The sum of the digits of a two-digit number is $9$. The number obtained by interchanging the digits exceeds the original number by $45$. Find the original number.

Ankita travels $14 \,km$ to her home partly by rickshaw and partly by bus. She takes half an $hour$ if she travels $2 \,km$ by rickshaw and the remaining distance by bus. On the other hand,if she travels $4 \,km$ by rickshaw and the remaining distance by bus,she takes $9 \,minutes$ longer. Find the speed of the rickshaw and of the bus (in $km/h$).

If in a rectangle,the length is increased and breadth reduced each by $2$ units,the area is reduced by $28$ square units. If,however,the length is reduced by $1$ unit and the breadth increased by $2$ units,the area increases by $33$ square units. Find the area of the rectangle.

The digit at the ten's place is $4$ in a two-digit number,and the product of both the digits is four times greater than the digit at the ten's place. Find the number.

The solution set of a pair of equations $x+y-1=0$ and $2x+2y-2=0$ is $\ldots \ldots \ldots \ldots$

Solve the following pair of equations by the method of cross-multiplication:
$x + 2y + 1 = 0$
$2x - 3y - 12 = 0$