Arrange the fractions `2/5, 3/10, 9/14, 16/35` in ascending order

Answer

To arrange the fractions \( \frac{2}{5}, \frac{3}{10}, \frac{9}{14}, \frac{16}{35} \) in ascending order, we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM) of the denominators
The denominators of the fractions are 5, 10, 14, and 35. We will find the LCM of these numbers.

- The prime factorization of each number is:
- \( 5 = 5^1 \)
- \( 10 = 2^1 \times 5^1 \)
- \( 14 = 2^1 \times 7^1 \)
- \( 35 = 5^1 \times 7^1 \)

To find the LCM, we take the highest power of each prime factor:
- The highest power of 2 is \( 2^1 \)
- The highest power of 5 is \( 5^1 \)
- The highest power of 7 is \( 7^1 \)

Thus, the LCM is:
\[ LCM = 2^1 \times 5^1 \times 7^1 = 70 \]

### Step 2: Convert each fraction to have the same denominator
Now we will convert each fraction to have a denominator of 70.

1. For \( \frac{2}{5} \):
\[ \frac{2}{5} = \frac{2 \times 14}{5 \times 14} = \frac{28}{70} \]

2. For \( \frac{3}{10} \):
\[ \frac{3}{10} = \frac{3 \times 7}{10 \times 7} = \frac{21}{70} \]

3. For \( \frac{9}{14} \):
\[ \frac{9}{14} = \frac{9 \times 5}{14 \times 5} = \frac{45}{70} \]

4. For \( \frac{16}{35} \):
\[ \frac{16}{35} = \frac{16 \times 2}{35 \times 2} = \frac{32}{70} \]

### Step 3: Arrange the fractions based on their numerators
Now we have the fractions as:
- \( \frac{21}{70} \)
- \( \frac{28}{70} \)
- \( \frac{32}{70} \)
- \( \frac{45}{70} \)

Arranging these in ascending order based on their numerators gives us:
1. \( \frac{21}{70} \) (which is \( \frac{3}{10} \))
2. \( \frac{28}{70} \) (which is \( \frac{2}{5} \))
3. \( \frac{32}{70} \) (which is \( \frac{16}{35} \))
4. \( \frac{45}{70} \) (which is \( \frac{9}{14} \))

### Final Answer
Thus, the fractions in ascending order are:
\[ \frac{3}{10}, \frac{2}{5}, \frac{16}{35}, \frac{9}{14} \]

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