The sum of the squares of two natural numbers is $116$. If the square of the larger is $25$ times the smaller, find the numbers.

The difference of squares of two natural numbers is $45$. The square of the smaller number is four times the larger number. Find the numbers.

The numerator of a fraction is one less than its denominator. If $3$ is added to each of the numerator and denominator, the fraction is increased by $\frac{3}{28}$: find the fraction. $28$

The area of a rectangular plot is $528 {\mathrm{m}}^2$. The length of the plot (in metres) is one metre more than twice its breadth. Find the length and breadth of the plot.

A rectangular garden $10 \mathrm{m}\times 16 \mathrm{m}$ is to be surrounded by a concrete walk of uniform width. If the area of walk is $120 {\mathrm{m}}^2$, find its width.