Which is the correct ascending order of the given number ?

Options

A. `(3)/(10), (4)/(15), (1)/(3)`
B. `(4)/(15), (1)/(3), (3)/(10)`
C. `(1)/(3), (3)/(10), (4)/(15)`
D. `(4)/(15), (3)/(10), (1)/(3)`

Answer

To determine the correct ascending order of the given fractions \( \frac{3}{10} \), \( \frac{4}{15} \), and \( \frac{1}{5} \), we will follow these steps:

### Step 1: Identify the fractions
The fractions given are:
- \( \frac{3}{10} \)
- \( \frac{4}{15} \)
- \( \frac{1}{5} \)

### Step 2: Find the Least Common Multiple (LCM) of the denominators
The denominators are 10, 15, and 5. We need to find the LCM of these numbers.

- The prime factorization of 10 is \( 2 \times 5 \).
- The prime factorization of 15 is \( 3 \times 5 \).
- The prime factorization of 5 is \( 5 \).

The LCM will be the product of the highest power of each prime:
- \( 2^1 \) from 10
- \( 3^1 \) from 15
- \( 5^1 \) from all three

Thus, the LCM is:
\[ LCM = 2^1 \times 3^1 \times 5^1 = 30 \]

### Step 3: Convert each fraction to have the same denominator (30)
Now we will convert each fraction to have a denominator of 30.

1. For \( \frac{3}{10} \):
\[ \frac{3}{10} = \frac{3 \times 3}{10 \times 3} = \frac{9}{30} \]

2. For \( \frac{4}{15} \):
\[ \frac{4}{15} = \frac{4 \times 2}{15 \times 2} = \frac{8}{30} \]

3. For \( \frac{1}{5} \):
\[ \frac{1}{5} = \frac{1 \times 6}{5 \times 6} = \frac{6}{30} \]

### Step 4: Compare the numerators
Now we have the fractions:
- \( \frac{9}{30} \)
- \( \frac{8}{30} \)
- \( \frac{6}{30} \)

We can compare the numerators since the denominators are the same:
- \( 6 < 8 < 9 \)

### Step 5: Write the fractions in ascending order
From the comparison, we can write the fractions in ascending order:
\[ \frac{1}{5} < \frac{4}{15} < \frac{3}{10} \]

### Final Answer
Thus, the correct ascending order of the given numbers is:
\[ \frac{1}{5}, \frac{4}{15}, \frac{3}{10} \]