Convert `(3)/(5),(7)/(10),(8)/(15),and (11)/(30)` into like fractions.

Answer

To convert the fractions \( \frac{3}{5}, \frac{7}{10}, \frac{8}{15}, \) and \( \frac{11}{30} \) into like fractions, we need to find a common denominator. Here are the steps to do that:

### Step 1: Identify the denominators
The denominators of the given fractions are:
- \( 5 \)
- \( 10 \)
- \( 15 \)
- \( 30 \)

### Step 2: Find the Least Common Multiple (LCM)
To convert these fractions into like fractions, we need to find the LCM of the denominators \( 5, 10, 15, \) and \( 30 \).

**Finding the LCM:**
- The prime factorization of each number:
- \( 5 = 5^1 \)
- \( 10 = 2^1 \times 5^1 \)
- \( 15 = 3^1 \times 5^1 \)
- \( 30 = 2^1 \times 3^1 \times 5^1 \)

- The LCM is found by taking the highest power of each prime factor:
- For \( 2 \): \( 2^1 \)
- For \( 3 \): \( 3^1 \)
- For \( 5 \): \( 5^1 \)

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

### Step 3: Convert each fraction to have the common denominator
Now that we have the common denominator \( 30 \), we will convert each fraction:

1. **Convert \( \frac{3}{5} \)**:
- Multiply both the numerator and denominator by \( 6 \) (because \( 5 \times 6 = 30 \)):
\[ \frac{3 \times 6}{5 \times 6} = \frac{18}{30} \]

2. **Convert \( \frac{7}{10} \)**:
- Multiply both the numerator and denominator by \( 3 \) (because \( 10 \times 3 = 30 \)):
\[ \frac{7 \times 3}{10 \times 3} = \frac{21}{30} \]

3. **Convert \( \frac{8}{15} \)**:
- Multiply both the numerator and denominator by \( 2 \) (because \( 15 \times 2 = 30 \)):
\[ \frac{8 \times 2}{15 \times 2} = \frac{16}{30} \]

4. **Convert \( \frac{11}{30} \)**:
- The denominator is already \( 30 \), so we keep it as:
\[ \frac{11}{30} \]

### Step 4: Write the like fractions
Now we can write all the fractions with a common denominator:
- \( \frac{3}{5} = \frac{18}{30} \)
- \( \frac{7}{10} = \frac{21}{30} \)
- \( \frac{8}{15} = \frac{16}{30} \)
- \( \frac{11}{30} = \frac{11}{30} \)

Thus, the like fractions are:
\[ \frac{18}{30}, \frac{21}{30}, \frac{16}{30}, \frac{11}{30} \]