A wholesaler has three variants of rice. He has 138kg of Grade 1, 222kg of Grade 2 and 252kg of Grade 3 variety.
He wants to pack the rice in bags in such a way that all bags have equal amount of rice and one bag will have only one kind of rice.
If he wants to pack the maximum amount of rice in each bag.. how many bags in all will he need?

Answer

To solve the problem of how many bags the wholesaler will need for packing rice, we will follow these steps:

### Step 1: Identify the quantities of rice
The wholesaler has:
- Grade 1: 138 kg
- Grade 2: 222 kg
- Grade 3: 252 kg

### Step 2: Find the Highest Common Factor (HCF)
To pack the rice in bags with equal amounts, we need to find the HCF of the three quantities (138, 222, and 252).

### Step 3: Use the Division Method to find HCF
1. **Divide the numbers by the smallest prime number (2)**:
- 138 ÷ 2 = 69
- 222 ÷ 2 = 111
- 252 ÷ 2 = 126

So we have:
- 69, 111, 126

2. **Now, divide again by 3 (the next smallest prime number)**:
- 69 ÷ 3 = 23
- 111 ÷ 3 = 37
- 126 ÷ 3 = 42

So we have:
- 23, 37, 42

3. **Now, we check for further common factors**:
- 23 and 37 are prime numbers, which means they cannot be divided further by any common factors.

### Step 4: Calculate HCF
The HCF from the divisions we performed is the product of the common prime factors we used:
- HCF = 2 × 3 = 6

### Step 5: Determine the number of bags needed
Now that we know the maximum amount of rice in each bag is 6 kg, we can calculate the number of bags needed for each variety of rice:
- For Grade 1: 138 kg ÷ 6 kg/bag = 23 bags
- For Grade 2: 222 kg ÷ 6 kg/bag = 37 bags
- For Grade 3: 252 kg ÷ 6 kg/bag = 42 bags

### Step 6: Total number of bags
Now, we sum the number of bags for each grade:
- Total bags = 23 + 37 + 42 = 102 bags

### Final Answer
The wholesaler will need a total of **102 bags**.

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