Which one of the following is the rational number lying between `6/7` and `7/8`?

Options

A. `3/4`
B. `9/122`
C. `95/112`
D. `97/112`

Answer

To find a rational number lying between \( \frac{6}{7} \) and \( \frac{7}{8} \), we can follow these steps:

### Step 1: Find the sum of the two fractions
We need to add \( \frac{6}{7} \) and \( \frac{7}{8} \).

**Hint:** When adding fractions, find a common denominator.

### Step 2: Determine the least common multiple (LCM) of the denominators
The denominators are 7 and 8. The LCM of 7 and 8 is 56.

**Hint:** The LCM is the smallest number that both denominators can divide into without leaving a remainder.

### Step 3: Convert the fractions to have the same denominator
Convert \( \frac{6}{7} \) and \( \frac{7}{8} \) to have a denominator of 56.

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

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

**Hint:** Multiply the numerator and denominator of each fraction by the same number to get equivalent fractions.

### Step 4: Add the two fractions
Now we can add the two fractions:
\[ \frac{48}{56} + \frac{49}{56} = \frac{48 + 49}{56} = \frac{97}{56} \]

**Hint:** When fractions have the same denominator, you can simply add the numerators.

### Step 5: Find the average of the two fractions
To find a rational number between \( \frac{6}{7} \) and \( \frac{7}{8} \), we can take the average:
\[ \text{Average} = \frac{1}{2} \times \frac{97}{56} = \frac{97}{112} \]

**Hint:** The average of two numbers is found by adding them and dividing by 2.

### Conclusion
The rational number lying between \( \frac{6}{7} \) and \( \frac{7}{8} \) is \( \frac{97}{112} \).

**Final Answer:** \( \frac{97}{112} \) (Option 4)