Find the sum
`3(2)/(3)+ 1(5)/(6)+2`

Answer

To find the sum of the mixed fractions and whole number given in the question \(3 \frac{2}{3} + 1 \frac{5}{6} + 2\), we will follow these steps:

### Step 1: Convert Mixed Fractions to Improper Fractions
First, we need to convert the mixed fractions into improper fractions.

1. **Convert \(3 \frac{2}{3}\)**:
- Multiply the whole number (3) by the denominator (3): \(3 \times 3 = 9\)
- Add the numerator (2): \(9 + 2 = 11\)
- So, \(3 \frac{2}{3} = \frac{11}{3}\)

2. **Convert \(1 \frac{5}{6}\)**:
- Multiply the whole number (1) by the denominator (6): \(1 \times 6 = 6\)
- Add the numerator (5): \(6 + 5 = 11\)
- So, \(1 \frac{5}{6} = \frac{11}{6}\)

### Step 2: Rewrite the Whole Number as a Fraction
The whole number \(2\) can be written as a fraction:
- \(2 = \frac{2}{1}\)

### Step 3: Find a Common Denominator
Now, we need to find a common denominator for the fractions \(\frac{11}{3}\), \(\frac{11}{6}\), and \(\frac{2}{1}\).

- The denominators are \(3\), \(6\), and \(1\).
- The least common multiple (LCM) of \(3\), \(6\), and \(1\) is \(6\).

### Step 4: Convert All Fractions to Have the Common Denominator
Now we convert each fraction to have the denominator of \(6\):

1. **Convert \(\frac{11}{3}\)**:
- Multiply the numerator and denominator by \(2\):
\[ \frac{11 \times 2}{3 \times 2} = \frac{22}{6} \]

2. **Convert \(\frac{11}{6}\)**:
- This fraction already has the denominator of \(6\), so it remains \(\frac{11}{6}\).

3. **Convert \(\frac{2}{1}\)**:
- Multiply the numerator and denominator by \(6\):
\[ \frac{2 \times 6}{1 \times 6} = \frac{12}{6} \]

### Step 5: Add the Fractions
Now we can add the fractions:
\[ \frac{22}{6} + \frac{11}{6} + \frac{12}{6} = \frac{22 + 11 + 12}{6} = \frac{45}{6} \]

### Step 6: Simplify the Result
Now we simplify \(\frac{45}{6}\):
- Divide both the numerator and denominator by \(3\):
\[ \frac{45 \div 3}{6 \div 3} = \frac{15}{2} \]

### Final Answer
The sum \(3 \frac{2}{3} + 1 \frac{5}{6} + 2\) is \(\frac{15}{2}\).

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