Arrange the following fractions in ascending order:
`5/(6), 7/(8), 11/(12), 3/(10)`

Answer

To arrange the fractions \( \frac{5}{6}, \frac{7}{8}, \frac{11}{12}, \frac{3}{10} \) in ascending order, we will follow these steps:

### Step 1: Find the LCM of the Denominators
The denominators of the fractions are 6, 8, 12, and 10. We need to find the Least Common Multiple (LCM) of these numbers.

- The prime factorization of the denominators is:
- \( 6 = 2 \times 3 \)
- \( 8 = 2^3 \)
- \( 12 = 2^2 \times 3 \)
- \( 10 = 2 \times 5 \)

To find the LCM, we take the highest power of each prime number:
- For 2: \( 2^3 \) (from 8)
- For 3: \( 3^1 \) (from 6 or 12)
- For 5: \( 5^1 \) (from 10)

Thus, the LCM is:
\[ LCM = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \]

### Step 2: Convert Each Fraction to Have the Same Denominator
Now we will convert each fraction to have a denominator of 120.

1. For \( \frac{5}{6} \):
\[ \frac{5}{6} = \frac{5 \times 20}{6 \times 20} = \frac{100}{120} \]

2. For \( \frac{7}{8} \):
\[ \frac{7}{8} = \frac{7 \times 15}{8 \times 15} = \frac{105}{120} \]

3. For \( \frac{11}{12} \):
\[ \frac{11}{12} = \frac{11 \times 10}{12 \times 10} = \frac{110}{120} \]

4. For \( \frac{3}{10} \):
\[ \frac{3}{10} = \frac{3 \times 12}{10 \times 12} = \frac{36}{120} \]

### Step 3: Compare the Numerators
Now that all fractions have the same denominator, we can compare their numerators:
- \( 36 \) (from \( \frac{3}{10} \))
- \( 100 \) (from \( \frac{5}{6} \))
- \( 105 \) (from \( \frac{7}{8} \))
- \( 110 \) (from \( \frac{11}{12} \))

### Step 4: Arrange in Ascending Order
The order of the numerators from smallest to largest is:
- \( 36 < 100 < 105 < 110 \)

Thus, the fractions in ascending order are:
\[ \frac{3}{10}, \frac{5}{6}, \frac{7}{8}, \frac{11}{12} \]

### Final Result
The fractions arranged in ascending order are:
\[ \frac{3}{10}, \frac{5}{6}, \frac{7}{8}, \frac{11}{12} \]

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