Find `7` rational numbers between `(5)/(6) "and" (7)/(9)`.

Answer

To find 7 rational numbers between \( \frac{5}{6} \) and \( \frac{7}{9} \), we can follow these steps:

### Step 1: Find a common denominator
The denominators of the two fractions are 6 and 9. The least common multiple (LCM) of 6 and 9 is 18. We will convert both fractions to have this common denominator.

### Step 2: Convert \( \frac{5}{6} \) to the common denominator
To convert \( \frac{5}{6} \) to a fraction with a denominator of 18:
\[ \frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18} \]

### Step 3: Convert \( \frac{7}{9} \) to the common denominator
To convert \( \frac{7}{9} \) to a fraction with a denominator of 18:
\[ \frac{7}{9} = \frac{7 \times 2}{9 \times 2} = \frac{14}{18} \]

### Step 4: Identify the range between the two fractions
Now we have:
\[ \frac{15}{18} \text{ and } \frac{14}{18} \]
However, we notice that \( \frac{15}{18} \) is greater than \( \frac{14}{18} \). So we need to correct our fractions:
\[ \frac{14}{18} < \frac{15}{18} \]

### Step 5: Find rational numbers between \( \frac{14}{18} \) and \( \frac{15}{18} \)
To find rational numbers between \( \frac{14}{18} \) and \( \frac{15}{18} \), we can use a larger denominator. Let's multiply both the numerator and denominator by 10 to get more fractions.

### Step 6: Multiply by 10
\[ \frac{14}{18} = \frac{14 \times 10}{18 \times 10} = \frac{140}{180} \]
\[ \frac{15}{18} = \frac{15 \times 10}{18 \times 10} = \frac{150}{180} \]

### Step 7: List the rational numbers
Now we can find rational numbers between \( \frac{140}{180} \) and \( \frac{150}{180} \). The rational numbers can be:
- \( \frac{141}{180} \)
- \( \frac{142}{180} \)
- \( \frac{143}{180} \)
- \( \frac{144}{180} \)
- \( \frac{145}{180} \)
- \( \frac{146}{180} \)
- \( \frac{147}{180} \)

### Conclusion
The 7 rational numbers between \( \frac{5}{6} \) and \( \frac{7}{9} \) are:
- \( \frac{141}{180} \)
- \( \frac{142}{180} \)
- \( \frac{143}{180} \)
- \( \frac{144}{180} \)
- \( \frac{145}{180} \)
- \( \frac{146}{180} \)
- \( \frac{147}{180} \)