If `(2)/(3), (23)/(30), (9)/(10), (11)/(15)` and `(4)/(5)` are written in ascending order, then the fraction in the middle most will be

Options

A. `(23)/(30)`
B. `(4)/(5)`
C. `(2)/(3)`
D. `(11)/(15)`

Answer

To solve the problem of finding the middle fraction when the fractions \( \frac{2}{3}, \frac{23}{30}, \frac{9}{10}, \frac{11}{15}, \) and \( \frac{4}{5} \) are arranged in ascending order, we will follow these steps:

### Step 1: Identify the fractions
The fractions given are:
- \( \frac{2}{3} \)
- \( \frac{23}{30} \)
- \( \frac{9}{10} \)
- \( \frac{11}{15} \)
- \( \frac{4}{5} \)

### Step 2: Find the Least Common Multiple (LCM) of the denominators
The denominators are \( 3, 30, 10, 15, \) and \( 5 \). We need to find the LCM of these numbers.

- The prime factorization of the denominators is:
- \( 3 = 3^1 \)
- \( 30 = 2^1 \times 3^1 \times 5^1 \)
- \( 10 = 2^1 \times 5^1 \)
- \( 15 = 3^1 \times 5^1 \)
- \( 5 = 5^1 \)

The LCM is obtained by taking the highest power of each prime:
- \( 2^1 \) from 30 and 10
- \( 3^1 \) from 3, 30, and 15
- \( 5^1 \) from 30, 10, 15, and 5

Thus, \( \text{LCM} = 2^1 \times 3^1 \times 5^1 = 30 \).

### Step 3: Convert each fraction to have a denominator of 30
Now, we will convert each fraction to have a common denominator of 30:

1. \( \frac{2}{3} = \frac{2 \times 10}{3 \times 10} = \frac{20}{30} \)
2. \( \frac{23}{30} = \frac{23}{30} \) (already has a denominator of 30)
3. \( \frac{9}{10} = \frac{9 \times 3}{10 \times 3} = \frac{27}{30} \)
4. \( \frac{11}{15} = \frac{11 \times 2}{15 \times 2} = \frac{22}{30} \)
5. \( \frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30} \)

### Step 4: List the converted fractions
The converted fractions are:
- \( \frac{20}{30} \)
- \( \frac{23}{30} \)
- \( \frac{27}{30} \)
- \( \frac{22}{30} \)
- \( \frac{24}{30} \)

### Step 5: Arrange the fractions in ascending order
Now, we will arrange these fractions in ascending order:
1. \( \frac{20}{30} \)
2. \( \frac{22}{30} \)
3. \( \frac{23}{30} \)
4. \( \frac{24}{30} \)
5. \( \frac{27}{30} \)

### Step 6: Identify the middle fraction
Since there are 5 fractions, the middle fraction is the 3rd one in the ordered list.

Thus, the middle fraction is:
- \( \frac{23}{30} \)

### Final Answer
The fraction in the middle most will be \( \frac{23}{30} \).

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