`5(1)/(5)+2(2)/(15)+3(2)/(3)=?`

Options

A. 15
B. 13
C. `(11)/(15)`
D. None of these

Explanation

To solve the expression \( 5\frac{1}{5} + 2\frac{2}{15} + 3\frac{2}{3} \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert \( 5\frac{1}{5} \):
\[ 5\frac{1}{5} = 5 + \frac{1}{5} = \frac{25}{5} + \frac{1}{5} = \frac{26}{5} \]

2. Convert \( 2\frac{2}{15} \):
\[ 2\frac{2}{15} = 2 + \frac{2}{15} = \frac{30}{15} + \frac{2}{15} = \frac{32}{15} \]

3. Convert \( 3\frac{2}{3} \):
\[ 3\frac{2}{3} = 3 + \frac{2}{3} = \frac{9}{3} + \frac{2}{3} = \frac{11}{3} \]

### Step 2: Write the Expression with Improper Fractions
Now, we can rewrite the original expression:
\[ \frac{26}{5} + \frac{32}{15} + \frac{11}{3} \]

### Step 3: Find the Least Common Multiple (LCM)
To add these fractions, we need to find the LCM of the denominators \( 5, 15, \) and \( 3 \). The LCM is \( 15 \).

### Step 4: Convert Each Fraction to Have the Same Denominator
1. Convert \( \frac{26}{5} \) to a denominator of \( 15 \):
\[ \frac{26}{5} = \frac{26 \times 3}{5 \times 3} = \frac{78}{15} \]

2. \( \frac{32}{15} \) already has a denominator of \( 15 \):
\[ \frac{32}{15} = \frac{32}{15} \]

3. Convert \( \frac{11}{3} \) to a denominator of \( 15 \):
\[ \frac{11}{3} = \frac{11 \times 5}{3 \times 5} = \frac{55}{15} \]

### Step 5: Add the Fractions
Now we can add all the fractions:
\[ \frac{78}{15} + \frac{32}{15} + \frac{55}{15} = \frac{78 + 32 + 55}{15} = \frac{165}{15} \]

### Step 6: Simplify the Result
Now, simplify \( \frac{165}{15} \):
\[ \frac{165}{15} = 11 \]

### Final Answer
Thus, the value of the expression \( 5\frac{1}{5} + 2\frac{2}{15} + 3\frac{2}{3} \) is:
\[ \boxed{11} \]