Fill in the boxes the correct symbol out of `gt,lt and =.`
`(-4)/(5) square (-5)/(7)`

Answer

To solve the problem of comparing the two rational numbers \(-\frac{4}{5}\) and \(-\frac{5}{7}\), we will follow these steps:

### Step 1: Identify the fractions
We have two fractions to compare:
- \(-\frac{4}{5}\)
- \(-\frac{5}{7}\)

### Step 2: Find the LCM of the denominators
The denominators of the two fractions are 5 and 7. To compare the fractions easily, we need a common denominator. The least common multiple (LCM) of 5 and 7 is:
\[ \text{LCM}(5, 7) = 5 \times 7 = 35 \]

### Step 3: Convert both fractions to have the same denominator
Now we will convert both fractions to have the denominator of 35.

For \(-\frac{4}{5}\):
- We multiply both the numerator and the denominator by 7:
\[ -\frac{4}{5} = -\frac{4 \times 7}{5 \times 7} = -\frac{28}{35} \]

For \(-\frac{5}{7}\):
- We multiply both the numerator and the denominator by 5:
\[ -\frac{5}{7} = -\frac{5 \times 5}{7 \times 5} = -\frac{25}{35} \]

### Step 4: Compare the two fractions
Now we have:
- \(-\frac{28}{35}\)
- \(-\frac{25}{35}\)

Since both fractions have the same denominator, we can compare the numerators:
- \(-28\) and \(-25\)

### Step 5: Determine the relationship
Since \(-28\) is less than \(-25\), we can conclude:
\[ -\frac{28}{35} < -\frac{25}{35} \]
This implies:
\[ -\frac{4}{5} < -\frac{5}{7} \]

### Final Answer
Thus, the correct symbol to fill in the box is:
\[ -\frac{4}{5} \, \text{(lt)} \, -\frac{5}{7} \]

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