`2 (4)/(5) - 1 (3)/(8) + 3 (1)/(2) = ?`

Options

A. `5 (17)/(40)`
B. `4 (37)/(40)`
C. `2 (27)/(40)`
D. `3 (11)/(40)`

Answer

To solve the expression \( 2 \frac{4}{5} - 1 \frac{3}{8} + 3 \frac{1}{2} \), we will first convert the mixed numbers into improper fractions, perform the operations, and then simplify the result.

### Step 1: Convert mixed numbers to improper fractions
1. **Convert \( 2 \frac{4}{5} \)**:
\[ 2 \frac{4}{5} = \frac{(2 \times 5) + 4}{5} = \frac{10 + 4}{5} = \frac{14}{5} \]

2. **Convert \( 1 \frac{3}{8} \)**:
\[ 1 \frac{3}{8} = \frac{(1 \times 8) + 3}{8} = \frac{8 + 3}{8} = \frac{11}{8} \]

3. **Convert \( 3 \frac{1}{2} \)**:
\[ 3 \frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2} \]

### Step 2: Rewrite the expression
Now we can rewrite the expression using the improper fractions:
\[ \frac{14}{5} - \frac{11}{8} + \frac{7}{2} \]

### Step 3: Find a common denominator
The denominators are \( 5, 8, \) and \( 2 \). The least common multiple (LCM) of these numbers is \( 40 \).

### Step 4: Convert each fraction to have the common denominator
1. **Convert \( \frac{14}{5} \)**:
\[ \frac{14}{5} = \frac{14 \times 8}{5 \times 8} = \frac{112}{40} \]

2. **Convert \( \frac{11}{8} \)**:
\[ \frac{11}{8} = \frac{11 \times 5}{8 \times 5} = \frac{55}{40} \]

3. **Convert \( \frac{7}{2} \)**:
\[ \frac{7}{2} = \frac{7 \times 20}{2 \times 20} = \frac{140}{40} \]

### Step 5: Substitute back into the expression
Now we can substitute these back into the expression:
\[ \frac{112}{40} - \frac{55}{40} + \frac{140}{40} \]

### Step 6: Combine the fractions
Combine the fractions:
\[ \frac{112 - 55 + 140}{40} = \frac{197}{40} \]

### Step 7: Simplify if necessary
The fraction \( \frac{197}{40} \) is already in its simplest form.

### Final Answer
Thus, the final answer is:
\[ \frac{197}{40} \]

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