`4(1)/(2)+(1div2(8)/(9))-3(1)/(13)=?`

Options

A. `1(9)/(26)`
B. `2(7)/(13)`
C. `1(11)/(26)`
D. `1(10)/(13)`

Explanation

To solve the expression \( 4 \frac{1}{2} + \frac{1}{2} \frac{8}{9} - 3 \frac{1}{13} \), we will break it down step by step.

### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert \( 4 \frac{1}{2} \) to an improper fraction:
\[ 4 \frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{8 + 1}{2} = \frac{9}{2} \]

2. Convert \( 3 \frac{1}{13} \) to an improper fraction:
\[ 3 \frac{1}{13} = \frac{3 \times 13 + 1}{13} = \frac{39 + 1}{13} = \frac{40}{13} \]

### Step 2: Convert the Second Term
3. The term \( \frac{1}{2} \frac{8}{9} \) can be simplified:
\[ \frac{1}{2} \cdot \frac{8}{9} = \frac{8}{18} = \frac{4}{9} \]

### Step 3: Rewrite the Expression
4. Now, rewrite the entire expression with improper fractions:
\[ \frac{9}{2} + \frac{4}{9} - \frac{40}{13} \]

### Step 4: Find the Least Common Multiple (LCM)
5. The denominators are 2, 9, and 13. The LCM of these numbers is:
\[ LCM(2, 9, 13) = 234 \]

### Step 5: Convert Each Fraction to Have the Same Denominator
6. Convert \( \frac{9}{2} \):
\[ \frac{9}{2} = \frac{9 \times 117}{2 \times 117} = \frac{1053}{234} \]

7. Convert \( \frac{4}{9} \):
\[ \frac{4}{9} = \frac{4 \times 26}{9 \times 26} = \frac{104}{234} \]

8. Convert \( \frac{40}{13} \):
\[ \frac{40}{13} = \frac{40 \times 18}{13 \times 18} = \frac{720}{234} \]

### Step 6: Combine the Fractions
9. Now combine the fractions:
\[ \frac{1053}{234} + \frac{104}{234} - \frac{720}{234} = \frac{1053 + 104 - 720}{234} = \frac{437}{234} \]

### Step 7: Simplify the Result
10. The fraction \( \frac{437}{234} \) cannot be simplified further. However, we can convert it to a mixed number:
\[ 437 \div 234 = 1 \quad \text{(with a remainder of 203)} \]
Thus,
\[ \frac{437}{234} = 1 \frac{203}{234} \]

### Final Answer
The final answer is:
\[ 1 \frac{203}{234} \]