A motorboat travels $30 \mathrm{km}$ upstream and $28 \mathrm{km}$ downstream in $7$ hours. It travels $21 \mathrm{km}$ upstream and return in $5$hours.

What is the speed of the stream?

In the picture below two cables supporting an electrical pole from either side of the pole, so that points where cables were fixed on the earth and foot of the pole are in a straight line.

The cable $AC$ and $AB$ are given by equation of straight lines, $12x-5y=0$ and $3x+y 27=0$.  Find the ratio of distances from $AO$ to that of $BO$.

The air traffic controller (ATC) is monitoring the flight path of two flights, flying on the same height. The position coordinates of flight $P$ at two different points of time was $A (23 \mathrm{km} \mathrm{E}, 32 \mathrm{km} \mathrm{N})$ and $B (55 \mathrm{km} \mathrm{E}, 43 \mathrm{km} \mathrm{N})$. The position coordinates of flight $Q$ was $C (26 \mathrm{km} \mathrm{E}, 72 \mathrm{km} \mathrm{N})$ and $D (58 \mathrm{km} \mathrm{E}, 83 \mathrm{km} \mathrm{N})$.

The pilots of the respective flights were told not to deviate from the current straight flight path without confirmation from the ATC, till further notice. Show that the flights will not collide in the event that either or both flights loose contact with the ATC.

Ramya, playing a game of treasure hunt through a maze whose sketch is given below. She knows that a straight line connects location $A$ with the treasure location $D$, and worked out line equation connecting these two locations. Then based on her current coordinates she worked out equations of straight line, to check if her current path is aligned with the planned path.

The equations were $3x-2y+1=0$, $4x-\left(\frac{8}{3}\right)y+\left(\frac{4}{3}\right) = 0$. Ramya got confused looking two different equations and thought she lost her way. Is it true?