Which of the following is true ?

Options

A. `7/8 gt 6/7 gt 3/4`
B. `3/4 gt 6/7 gt 7/8`
C. `6/7 gt 3/4 gt 7/8`
D. `6/7 gt 7/8 gt 3/4`

Answer

To determine which of the given options is true regarding the rational numbers \( \frac{7}{8} \), \( \frac{6}{7} \), and \( \frac{3}{4} \), we will compare these fractions step by step.

### Step 1: Identify the fractions
We have three fractions:
1. \( \frac{7}{8} \)
2. \( \frac{6}{7} \)
3. \( \frac{3}{4} \)

### Step 2: Find the Least Common Multiple (LCM) of the denominators
The denominators are 8, 7, and 4. We need to find the LCM of these numbers to compare the fractions easily.
- The prime factorization is:
- \( 8 = 2^3 \)
- \( 7 = 7^1 \)
- \( 4 = 2^2 \)
- The LCM is found by taking the highest power of each prime:
- LCM = \( 2^3 \times 7^1 = 56 \)

### Step 3: Convert each fraction to have the same denominator
Now we will convert each fraction to have a denominator of 56.

1. For \( \frac{7}{8} \):
\[ \frac{7}{8} = \frac{7 \times 7}{8 \times 7} = \frac{49}{56} \]

2. For \( \frac{6}{7} \):
\[ \frac{6}{7} = \frac{6 \times 8}{7 \times 8} = \frac{48}{56} \]

3. For \( \frac{3}{4} \):
\[ \frac{3}{4} = \frac{3 \times 14}{4 \times 14} = \frac{42}{56} \]

### Step 4: Compare the fractions
Now we compare the numerators of the fractions with a common denominator:
- \( \frac{49}{56} \) (which is \( \frac{7}{8} \))
- \( \frac{48}{56} \) (which is \( \frac{6}{7} \))
- \( \frac{42}{56} \) (which is \( \frac{3}{4} \))

From this, we can conclude:
- \( 49 > 48 > 42 \)

Thus, we have:
\[ \frac{7}{8} > \frac{6}{7} > \frac{3}{4} \]

### Step 5: Identify the true statement
We need to check which option correctly represents this relationship. The correct statement is:
\[ \frac{7}{8} > \frac{6}{7} > \frac{3}{4} \]
This corresponds to option 1.

### Conclusion
The correct answer is **Option 1**.