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

Answer

To simplify the expression \((1 \frac{1}{4} + 2 \frac{3}{4}) \div (2 \frac{2}{3} + 3 \frac{1}{3})\), 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 \(1 \frac{1}{4}\):
\[ 1 \frac{1}{4} = \frac{1 \times 4 + 1}{4} = \frac{5}{4} \]

2. For \(2 \frac{3}{4}\):
\[ 2 \frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{11}{4} \]

3. For \(2 \frac{2}{3}\):
\[ 2 \frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{8}{3} \]

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

### Step 2: Rewrite the Expression
Now we can rewrite the expression with improper fractions:
\[ \left(\frac{5}{4} + \frac{11}{4}\right) \div \left(\frac{8}{3} + \frac{10}{3}\right) \]

### Step 3: Add the Fractions
Next, we add the fractions in the numerator and the denominator.

1. For the numerator:
\[ \frac{5}{4} + \frac{11}{4} = \frac{5 + 11}{4} = \frac{16}{4} = 4 \]

2. For the denominator:
\[ \frac{8}{3} + \frac{10}{3} = \frac{8 + 10}{3} = \frac{18}{3} = 6 \]

### Step 4: Rewrite the Division as Multiplication
Now we can rewrite the division:
\[ 4 \div 6 = 4 \times \frac{1}{6} = \frac{4}{6} \]

### Step 5: Simplify the Fraction
Finally, we simplify \(\frac{4}{6}\):
\[ \frac{4}{6} = \frac{2}{3} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{2}{3} \]

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