Two numbers are in the ratio $5:6$. If $8$ is subtracted from each of the numbers,the ratio becomes $4:5$. Find the numbers.

Pathik travels $600 \, km$ to his home partly by train and partly by car. He takes $8$ hours if he travels $120 \, km$ by train and the rest by car. He takes $20$ minutes longer if he travels $200 \, km$ by train and the rest by car. Find the speed of the train and the car.

Formulate the following problems as a pair of equations, and hence find their solutions:
Ritu can row downstream $20\, km$ in $2\, hours,$ and upstream $4\, km$ in $2\, hours.$ Find her speed of rowing in still water and the speed of the current.

In a competitive examination,one mark is awarded for each correct answer while $\frac{1}{2}$ mark is deducted for every wrong answer. Jayanti answered $120$ questions and got $90$ marks. How many questions did she answer correctly?

$A$ boat goes $32 \, km$ upstream and $36 \, km$ downstream in $7$ hours. In $9$ hours,it can go $40 \, km$ upstream and $48 \, km$ downstream. Determine the speed of the stream and that of the boat in still water. (Speed of boat in still water is more than the speed of the stream of river.)

Are the following pair of linear equations consistent? Justify your answer.
$2ax + by = a$
$4ax + 2by - 2a = 0$; $a, b \neq 0$