A business person buys potatoes of two different varieties $P$ and $Q$, mixes them in a certain ratio and sells them at ₹192 per kg.
The cost of the variety P is $Rs\,800$ for 5 kg .
The cost of the variety Q is $Rs\, 800$ for 4 kg .
If the person gets $8 \%$ profit, what is the $\mathrm{P}: Q$ ratio (by weight)?
Three villages $P, Q$, and $R$ are located in such a way that the distance $P Q=13 \mathrm{~km}$, $Q R=14 \mathrm{~km}$, and $R P=15 \mathrm{~km}$, as shown in the figure. A straight road joins $Q$ and $R$. It is proposed to connect $P$ to this road $Q R$ by constructing another road. What is the minimum possible length (in km ) of this connecting road?
Note: The figure shown is representative.
A circle with center at $(x, y)=(0.5,0)$ and radius $=0.5$ intersects with another circle with center at $(x, \mathrm{y})=(1,1)$ and radius $=1$ at two points. One of the points of intersection $(x, \mathrm{y})$ is: