Simplify the following (1 to 3) :
`3/5" of "1""1/9+3""1/2`

Answer

To simplify the expression \( \frac{3}{5} \) of \( 1 \frac{1}{9} + 3 \frac{1}{2} \), we will follow these steps:

### Step 1: Convert mixed numbers to improper fractions
First, we need to convert the mixed numbers \( 1 \frac{1}{9} \) and \( 3 \frac{1}{2} \) into improper fractions.

- For \( 1 \frac{1}{9} \):
\[ 1 \frac{1}{9} = \frac{1 \times 9 + 1}{9} = \frac{9 + 1}{9} = \frac{10}{9} \]

- For \( 3 \frac{1}{2} \):
\[ 3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2} \]

### Step 2: Add the improper fractions
Now, we will add \( \frac{10}{9} \) and \( \frac{7}{2} \). To do this, we need a common denominator. The least common multiple (LCM) of 9 and 2 is 18.

- Convert \( \frac{10}{9} \) to have a denominator of 18:
\[ \frac{10}{9} = \frac{10 \times 2}{9 \times 2} = \frac{20}{18} \]

- Convert \( \frac{7}{2} \) to have a denominator of 18:
\[ \frac{7}{2} = \frac{7 \times 9}{2 \times 9} = \frac{63}{18} \]

Now we can add them:
\[ \frac{20}{18} + \frac{63}{18} = \frac{20 + 63}{18} = \frac{83}{18} \]

### Step 3: Multiply by \( \frac{3}{5} \)
Next, we need to multiply \( \frac{3}{5} \) by \( \frac{83}{18} \):
\[ \frac{3}{5} \times \frac{83}{18} = \frac{3 \times 83}{5 \times 18} = \frac{249}{90} \]

### Step 4: Simplify the fraction
Now we will simplify \( \frac{249}{90} \). We can find the greatest common divisor (GCD) of 249 and 90.

- The GCD of 249 and 90 is 3. Therefore, we divide both the numerator and the denominator by 3:
\[ \frac{249 \div 3}{90 \div 3} = \frac{83}{30} \]

### Step 5: Convert to mixed number (if necessary)
Now we can convert \( \frac{83}{30} \) into a mixed number:
- Divide 83 by 30:
\[ 83 \div 30 = 2 \quad \text{(remainder 23)} \]
Thus, we can write:
\[ \frac{83}{30} = 2 \frac{23}{30} \]

### Final Answer
The simplified form of \( \frac{3}{5} \) of \( 1 \frac{1}{9} + 3 \frac{1}{2} \) is:
\[ 2 \frac{23}{30} \]