Problem 1. The sum of the numerator and the denominator of a fraction is 95. After reducing the fraction, the obtained fraction is $\frac{7}{12}$. Determine that fraction.
Solution. Let the fraction be $\frac{a}{b}$. From the first condition of the problem, we have: $a+b=95$. From the second condition of the problem, we have: $\frac{a: k}{b: k}=\frac{7}{12}$, from which $a=7 k$ and $b=12 k$. If we substitute $a$ and $b$ in $a+b=95$, we get $7 k+12 k=95, k=5$. The numerator of the fraction is 35, and the denominator of the fraction is 60. Therefore, the required fraction is $\frac{35}{60}$.

Part 2. Let $P Q$ and $Q R$ be segments on a straight line such that

![](https://cdn.mathpix.com/cropped/2024_06_05_4fa80d9916883368a7a6g-3.jpg?height=55&width=1240&top_left_y=1770&top_left_x=245)
the midpoint of segment $P Q$ and $N$ is the midpoint of segment $P R$, prove that $\overline{\|}$ : $\bar{R}=2 \cdot \bar{\pi} i i i \bar{\Lambda}$

![](https://cdn.mathpix.com/cropped/2024_06_05_4fa80d9916883368a7a6g-3.jpg?height=58&width=1156&top_left_y=1939&top_left_x=287)

![](https://cdn.mathpix.com/cropped/2024_06_05_4fa80d9916883368a7a6g-3.jpg?height=53&width=608&top_left_y=1995&top_left_x=561)

![](https://cdn.mathpix.com/cropped/2024_06_05_4fa80d9916883368a7a6g-3.jpg?height=53&width=589&top_left_y=2045&top_left_x=571)

![](https://cdn.mathpix.com/cropped/2024_06_05_4fa80d9916883368a7a6g-3.jpg?height=55&width=613&top_left_y=2097&top_left_x=559)