Compare and fill in the box with `lt , gt` or =
`18.8 + 3"" (1)/(5) + 7 "" (2)/(3) square 16.2 + 4"" (2)/(7) + 3"" (1)/(8)`

Options

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

Answer

To compare the two expressions given in the question, we will break down each expression step by step and then compare the results.

**Step 1: Simplify the first expression**

The first expression is:
\[ 18.8 + 3 \frac{1}{5} + 7 \frac{2}{3} \]

1. Convert \(18.8\) to a fraction:
\[ 18.8 = \frac{188}{10} \]

2. Convert \(3 \frac{1}{5}\) to an improper fraction:
\[ 3 \frac{1}{5} = 3 + \frac{1}{5} = \frac{15}{5} + \frac{1}{5} = \frac{16}{5} \]

3. Convert \(7 \frac{2}{3}\) to an improper fraction:
\[ 7 \frac{2}{3} = 7 + \frac{2}{3} = \frac{21}{3} + \frac{2}{3} = \frac{23}{3} \]

Now, we have:
\[ \frac{188}{10} + \frac{16}{5} + \frac{23}{3} \]

**Step 2: Find the LCM of the denominators**

The denominators are \(10\), \(5\), and \(3\). The LCM of these numbers is \(30\).

**Step 3: Convert each fraction to have a common denominator**

1. Convert \(\frac{188}{10}\):
\[ \frac{188}{10} = \frac{188 \times 3}{10 \times 3} = \frac{564}{30} \]

2. Convert \(\frac{16}{5}\):
\[ \frac{16}{5} = \frac{16 \times 6}{5 \times 6} = \frac{96}{30} \]

3. Convert \(\frac{23}{3}\):
\[ \frac{23}{3} = \frac{23 \times 10}{3 \times 10} = \frac{230}{30} \]

Now, we can add them:
\[ \frac{564}{30} + \frac{96}{30} + \frac{230}{30} = \frac{564 + 96 + 230}{30} = \frac{890}{30} \]

**Step 4: Simplify the first expression**

Now, we simplify:
\[ \frac{890}{30} = \frac{89}{3} \]

**Step 5: Simplify the second expression**

The second expression is:
\[ 16.2 + 4 \frac{2}{7} + 3 \frac{1}{8} \]

1. Convert \(16.2\) to a fraction:
\[ 16.2 = \frac{162}{10} \]

2. Convert \(4 \frac{2}{7}\) to an improper fraction:
\[ 4 \frac{2}{7} = 4 + \frac{2}{7} = \frac{28}{7} + \frac{2}{7} = \frac{30}{7} \]

3. Convert \(3 \frac{1}{8}\) to an improper fraction:
\[ 3 \frac{1}{8} = 3 + \frac{1}{8} = \frac{24}{8} + \frac{1}{8} = \frac{25}{8} \]

Now, we have:
\[ \frac{162}{10} + \frac{30}{7} + \frac{25}{8} \]

**Step 6: Find the LCM of the denominators**

The denominators are \(10\), \(7\), and \(8\). The LCM of these numbers is \(280\).

**Step 7: Convert each fraction to have a common denominator**

1. Convert \(\frac{162}{10}\):
\[ \frac{162}{10} = \frac{162 \times 28}{10 \times 28} = \frac{4536}{280} \]

2. Convert \(\frac{30}{7}\):
\[ \frac{30}{7} = \frac{30 \times 40}{7 \times 40} = \frac{1200}{280} \]

3. Convert \(\frac{25}{8}\):
\[ \frac{25}{8} = \frac{25 \times 35}{8 \times 35} = \frac{875}{280} \]

Now, we can add them:
\[ \frac{4536}{280} + \frac{1200}{280} + \frac{875}{280} = \frac{4536 + 1200 + 875}{280} = \frac{6611}{280} \]

**Step 8: Compare the two expressions**

Now we need to compare:
\[ \frac{89}{3} \quad \text{and} \quad \frac{6611}{280} \]

**Step 9: Cross-multiply to compare**

Cross-multiply:
\[ 89 \times 280 \quad \text{and} \quad 6611 \times 3 \]

Calculating:
1. \(89 \times 280 = 24920\)
2. \(6611 \times 3 = 19833\)

Since \(24920 > 19833\), we conclude:
\[ \frac{89}{3} > \frac{6611}{280} \]

**Final Answer:**
The box should be filled with \(>\).

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