Arrange `(7)/(12), ( 2)/(3)` and `(3)/(8)` in the ascending order.

Options

A. `(3)/(8) lt ( 7 )/( 12) lt ( 2)/( 3)`
B. `( 2)/( 3) lt ( 7 )/( 12) lt ( 3)/( 8) `
C. `( 7)/(12) lt ( 2)/( 3) lt ( 3)/( 8 )`
D. `(3)/(8) lt ( 2)/(3) lt ( 7)/( 12)`

Answer

To arrange the fractions \( \frac{7}{12}, \frac{2}{3}, \) and \( \frac{3}{8} \) in ascending order, we need to compare them. The best way to compare fractions is to convert them to have a common denominator.

### Step-by-Step Solution:

1. **Identify the denominators**: The denominators of the fractions are 12, 3, and 8.

2. **Find the Least Common Multiple (LCM)**: We need to find the LCM of the denominators (12, 3, and 8).
- The prime factorization of 12 is \( 2^2 \times 3^1 \).
- The prime factorization of 3 is \( 3^1 \).
- The prime factorization of 8 is \( 2^3 \).
- The LCM is obtained by taking the highest power of each prime: \( 2^3 \) (from 8) and \( 3^1 \) (from 12) gives us \( 8 \times 3 = 24 \).

3. **Convert each fraction to have the common denominator of 24**:
- For \( \frac{7}{12} \):
\[ \frac{7}{12} = \frac{7 \times 2}{12 \times 2} = \frac{14}{24} \]
- For \( \frac{2}{3} \):
\[ \frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24} \]
- For \( \frac{3}{8} \):
\[ \frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24} \]

4. **Compare the numerators**: Now we have:
- \( \frac{14}{24} \)
- \( \frac{16}{24} \)
- \( \frac{9}{24} \)

The numerators are 14, 16, and 9. The smallest numerator is 9.

5. **Arrange in ascending order**:
- The smallest fraction is \( \frac{3}{8} \) (which corresponds to \( \frac{9}{24} \)).
- Next is \( \frac{7}{12} \) (which corresponds to \( \frac{14}{24} \)).
- The largest fraction is \( \frac{2}{3} \) (which corresponds to \( \frac{16}{24} \)).

Therefore, the fractions in ascending order are:
\[ \frac{3}{8}, \frac{7}{12}, \frac{2}{3} \]

### Final Answer:
The fractions arranged in ascending order are:
\[ \frac{3}{8}, \frac{7}{12}, \frac{2}{3} \]