The greatest fraction among `(2)/(3),(5)/(6),(11)/(15) and (7)/(8)`

Options

A. `(7)/(8)`
B. `(11)/(15)`
C. `(5)/(6)`
D. `(2)/(3)`

Answer

To find the greatest fraction among \( \frac{2}{3}, \frac{5}{6}, \frac{11}{15}, \) and \( \frac{7}{8} \), we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM) of the Denominators
The denominators are 3, 6, 15, and 8. We need to find the LCM of these numbers.

- **Prime factorization:**
- \( 3 = 3^1 \)
- \( 6 = 2^1 \times 3^1 \)
- \( 15 = 3^1 \times 5^1 \)
- \( 8 = 2^3 \)

- **Taking the highest power of each prime:**
- From \( 2 \): \( 2^3 \)
- From \( 3 \): \( 3^1 \)
- From \( 5 \): \( 5^1 \)

Thus, the LCM is:
\[ LCM = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \]

### Step 2: Convert Each Fraction to Have the Same Denominator
Now we will convert each fraction to have a denominator of 120.

1. **For \( \frac{2}{3} \):**
\[ \frac{2}{3} = \frac{2 \times 40}{3 \times 40} = \frac{80}{120} \]

2. **For \( \frac{5}{6} \):**
\[ \frac{5}{6} = \frac{5 \times 20}{6 \times 20} = \frac{100}{120} \]

3. **For \( \frac{11}{15} \):**
\[ \frac{11}{15} = \frac{11 \times 8}{15 \times 8} = \frac{88}{120} \]

4. **For \( \frac{7}{8} \):**
\[ \frac{7}{8} = \frac{7 \times 15}{8 \times 15} = \frac{105}{120} \]

### Step 3: Compare the Numerators
Now we compare the numerators of the fractions with the common denominator of 120:
- \( \frac{80}{120} \)
- \( \frac{100}{120} \)
- \( \frac{88}{120} \)
- \( \frac{105}{120} \)

The numerators are 80, 100, 88, and 105. The greatest numerator is 105.

### Step 4: Identify the Greatest Fraction
Since \( \frac{105}{120} \) corresponds to \( \frac{7}{8} \), we conclude that the greatest fraction among the given options is:
\[ \frac{7}{8} \]

### Final Answer
The greatest fraction among \( \frac{2}{3}, \frac{5}{6}, \frac{11}{15}, \) and \( \frac{7}{8} \) is \( \frac{7}{8} \).

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