Divide 468 g of rice into three heaps containing the quantities in the ratio `(1)/(6) : (1)/(8) : (1)/(12)`

Answer

To divide 468 grams of rice into three heaps in the ratio \( \frac{1}{6} : \frac{1}{8} : \frac{1}{12} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Ratios**:
The given ratios are \( \frac{1}{6} \), \( \frac{1}{8} \), and \( \frac{1}{12} \).

2. **Find a Common Factor**:
To work with these fractions, we can express them in terms of a common variable \( x \):
- First heap: \( \frac{1}{6}x \)
- Second heap: \( \frac{1}{8}x \)
- Third heap: \( \frac{1}{12}x \)

3. **Set Up the Equation**:
The total amount of rice is 468 grams, so we can write the equation:
\[ \frac{1}{6}x + \frac{1}{8}x + \frac{1}{12}x = 468 \]

4. **Find the Least Common Multiple (LCM)**:
The LCM of the denominators (6, 8, and 12) is 24. We will use this to combine the fractions:
\[ \frac{4}{24}x + \frac{3}{24}x + \frac{2}{24}x = 468 \]

5. **Combine the Fractions**:
Now, combine the fractions on the left side:
\[ \frac{4x + 3x + 2x}{24} = 468 \]
This simplifies to:
\[ \frac{9x}{24} = 468 \]

6. **Clear the Fraction**:
Multiply both sides by 24 to eliminate the denominator:
\[ 9x = 468 \times 24 \]

7. **Calculate \( x \)**:
Now, divide both sides by 9:
\[ x = \frac{468 \times 24}{9} \]
Simplifying the right side:
\[ x = 52 \times 24 = 1248 \]

8. **Find Each Heap**:
Now we can find the quantities for each heap:
- First heap:
\[ \frac{1}{6}x = \frac{1}{6} \times 1248 = 208 \text{ grams} \]
- Second heap:
\[ \frac{1}{8}x = \frac{1}{8} \times 1248 = 156 \text{ grams} \]
- Third heap:
\[ \frac{1}{12}x = \frac{1}{12} \times 1248 = 104 \text{ grams} \]

9. **Final Result**:
Therefore, the rice is divided into three heaps of:
- 208 grams
- 156 grams
- 104 grams

### Summary:
The three heaps contain 208 grams, 156 grams, and 104 grams of rice.