Which of the following fractions is the greatest ?

Options

A. `(5)/(14)`
B. `(29)/(77)`
C. `(25)/(66)`
D. `(8)/(21)`

Answer

To determine which of the given fractions is the greatest, we will follow these steps:

### Step 1: Identify the Fractions
The fractions given are:
1. \( \frac{5}{14} \)
2. \( \frac{29}{77} \)
3. \( \frac{25}{66} \)
4. \( \frac{8}{21} \)

### Step 2: Find the Least Common Multiple (LCM) of the Denominators
We need to find the LCM of the denominators: 14, 77, 66, and 21.

- **Prime Factorization**:
- \( 14 = 2 \times 7 \)
- \( 77 = 7 \times 11 \)
- \( 66 = 2 \times 3 \times 11 \)
- \( 21 = 3 \times 7 \)

- **LCM Calculation**:
- The LCM is found by taking the highest power of each prime factor:
- \( 2^1 \) from 14 and 66
- \( 3^1 \) from 66 and 21
- \( 7^1 \) from 14, 77, and 21
- \( 11^1 \) from 77 and 66
- Therefore, \( \text{LCM} = 2^1 \times 3^1 \times 7^1 \times 11^1 = 462 \)

### Step 3: Convert Each Fraction to Have the Same Denominator
Now we will convert each fraction to have a denominator of 462.

1. For \( \frac{5}{14} \):
- \( \frac{5}{14} = \frac{5 \times 33}{14 \times 33} = \frac{165}{462} \)

2. For \( \frac{29}{77} \):
- \( \frac{29}{77} = \frac{29 \times 6}{77 \times 6} = \frac{174}{462} \)

3. For \( \frac{25}{66} \):
- \( \frac{25}{66} = \frac{25 \times 7}{66 \times 7} = \frac{175}{462} \)

4. For \( \frac{8}{21} \):
- \( \frac{8}{21} = \frac{8 \times 22}{21 \times 22} = \frac{176}{462} \)

### Step 4: Compare the Numerators
Now we compare the numerators of the fractions with the common denominator of 462:
- \( 165 \) (from \( \frac{5}{14} \))
- \( 174 \) (from \( \frac{29}{77} \))
- \( 175 \) (from \( \frac{25}{66} \))
- \( 176 \) (from \( \frac{8}{21} \))

### Step 5: Identify the Greatest Fraction
The greatest numerator is \( 176 \), which corresponds to the fraction \( \frac{8}{21} \).

### Conclusion
Thus, the greatest fraction among the given options is:
\[ \frac{8}{21} \]