The fluid cortained in a bucket can fill four large bottles or seven small bottles. A full large bottle is ussed to fill an empty small bottle. What fraction of the fluid is left over in the large battle when the small one is full ?

Options

A. `(2)/(7)`
B. `(3)/(7)`
C. `(4)/(7)`
D. `(5)/(7)`

Answer

To solve the problem step by step, we will follow these instructions:

1. **Define the Capacity of the Bucket**:
Let the total capacity of the bucket be \( x \) liters.

2. **Determine the Capacity of One Large Bottle**:
Since the bucket can fill 4 large bottles, the capacity of one large bottle is:
\[ \text{Capacity of one large bottle} = \frac{x}{4} \]

3. **Determine the Capacity of One Small Bottle**:
Since the bucket can fill 7 small bottles, the capacity of one small bottle is:
\[ \text{Capacity of one small bottle} = \frac{x}{7} \]

4. **Calculate the Fluid Used to Fill the Small Bottle**:
When a full large bottle is used to fill an empty small bottle, the amount of fluid transferred is equal to the capacity of one large bottle:
\[ \text{Fluid used} = \frac{x}{4} \]

5. **Calculate the Remaining Fluid in the Large Bottle**:
To find out how much fluid is left in the large bottle after filling the small bottle, we subtract the capacity of the small bottle from the capacity of the large bottle:
\[ \text{Remaining fluid} = \frac{x}{4} - \frac{x}{7} \]

6. **Find a Common Denominator**:
The least common multiple (LCM) of 4 and 7 is 28. We will convert both fractions to have a common denominator:
\[ \frac{x}{4} = \frac{7x}{28} \quad \text{and} \quad \frac{x}{7} = \frac{4x}{28} \]

7. **Subtract the Two Fractions**:
Now we can subtract the two fractions:
\[ \text{Remaining fluid} = \frac{7x}{28} - \frac{4x}{28} = \frac{3x}{28} \]

8. **Calculate the Fraction of Fluid Left in the Large Bottle**:
To find the fraction of the fluid left in the large bottle, we divide the remaining fluid by the total capacity of the large bottle:
\[ \text{Fraction left} = \frac{\text{Remaining fluid}}{\text{Capacity of one large bottle}} = \frac{\frac{3x}{28}}{\frac{x}{4}} \]

9. **Simplify the Fraction**:
When we simplify this, we have:
\[ \text{Fraction left} = \frac{3x}{28} \times \frac{4}{x} = \frac{3 \cdot 4}{28} = \frac{12}{28} = \frac{3}{7} \]

10. **Final Answer**:
Thus, the fraction of the fluid left over in the large bottle when the small one is full is:
\[ \frac{3}{7} \]