Find the sum
`2(1)/(3)+1(1)/(4)+2(5)/(6)+3(7)/(12)`

Answer

To find the sum of the fractions \(2\frac{1}{3} + 1\frac{1}{4} + 2\frac{5}{6} + 3\frac{7}{12}\), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert \(2\frac{1}{3}\) to an improper fraction:
\[ 2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3} \]

2. Convert \(1\frac{1}{4}\) to an improper fraction:
\[ 1\frac{1}{4} = \frac{1 \times 4 + 1}{4} = \frac{4 + 1}{4} = \frac{5}{4} \]

3. Convert \(2\frac{5}{6}\) to an improper fraction:
\[ 2\frac{5}{6} = \frac{2 \times 6 + 5}{6} = \frac{12 + 5}{6} = \frac{17}{6} \]

4. Convert \(3\frac{7}{12}\) to an improper fraction:
\[ 3\frac{7}{12} = \frac{3 \times 12 + 7}{12} = \frac{36 + 7}{12} = \frac{43}{12} \]

### Step 2: Write the Sum of Improper Fractions
Now we can write the sum:
\[ \frac{7}{3} + \frac{5}{4} + \frac{17}{6} + \frac{43}{12} \]

### Step 3: Find the Least Common Multiple (LCM)
To add these fractions, we need a common denominator. The denominators are \(3, 4, 6, 12\). The LCM of these numbers is \(12\).

### Step 4: Convert Each Fraction to Have a Common Denominator
1. Convert \(\frac{7}{3}\) to a denominator of \(12\):
\[ \frac{7}{3} = \frac{7 \times 4}{3 \times 4} = \frac{28}{12} \]

2. Convert \(\frac{5}{4}\) to a denominator of \(12\):
\[ \frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12} \]

3. Convert \(\frac{17}{6}\) to a denominator of \(12\):
\[ \frac{17}{6} = \frac{17 \times 2}{6 \times 2} = \frac{34}{12} \]

4. \(\frac{43}{12}\) already has a denominator of \(12\).

### Step 5: Add the Converted Fractions
Now we can add the fractions:
\[ \frac{28}{12} + \frac{15}{12} + \frac{34}{12} + \frac{43}{12} = \frac{28 + 15 + 34 + 43}{12} \]

### Step 6: Calculate the Numerator
Calculate the sum of the numerators:
\[ 28 + 15 + 34 + 43 = 120 \]

### Step 7: Write the Final Result
So, the sum is:
\[ \frac{120}{12} = 10 \]

### Final Answer
The sum of \(2\frac{1}{3} + 1\frac{1}{4} + 2\frac{5}{6} + 3\frac{7}{12}\) is \(10\).

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