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

### Concept & Logic This is a comparative volume problem solvable through algebraic substitution or the LCM ratio method. By equating the total volume of the bucket in terms of both large and small bottles, we can establish a direct volume relationship (ratio) between one large bottle and one small bottle. ### Step-by-Step Solution * **Given:** 1 Bucket = 4 Large Bottles ($L$) 1 Bucket = 7 Small Bottles ($S$) Therefore, $4L = 7S$. * **Calculation:** Express the volume of a Large bottle in terms of a Small bottle: $$ L = \frac{7}{4}S $$ * A full large bottle ($L$) is poured into an empty small bottle until the small bottle is full (volume $S$ removed). Find the remaining volume in the large bottle: $$ \text{Remaining Volume} = L - S $$ $$ \text{Remaining Volume} = \frac{7}{4}S - S = \frac{3}{4}S $$ * Calculate what fraction this remaining fluid is relative to the original large bottle's capacity ($L$): $$ \text{Fraction Left} = \frac{\text{Remaining Volume}}{\text{Original Volume } L} $$ $$ \text{Fraction Left} = \frac{\frac{3}{4}S}{\frac{7}{4}S} $$ $$ \text{Fraction Left} = \frac{3}{4} \times \frac{4}{7} = \frac{3}{7} $$ ### Exam Strategy & Shortcut Use the **LCM Value Assumption Method**. Let the capacity of the bucket be the LCM of 4 and 7, which is 28 units. Capacity of 1 Large bottle = $\frac{28}{4} = 7$ units. Capacity of 1 Small bottle = $\frac{28}{7} = 4$ units. If you pour from the Large (7 units) to fill the Small (4 units), you are left with $7 - 4 = 3$ units in the Large bottle. The fraction left over in the large bottle is simply $\frac{\text{Leftover}}{\text{Total Large}} = \frac{3}{7}$. This avoids fractions entirely! ### Common Pitfall A common misstep is calculating the fraction relative to the small bottle or the whole bucket, rather than the initial volume of the *large bottle*. The question asks: "What fraction of the fluid is left over *in the large bottle*". The denominator must be the large bottle's capacity. ### Final Answer Therefore, the correct answer is **$\frac{3}{7}$**.