A basket contains three types of fruits weighing `19(1)/(3)kg` in all. If `8(1)/(9)kg` of these be apples, `3(1)/(6)kg` be oranges and the rest pears, what is the weight of the pears in the basket?

Answer

To find the weight of pears in the basket, we will follow these steps:

### Step 1: Convert mixed fractions to improper fractions
1. **Weight of apples**: \(8 \frac{1}{9} = \frac{8 \times 9 + 1}{9} = \frac{72 + 1}{9} = \frac{73}{9} \text{ kg}\)
2. **Weight of oranges**: \(3 \frac{1}{6} = \frac{3 \times 6 + 1}{6} = \frac{18 + 1}{6} = \frac{19}{6} \text{ kg}\)
3. **Total weight of fruits**: \(19 \frac{1}{3} = \frac{19 \times 3 + 1}{3} = \frac{57 + 1}{3} = \frac{58}{3} \text{ kg}\)

### Step 2: Set up the equation
Let the weight of pears be \(x\) kg. According to the problem, the total weight of the fruits can be expressed as:
\[ \text{Weight of apples} + \text{Weight of oranges} + \text{Weight of pears} = \text{Total weight} \]
This can be written as:
\[ \frac{73}{9} + \frac{19}{6} + x = \frac{58}{3} \]

### Step 3: Find a common denominator
The denominators are 9, 6, and 3. The least common multiple (LCM) of these numbers is 18. We will convert each fraction to have a denominator of 18:
- Convert \(\frac{73}{9}\):
\[ \frac{73}{9} = \frac{73 \times 2}{9 \times 2} = \frac{146}{18} \]
- Convert \(\frac{19}{6}\):
\[ \frac{19}{6} = \frac{19 \times 3}{6 \times 3} = \frac{57}{18} \]
- Convert \(\frac{58}{3}\):
\[ \frac{58}{3} = \frac{58 \times 6}{3 \times 6} = \frac{348}{18} \]

### Step 4: Substitute back into the equation
Now we substitute these values into our equation:
\[ \frac{146}{18} + \frac{57}{18} + x = \frac{348}{18} \]

### Step 5: Combine the fractions
Combine the fractions on the left:
\[ \frac{146 + 57}{18} + x = \frac{348}{18} \]
This simplifies to:
\[ \frac{203}{18} + x = \frac{348}{18} \]

### Step 6: Solve for \(x\)
To isolate \(x\), subtract \(\frac{203}{18}\) from both sides:
\[ x = \frac{348}{18} - \frac{203}{18} = \frac{348 - 203}{18} = \frac{145}{18} \]

### Step 7: Convert back to a mixed fraction
To convert \(\frac{145}{18}\) back to a mixed fraction:
- Divide 145 by 18, which gives 8 with a remainder of 1.
Thus,
\[ \frac{145}{18} = 8 \frac{1}{18} \text{ kg} \]

### Final Answer
The weight of the pears in the basket is \(8 \frac{1}{18} \text{ kg}\).

---