Compare and fill in the box with `lt, gt` or =.
`18.8 + 3 ( 1)/( 5) + 7 ( 3)/( 5) square 16.2 + 4 ( 7 )/( 25) + 3 ( 1)/( 8)`

Options

A. =
B. `gt`
C. `lt`
D. Can't be determined

Answer

To solve the problem, we need to compare the two expressions:

**Expression 1:** \( 18.8 + 3 \frac{1}{5} + 7 \frac{3}{5} \)

**Expression 2:** \( 16.2 + 4 \frac{7}{25} + 3 \frac{1}{8} \)

### Step 1: Convert mixed numbers to improper fractions
1. **For Expression 1:**
- \( 3 \frac{1}{5} = 3 + \frac{1}{5} = \frac{15}{5} + \frac{1}{5} = \frac{16}{5} \)
- \( 7 \frac{3}{5} = 7 + \frac{3}{5} = \frac{35}{5} + \frac{3}{5} = \frac{38}{5} \)

So, Expression 1 becomes:
\[ 18.8 + \frac{16}{5} + \frac{38}{5} \]

2. **For Expression 2:**
- \( 4 \frac{7}{25} = 4 + \frac{7}{25} = \frac{100}{25} + \frac{7}{25} = \frac{107}{25} \)
- \( 3 \frac{1}{8} = 3 + \frac{1}{8} = \frac{24}{8} + \frac{1}{8} = \frac{25}{8} \)

So, Expression 2 becomes:
\[ 16.2 + \frac{107}{25} + \frac{25}{8} \]

### Step 2: Convert decimals to fractions
- \( 18.8 = \frac{188}{10} \)
- \( 16.2 = \frac{162}{10} \)

### Step 3: Find a common denominator for the fractions
- The least common multiple (LCM) of 5, 25, and 8 is 200.

### Step 4: Convert all fractions to have a common denominator
1. **For Expression 1:**
- \( \frac{16}{5} = \frac{16 \times 40}{5 \times 40} = \frac{640}{200} \)
- \( \frac{38}{5} = \frac{38 \times 40}{5 \times 40} = \frac{1520}{200} \)

So, Expression 1 becomes:
\[ \frac{1880}{200} + \frac{640}{200} + \frac{1520}{200} = \frac{1880 + 640 + 1520}{200} = \frac{4040}{200} \]

2. **For Expression 2:**
- \( \frac{107}{25} = \frac{107 \times 8}{25 \times 8} = \frac{856}{200} \)
- \( \frac{25}{8} = \frac{25 \times 25}{8 \times 25} = \frac{625}{200} \)

So, Expression 2 becomes:
\[ \frac{1620}{200} + \frac{856}{200} + \frac{625}{200} = \frac{1620 + 856 + 625}{200} = \frac{3101}{200} \]

### Step 5: Compare the two expressions
Now we compare:
\[ \frac{4040}{200} \quad \text{and} \quad \frac{3101}{200} \]

Since \( 4040 > 3101 \), we can conclude:
\[ 18.8 + 3 \frac{1}{5} + 7 \frac{3}{5} > 16.2 + 4 \frac{7}{25} + 3 \frac{1}{8} \]

### Final Answer
Thus, we fill in the box with:
\[ \text{gt} \]

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