Compare `4 : 5` and `7 : 9`

Answer

To compare the ratios \(4 : 5\) and \(7 : 9\), we can follow these steps:

### Step 1: Convert the ratios to fractions
We convert the ratios into fractions:
- \(4 : 5\) can be written as \(\frac{4}{5}\)
- \(7 : 9\) can be written as \(\frac{7}{9}\)

### Step 2: Find the Least Common Multiple (LCM) of the denominators
The denominators are \(5\) and \(9\). We need to find the LCM of these two numbers.
- The multiples of \(5\) are \(5, 10, 15, 20, 25, 30, 35, 40, 45, \ldots\)
- The multiples of \(9\) are \(9, 18, 27, 36, 45, \ldots\)
- The smallest common multiple is \(45\).

### Step 3: Convert each fraction to have the same denominator
Now we will convert both fractions to have the denominator \(45\).

For \(\frac{4}{5}\):
- To convert \(5\) to \(45\), we multiply by \(9\) (since \(5 \times 9 = 45\)).
- Therefore, we also multiply the numerator by \(9\):
\[ \frac{4}{5} = \frac{4 \times 9}{5 \times 9} = \frac{36}{45} \]

For \(\frac{7}{9}\):
- To convert \(9\) to \(45\), we multiply by \(5\) (since \(9 \times 5 = 45\)).
- Therefore, we also multiply the numerator by \(5\):
\[ \frac{7}{9} = \frac{7 \times 5}{9 \times 5} = \frac{35}{45} \]

### Step 4: Compare the numerators
Now we have:
- \(\frac{36}{45}\) for \(4 : 5\)
- \(\frac{35}{45}\) for \(7 : 9\)

Since the denominators are the same, we can compare the numerators directly:
- \(36\) (from \(\frac{36}{45}\)) is greater than \(35\) (from \(\frac{35}{45}\)).

### Conclusion
Thus, we conclude that:
\[ 4 : 5 > 7 : 9 \]