The length of a rectangle exceeds its breadth by $7 \mathrm{cm}$. If the length is decreased by $4 \mathrm{cm}$ and the breadth is increased by $3 \mathrm{cm}$, the area of the new rectangle is the same as the area of the original rectangle. Find the length and the breadth of the original rectangle.

The width of a rectangle is two-thirds its length. If the perimeter is $180$ $\mathrm{metres}$, find the dimensions of the rectangle.

An altitude of a triangle is five-thirds the length of its corresponding base. If the altitude be increased by $4 \mathrm{cm}$ and the base decreased by $2 \mathrm{cm}$, the area of the triangle remains the same. Find the base and the altitude of the triangle.

Two angles of a triangle are in the ratio $4:5$. If the sum of these angles is equal to the third angle, find the angles of the triangle.

A steamer goes downstream from one port to another in $9$ hours. He covers the same distance upstream in $10$ hours. If the speed of the stream is $1$ $\mathrm{km}/\mathrm{h}$, find the speed of the steamer in still water and the distance between the ports.

The distance between two stations is $300 \mathrm{km}$. Two motorcyclists start simultaneously from these stations and move towards each other. The speed of one of them is $7 \mathrm{km}/\mathrm{h}$ more than that of the other. If the distance between them after $2$ hours of their start is $34 \mathrm{km}$, find the speed of each motorcyclist. Check your solution.

Divide $150$ into three parts such that the second number is five-sixths the first and the third number is four-fifths the second.

Divide $4500$ into two parts such that $5\%$ of the first part is equal to $10\%$ of the second part.

Rakhi's mother is four times as old as Rakhi. After$5$years, her mother will be three times as old as she will be then. Find their present ages.