Find the sum `2(4)/(5)+1(3)/(10)+3(1)/(15)`

Answer

To find the sum of the fractions \(2 \frac{4}{5} + 1 \frac{3}{10} + 3 \frac{1}{15}\), we will follow these steps:

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

1. For \(2 \frac{4}{5}\):
\[ 2 \frac{4}{5} = \frac{2 \times 5 + 4}{5} = \frac{10 + 4}{5} = \frac{14}{5} \]

2. For \(1 \frac{3}{10}\):
\[ 1 \frac{3}{10} = \frac{1 \times 10 + 3}{10} = \frac{10 + 3}{10} = \frac{13}{10} \]

3. For \(3 \frac{1}{15}\):
\[ 3 \frac{1}{15} = \frac{3 \times 15 + 1}{15} = \frac{45 + 1}{15} = \frac{46}{15} \]

### Step 2: Find the Least Common Multiple (LCM)
Next, we need to find the LCM of the denominators \(5\), \(10\), and \(15\).

- The multiples of \(5\) are \(5, 10, 15, 20, 25, 30, \ldots\)
- The multiples of \(10\) are \(10, 20, 30, 40, \ldots\)
- The multiples of \(15\) are \(15, 30, 45, \ldots\)

The smallest common multiple is \(30\).

### Step 3: Convert Each Fraction to Have a Common Denominator
Now we will convert each fraction to have the common denominator of \(30\).

1. For \(\frac{14}{5}\):
\[ \frac{14}{5} = \frac{14 \times 6}{5 \times 6} = \frac{84}{30} \]

2. For \(\frac{13}{10}\):
\[ \frac{13}{10} = \frac{13 \times 3}{10 \times 3} = \frac{39}{30} \]

3. For \(\frac{46}{15}\):
\[ \frac{46}{15} = \frac{46 \times 2}{15 \times 2} = \frac{92}{30} \]

### Step 4: Add the Fractions
Now we can add the fractions:
\[ \frac{84}{30} + \frac{39}{30} + \frac{92}{30} = \frac{84 + 39 + 92}{30} = \frac{215}{30} \]

### Step 5: Simplify the Result
Now we will simplify \(\frac{215}{30}\).

Both \(215\) and \(30\) can be divided by \(5\):
\[ \frac{215 \div 5}{30 \div 5} = \frac{43}{6} \]

### Step 6: Convert to Mixed Number
Finally, we can convert \(\frac{43}{6}\) to a mixed number:
\[ 43 \div 6 = 7 \quad \text{(remainder } 1\text{)} \]
So, \(\frac{43}{6} = 7 \frac{1}{6}\).

### Final Answer
The sum \(2 \frac{4}{5} + 1 \frac{3}{10} + 3 \frac{1}{15}\) is:
\[ \boxed{7 \frac{1}{6}} \]