Which of the following fractions `(3)/(8), (4)/(5), (31)/(40), (9)/(20)` and ` (7)/(10)` is greater than `(1)/(2)` and less than `(3)/(4)` ?

Options

A. `4/5`
B. `(7)/(10)`
C. `(31)/(40)`
D. `(9)/(20)`

Answer

To solve the problem of finding which of the fractions \( \frac{3}{8}, \frac{4}{5}, \frac{31}{40}, \frac{9}{20}, \frac{7}{10} \) is greater than \( \frac{1}{2} \) and less than \( \frac{3}{4} \), we can follow these steps:

### Step 1: Identify the fractions to compare
We have the following fractions to compare:
- \( \frac{3}{8} \)
- \( \frac{4}{5} \)
- \( \frac{31}{40} \)
- \( \frac{9}{20} \)
- \( \frac{7}{10} \)

### Step 2: Convert \( \frac{1}{2} \) and \( \frac{3}{4} \) to a common denominator
To compare these fractions effectively, we will convert \( \frac{1}{2} \) and \( \frac{3}{4} \) to a common denominator. The least common multiple (LCM) of the denominators of the given fractions (8, 5, 40, 20, 10) is 40.

- Convert \( \frac{1}{2} \):
\[ \frac{1}{2} = \frac{1 \times 20}{2 \times 20} = \frac{20}{40} \]

- Convert \( \frac{3}{4} \):
\[ \frac{3}{4} = \frac{3 \times 10}{4 \times 10} = \frac{30}{40} \]

### Step 3: Convert the other fractions to the common denominator of 40
Now we will convert each of the fractions to have a denominator of 40:

- \( \frac{3}{8} \):
\[ \frac{3}{8} = \frac{3 \times 5}{8 \times 5} = \frac{15}{40} \]

- \( \frac{4}{5} \):
\[ \frac{4}{5} = \frac{4 \times 8}{5 \times 8} = \frac{32}{40} \]

- \( \frac{31}{40} \):
\[ \frac{31}{40} \text{ (already has a denominator of 40)} \]

- \( \frac{9}{20} \):
\[ \frac{9}{20} = \frac{9 \times 2}{20 \times 2} = \frac{18}{40} \]

- \( \frac{7}{10} \):
\[ \frac{7}{10} = \frac{7 \times 4}{10 \times 4} = \frac{28}{40} \]

### Step 4: Compare the fractions
Now we have the following fractions with a common denominator of 40:
- \( \frac{3}{8} = \frac{15}{40} \)
- \( \frac{4}{5} = \frac{32}{40} \)
- \( \frac{31}{40} \)
- \( \frac{9}{20} = \frac{18}{40} \)
- \( \frac{7}{10} = \frac{28}{40} \)

We need to find which of these fractions is greater than \( \frac{20}{40} \) and less than \( \frac{30}{40} \).

### Step 5: Identify the valid fraction
- \( \frac{15}{40} < \frac{20}{40} \) (not valid)
- \( \frac{18}{40} > \frac{20}{40} \) and \( < \frac{30}{40} \) (valid)
- \( \frac{28}{40} > \frac{20}{40} \) and \( < \frac{30}{40} \) (valid)
- \( \frac{31}{40} > \frac{30}{40} \) (not valid)
- \( \frac{32}{40} > \frac{30}{40} \) (not valid)

### Conclusion
The fractions that are greater than \( \frac{1}{2} \) and less than \( \frac{3}{4} \) are:
- \( \frac{9}{20} \)
- \( \frac{7}{10} \)

### Final Answer
Both \( \frac{9}{20} \) and \( \frac{7}{10} \) are valid answers, but since the question asks for which of the options is valid, the answer is:
- \( \frac{7}{10} \)