$A$ fraction becomes $\frac{1}{4}$ if $2$ is subtracted from the numerator and $3$ is added to the denominator. If $6$ is added to the numerator and the denominator is multiplied by $3$,it becomes $\frac{2}{3}$. Find the fraction.

$A$ pair of linear equations which has a unique solution $x=2, y=-3$ is

$A$ two-digit number is obtained by either multiplying the sum of the digits by $8$ and adding $1$,or by multiplying the difference of the digits by $13$ and adding $2$. Find the number.

For a pair of equations $13x - 7y = 19$ and $7x - 13y = 31$,find the value of $x + y$.

Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method.
$A$ part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student $A$ takes food for $20$ days she has to pay ₹ $1000$ as hostel charges,whereas a student $B$,who takes food for $26$ days,pays ₹ $1180$ as hostel charges. Find the fixed charges and the cost of food per day.

Four years ago,the sum of the ages of four persons was $40$ years. After two years,the sum of the ages of the same four persons will be ............ years.