1. A butcher cuts a sausage into pieces. First, he cuts off a third of the whole sausage. The two resulting pieces he also cuts: from the larger piece, he cuts off a fifth, and from the smaller piece, he cuts off a quarter. What is the length of the smallest piece of the sausage if the length of the largest piece is $48 \mathrm{~cm}$?
First solution.

After the first cut, we have two parts of the sausage:

![](https://cdn.mathpix.com/cropped/2024_05_30_2e894aa57f8bdfbd97cfg-13.jpg?height=105&width=665&top_left_y=1164&top_left_x=250)
$\frac{1}{3}$
$\frac{1}{3}$

After the second cut, we have four parts of the sausage:

![](https://cdn.mathpix.com/cropped/2024_05_30_2e894aa57f8bdfbd97cfg-13.jpg?height=137&width=522&top_left_y=1553&top_left_x=276)

$\frac{4}{5} \cdot \frac{2}{3}$

![](https://cdn.mathpix.com/cropped/2024_05_30_2e894aa57f8bdfbd97cfg-13.jpg?height=69&width=149&top_left_y=1619&top_left_x=868)

$\frac{1}{5} \cdot \frac{2}{3}$

![](https://cdn.mathpix.com/cropped/2024_05_30_2e894aa57f8bdfbd97cfg-13.jpg?height=72&width=260&top_left_y=1620&top_left_x=1201)

$\frac{3}{4} \cdot \frac{1}{3} \quad \frac{1}{4} \cdot \frac{1}{3}$

4 POINTS (1 POINT for each part)

The largest part constitutes $\frac{4}{5} \cdot \frac{2}{3} = \frac{8}{15}$ of the entire sausage and has a length of $48 \mathrm{~cm}$.

1 POINT

The length of the entire sausage is $48 \div \frac{8}{15} = 48 \cdot \frac{15}{8} = 90 \mathrm{~cm}$.

2 POINTS

The two smallest parts of the sausage constitute $\frac{1}{5} \cdot \frac{2}{3} = \frac{2}{15}$ of the sausage and $\frac{1}{4} \cdot \frac{1}{3} = \frac{1}{12}$ of the sausage.

It is true that $\frac{2}{15} > \frac{1}{12}$ because $2 \cdot 12 > 15 \cdot 1$.

2 POINTS

The smallest obtained part has a length of $\frac{1}{12} \cdot 90 = \frac{30}{4} = \frac{15}{2} = 7.5 \mathrm{~cm}$.

1 POINT