frac{2 \times 91}{9 \times 91} = \frac{182}{819} \]

### Step 4: Compare the converted fractions
Now we have the fractions:
- \( \frac{546}{819} \)
- \( \frac{189}{819} \)
- \( \frac{585}{819} \)
- \( \frac{182}{819} \)

### Step 5: Order the fractions
Now we can compare the numerators since they all have the same denominator:
- \( 182 < 189 < 546 < 585 \)

Thus, the order from least to greatest is:
- \( \frac{2}{9} < \frac{3}{13} < \frac{2}{3} < \frac{5}{7} \)

### Final Answer
The fractions in order from least to greatest are:
\[ \frac{2}{9}, \frac{3}{13}, \frac{2}{3}, \frac{5}{7} \]