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

Options

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

Explanation

To solve the expression \( 2\frac{4}{5} - 1\frac{3}{8} + 3\frac{1}{2} \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert each mixed number into an improper fraction.

1. \( 2\frac{4}{5} = \frac{2 \times 5 + 4}{5} = \frac{10 + 4}{5} = \frac{14}{5} \)
2. \( 1\frac{3}{8} = \frac{1 \times 8 + 3}{8} = \frac{8 + 3}{8} = \frac{11}{8} \)
3. \( 3\frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2} \)

So, we rewrite the expression as:
\[ \frac{14}{5} - \frac{11}{8} + \frac{7}{2} \]

### Step 2: Find the Least Common Multiple (LCM)
Next, we need to find the LCM of the denominators (5, 8, and 2) to combine the fractions.

The LCM of 5, 8, and 2 is 40.

### Step 3: Convert Each Fraction to Have the Same Denominator
Now, we convert each fraction to have a denominator of 40.

1. For \( \frac{14}{5} \):
\[ \frac{14}{5} = \frac{14 \times 8}{5 \times 8} = \frac{112}{40} \]

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

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

Now, we can rewrite the expression as:
\[ \frac{112}{40} - \frac{55}{40} + \frac{140}{40} \]

### Step 4: Combine the Fractions
Now that all fractions have the same denominator, we can combine them:
\[ \frac{112 - 55 + 140}{40} = \frac{112 - 55 + 140}{40} = \frac{197}{40} \]

### Step 5: Convert to Mixed Number
Finally, we can convert \( \frac{197}{40} \) into a mixed number:
\[ 197 \div 40 = 4 \quad \text{(remainder } 37\text{)} \]
Thus, \( \frac{197}{40} = 4\frac{37}{40} \).

### Final Answer
The final answer is:
\[ 4\frac{37}{40} \]

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