The sum of a two-digit number and the number obtained by interchanging the digits is $154$. If the ten's digit is $2$ more than unit's digit, find the original number.

The sum of the digits of a two-digit number is $8$. If the digits are interchanged, new number is $10$ more than the double the original number. Find the original number.

A two-digit number becomes $\frac{5}{6}$ of the reversed number when the digits are interchanged. The difference between the digits is $1$. Find the number.

$ABC$ is an equilateral triangle. If $AB=2x-1, BC=y+7, AC=2y+3$. Find $x$ and $y$.

If the length of a rectangle is increased by $12 cm$ and the width is decreased by $8 cm$, the area is unchanged. If the original length is increased by $5 cm$ and the original width is decreased by $4 cm$, also the area remains the same. Find the original dimensions.

In $△ABC, \angle B=90°, AB=20 cm, BC=a cm$ and $AC=b cm$. If $a+b=40 cm$, find the values of $a$ and $b$.

The length of a rectangle is greater than $4$ times its breadth by $5 cm$. If the length is reduced by $2 cm$ and the breadth is increased by $2 cm$, then the area of the rectangle increases by $36 c{m}^2$. Find the length and breadth of the original rectangle.

The area of a rectangle decreases by $10 c{m}^2$ if the length is decreased by $5 cm$ and the breadth is increased by $3 cm$.If the length is increased by $5 cm$ and the breadth is increased by $2 cm$, then the area increases by $80 c{m}^2$. Find the perimeter of the rectangle.

Find the fraction which becomes $\frac{1}{2}$ when the denominator is increased by $1$ and is equal to $\frac{2}{3}$ when both the numerator and denominator are increased by $4$.