Compare `5/6` and `7/9`

Answer

To compare the fractions \( \frac{5}{6} \) and \( \frac{7}{9} \), we can follow these steps:

### Step 1: Find the Least Common Multiple (LCM) of the Denominators
The denominators are 6 and 9. We need to find the LCM of these two numbers.

- The prime factorization of 6 is \( 2 \times 3 \).
- The prime factorization of 9 is \( 3^2 \).

To find the LCM, we take the highest power of each prime factor:
- For 2, the highest power is \( 2^1 \).
- For 3, the highest power is \( 3^2 \).

Thus, the LCM is:
\[ LCM = 2^1 \times 3^2 = 2 \times 9 = 18 \]

### Step 2: Convert Each Fraction to Have the Same Denominator
Now that we have the LCM, we will convert both fractions to have a denominator of 18.

- For \( \frac{5}{6} \):
- To convert 6 to 18, we multiply by 3.
- Therefore, we also multiply the numerator by 3:
\[ \frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18} \]

- For \( \frac{7}{9} \):
- To convert 9 to 18, we multiply by 2.
- Therefore, we also multiply the numerator by 2:
\[ \frac{7}{9} = \frac{7 \times 2}{9 \times 2} = \frac{14}{18} \]

### Step 3: Compare the Two Fractions
Now we can compare \( \frac{15}{18} \) and \( \frac{14}{18} \):
- Since both fractions have the same denominator, we can compare the numerators directly:
\[ 15 > 14 \]

### Conclusion
Thus, we conclude that:
\[ \frac{5}{6} > \frac{7}{9} \]

### Final Answer
The final answer is:
\[ \frac{5}{6} \text{ is greater than } \frac{7}{9}. \]

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