Arrange 3/4,9/13,12/17 and 1/2 in the ascending order.

Options

A. 1/2,9/13,3/4,12/17
B. 3/4,9/13,1/2,12/17
C. 1/2,3/4,9/13,12/17
D. 1/2,9/13,12/17,3/4

Answer

To arrange the fractions \( \frac{3}{4}, \frac{9}{13}, \frac{12}{17}, \) and \( \frac{1}{2} \) in ascending order, we can follow these steps:

### Step 1: Identify the fractions
The fractions we need to arrange are:
- \( \frac{3}{4} \)
- \( \frac{9}{13} \)
- \( \frac{12}{17} \)
- \( \frac{1}{2} \)

### Step 2: Find the least common multiple (LCM) of the denominators
The denominators are \( 4, 13, 17, \) and \( 2 \).
To find the LCM:
- The prime factorization of \( 4 = 2^2 \)
- \( 13 \) is prime
- \( 17 \) is prime
- The prime factorization of \( 2 = 2^1 \)

The LCM is \( 2^2 \times 13 \times 17 = 4 \times 13 \times 17 \).

### Step 3: Calculate the LCM
Calculating \( 4 \times 13 = 52 \) and then \( 52 \times 17 = 884 \).
Thus, the LCM of \( 4, 13, 17, \) and \( 2 \) is \( 884 \).

### Step 4: Convert each fraction to have the same denominator
Now we convert each fraction to have a denominator of \( 884 \):

1. For \( \frac{3}{4} \):
\[ \frac{3}{4} = \frac{3 \times 221}{4 \times 221} = \frac{663}{884} \]

2. For \( \frac{9}{13} \):
\[ \frac{9}{13} = \frac{9 \times 68}{13 \times 68} = \frac{612}{884} \]

3. For \( \frac{12}{17} \):
\[ \frac{12}{17} = \frac{12 \times 52}{17 \times 52} = \frac{624}{884} \]

4. For \( \frac{1}{2} \):
\[ \frac{1}{2} = \frac{1 \times 442}{2 \times 442} = \frac{442}{884} \]

### Step 5: List the converted fractions
Now we have:
- \( \frac{663}{884} \) (from \( \frac{3}{4} \))
- \( \frac{612}{884} \) (from \( \frac{9}{13} \))
- \( \frac{624}{884} \) (from \( \frac{12}{17} \))
- \( \frac{442}{884} \) (from \( \frac{1}{2} \))

### Step 6: Compare the numerators
Now we can compare the numerators:
- \( 442 \) (from \( \frac{1}{2} \))
- \( 612 \) (from \( \frac{9}{13} \))
- \( 624 \) (from \( \frac{12}{17} \))
- \( 663 \) (from \( \frac{3}{4} \))

### Step 7: Arrange in ascending order
The order from smallest to largest is:
1. \( \frac{1}{2} \)
2. \( \frac{9}{13} \)
3. \( \frac{12}{17} \)
4. \( \frac{3}{4} \)

### Final Answer
Thus, the fractions in ascending order are:
\[ \frac{1}{2}, \frac{9}{13}, \frac{12}{17}, \frac{3}{4} \]