The area of a rectangle gets reduced by $9$ square units if its length is reduced by $5$ units and breadth is increased by $3$ units. If we increase the length by $3$ units and breadth by $2$ units, the area increases by $67$ square units. Find the dimensions of the rectangle.

Open in App

Solution

Step 1: Use the given data and obtain the first equation:
Let the area of the rectangle be $A$ square units.
Let the length of the rectangle be $x$ units.
Let the breadth of the rectangle be $y$ units.
$∴$ Area of the rectangle, $A = x y$ .
It is given that the area of a rectangle gets reduced by $9$ square units if its length is reduced by $5$ units and breadth is increased by $3$ units.
The new area will be $x y - 9$ square units, if the length $= x - 5$ units, and breadth $= y + 3$ units.
Thus,
$\begin{array}{rcl} x y - 9 & = & (x - 5) (y + 3) \\ x y - 9 & = & x y + 3 x - 5 y - 15 \\ - 9 + 15 & = & 3 x - 5 y \\ 6 & = & 3 x - 5 y \\ 3 x - 5 y - 6 & = & 0 . . . . (1) \end{array}$
Step 2: Find the second equation by using another given condition:
It is given that if we increase the length by $3$ units and breadth by $2$ units, then the area increases by $67$ square units.
The new area will be $x y + 67$ square units, if the length $= x + 3$ units, and breadth $= y + 2$ units.
Thus,
$\begin{array}{rcl} x y + 67 & = & (x + 3) (y + 2) \\ x y + 67 & = & x y + 2 x + 3 y + 6 \\ 67 - 6 & = & 2 x + 3 y \\ 61 & = & 2 x + 3 y \\ 2 x + 3 y - 61 & = & 0 . . . . (2) \end{array}$
Thus, we get the following system of linear equations:
$3 x - 5 y - 6 = 0 2 x + 3 y - 61 = 0$
Step 3: Solve this system of linear equations using the cross-multiplication method:
$\begin{array}{rcl} \frac{x}{|\begin{matrix} - 5 & - 6 \\ 3 & - 61 \end{matrix}|} & = & \frac{- y}{|\begin{matrix} - 6 & 3 \\ - 61 & 2 \end{matrix}|} = \frac{1}{|\begin{matrix} 3 & - 5 \\ 2 & 3 \end{matrix}|} \\ \frac{x}{305 + 18} & = & \frac{y}{- 12 + 183} = \frac{1}{9 + 10} \\ \frac{x}{323} & = & \frac{y}{171} = \frac{1}{19} \\ x & = & \frac{323}{19}, y = \frac{171}{19} \\ x & = & 17, y = 9 \end{array}$
Hence, the dimensions of the rectangle are as follows:
$\begin{array}{rcl} Length & = & 17 units \\ Breadth & = & 9 units \end{array}$

Suggest Corrections

Similar questions

The area of a rectangle gets reduced by 9 square units. if its length is reduced by 5 units and breadth is increased by 3 units. However, if the length of this rectangle increases by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.

Let the length of the rectangle = x units and its breadth = y units

According to first condition :
$(x - 5) (y + 3) = x y - 9$ and

According to second condition:
$(x + 3) (y + 2) = x y + 67$

Q. The area of a rectangle gets reduced by

9

square units, if its length is reduced by

5

units and breadth is increased by

3

units. If we increase the length by

3

units and the breadth by

2

units, the area increases by

67

square units. Find the dimensions of the rectangle.

Q. The area of a rectangle gets reduced by

9

square units if its length is reduced by

5

units and the breadth is increased by

3

units. If we increase the length by

3

units and breadth by

2

units, the area is increased by

67

square units. Find the length and breadth of the rectangle.

State true or false:

The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. The dimensions of the rectangle are 17 units and 9 units.

Q. If the length of a rectangle is reduced by 5 units and its breadth is increased by 3 units, then the area of the rectangle is reduced by 8 square units. If length is reduced by 3 units and breadth is increased by 2 units, then the area of the rectangle will increase by 67 square units. Then find the length and breadth of the rectangle.

Join BYJU'S Learning Program

The area of a rectangle gets reduced by 9 square units if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.

The area of a rectangle gets reduced by $9$ square units if its length is reduced by $5$ units and breadth is increased by $3$ units. If we increase the length by $3$ units and breadth by $2$ units, the area increases by $67$ square units. Find the dimensions of the rectangle.