Simplify the following :
`1""2/3+2""1/2+3/4`

Answer

To simplify the expression \(1 \frac{2}{3} + 2 \frac{1}{2} + \frac{3}{4}\), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert \(1 \frac{2}{3}\) to an improper fraction:
\[ 1 \frac{2}{3} = \frac{3 \times 1 + 2}{3} = \frac{3 + 2}{3} = \frac{5}{3} \]

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

So, we now have:
\[ \frac{5}{3} + \frac{5}{2} + \frac{3}{4} \]

### Step 2: Find the Least Common Multiple (LCM) of the Denominators
The denominators are 3, 2, and 4. The LCM of these numbers is 12.

### Step 3: Convert Each Fraction to Have a Common Denominator
1. Convert \(\frac{5}{3}\):
\[ \frac{5}{3} = \frac{5 \times 4}{3 \times 4} = \frac{20}{12} \]

2. Convert \(\frac{5}{2}\):
\[ \frac{5}{2} = \frac{5 \times 6}{2 \times 6} = \frac{30}{12} \]

3. Convert \(\frac{3}{4}\):
\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \]

Now we can rewrite the expression as:
\[ \frac{20}{12} + \frac{30}{12} + \frac{9}{12} \]

### Step 4: Add the Fractions
Now that all fractions have the same denominator, we can add them:
\[ \frac{20 + 30 + 9}{12} = \frac{59}{12} \]

### Step 5: Convert the Improper Fraction to a Mixed Number
To convert \(\frac{59}{12}\) to a mixed number:
1. Divide 59 by 12. The quotient is 4 and the remainder is 11.
2. Thus, we can write:
\[ \frac{59}{12} = 4 \frac{11}{12} \]

### Final Answer
The simplified form of \(1 \frac{2}{3} + 2 \frac{1}{2} + \frac{3}{4}\) is:
\[ 4 \frac{11}{12} \]