`3(7)/(11)+7(3)/(11)xx1(1)/(2)=?`

Options

A. `13(10)/(11)`
B. `14(6)/(11)`
C. `14(9)/(11)`
D. `10(17)/(22)`

Explanation

To solve the expression \( \frac{3 \cdot 7}{11} + \frac{7 \cdot 3}{11} \cdot 1 \frac{1}{2} \), we will break it down step by step.

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed number \( 1 \frac{1}{2} \) into an improper fraction.

\[ 1 \frac{1}{2} = \frac{2 \cdot 1 + 1}{2} = \frac{3}{2} \]

**Hint:** To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place that over the denominator.

### Step 2: Rewrite the Expression
Now, we can rewrite the original expression using the improper fraction:

\[ \frac{3 \cdot 7}{11} + \frac{7 \cdot 3}{11} \cdot \frac{3}{2} \]

### Step 3: Simplify Each Term
Next, we simplify each term.

The first term is:

\[ \frac{3 \cdot 7}{11} = \frac{21}{11} \]

The second term can be simplified as follows:

\[ \frac{7 \cdot 3}{11} \cdot \frac{3}{2} = \frac{21}{11} \cdot \frac{3}{2} = \frac{21 \cdot 3}{11 \cdot 2} = \frac{63}{22} \]

**Hint:** When multiplying fractions, multiply the numerators together and the denominators together.

### Step 4: Find a Common Denominator
To add the two fractions \( \frac{21}{11} \) and \( \frac{63}{22} \), we need a common denominator. The least common multiple of 11 and 22 is 22.

Convert \( \frac{21}{11} \) to a fraction with a denominator of 22:

\[ \frac{21}{11} = \frac{21 \cdot 2}{11 \cdot 2} = \frac{42}{22} \]

### Step 5: Add the Two Fractions
Now we can add the two fractions:

\[ \frac{42}{22} + \frac{63}{22} = \frac{42 + 63}{22} = \frac{105}{22} \]

### Step 6: Simplify the Result
The fraction \( \frac{105}{22} \) cannot be simplified further, so we can leave it as is or convert it to a mixed number if needed:

\[ \frac{105}{22} = 4 \frac{17}{22} \]

### Final Answer
Thus, the final answer is:

\[ \frac{105}{22} \text{ or } 4 \frac{17}{22} \]

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