The rational number lying between `5/6` and `6/7` is

Options

A. `1/2`
B. `15/21`
C. `35/42`
D. `71/84`

Answer

To find a rational number that lies between \( \frac{5}{6} \) and \( \frac{6}{7} \), we can follow these steps:

### Step 1: Identify the two fractions
We have two fractions:
- \( \frac{5}{6} \)
- \( \frac{6}{7} \)

### Step 2: Find a common denominator
To compare these fractions easily, we need to find a common denominator. The denominators are 6 and 7. The least common multiple (LCM) of 6 and 7 is 42.

### Step 3: Convert the fractions to have the common denominator
Now, we convert both fractions to have the denominator of 42.

For \( \frac{5}{6} \):
\[ \frac{5}{6} = \frac{5 \times 7}{6 \times 7} = \frac{35}{42} \]

For \( \frac{6}{7} \):
\[ \frac{6}{7} = \frac{6 \times 6}{7 \times 6} = \frac{36}{42} \]

### Step 4: Find a rational number between the two fractions
Now that we have:
- \( \frac{5}{6} = \frac{35}{42} \)
- \( \frac{6}{7} = \frac{36}{42} \)

We can see that \( \frac{35}{42} < \frac{36}{42} \). A simple way to find a rational number between them is to take the average of these two fractions.

### Step 5: Calculate the average
To find the average:
\[ \text{Average} = \frac{\frac{35}{42} + \frac{36}{42}}{2} = \frac{71}{42} \div 2 = \frac{71}{84} \]

### Conclusion
Thus, the rational number lying between \( \frac{5}{6} \) and \( \frac{6}{7} \) is \( \frac{71}{84} \).

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