To compare the two fractions \(\frac{5}{7}\) and \(\frac{7}{9}\), we can find a common denominator. The least common multiple (LCM) of the denominators 7 and 9 will serve as our common denominator.
The multiples of 7 are: 7, 14, 21, 28, 35, 42, and so on. The multiples of 9 are: 9, 18, 27, 36, 45, 54, and so on.
The LCM of 7 and 9 is 63.
Now we will convert both fractions to have this common denominator:
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For \(\frac{5}{7}\): \[ \frac{5}{7} \times \frac{9}{9} = \frac{5 \cdot 9}{7 \cdot 9} = \frac{45}{63} \]
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For \(\frac{7}{9}\): \[ \frac{7}{9} \times \frac{7}{7} = \frac{7 \cdot 7}{9 \cdot 7} = \frac{49}{63} \]
Now, we can compare \(\frac{45}{63}\) and \(\frac{49}{63}\). Since the denominators are the same, we only need to compare the numerators:
- \(45 < 49\)
Thus, \(\frac{45}{63} < \frac{49}{63}\), which means:
\[ \frac{5}{7} < \frac{7}{9} \]
Therefore, \(\frac{7}{9}\) is the larger fraction. The comparison result is:
\(\frac{7}{9}\) > \(\frac{5}{7}\)