Question

The length of a rectangle is \[16\,m\] less than two times its breadth. If the perimeter of the rectangle is \[112 m\], find the length and the breadth.

Answer 1

Hint: We will use the concept of solving linear equations in two variables. Using the given condition “The length of the rectangle is\[16\,m\] less than two times its breadth” and the formula of perimeter, we will get two equations. By solving these equations we can find the length and breadth.

Formula used:
Here, we will be using the formula for the perimeter of the rectangle \[P = 2\left( {l + b} \right)\], where $l$ and $b$ are the length and breadth of the rectangle.

Complete step-by-step answer:
Let \[l\] be the length of the rectangle and \[b\] be the breadth.
 Now, we have given that the length of the rectangle is \[16\,m\] less than two times its breadth. Then,
\[ \Rightarrow l = 2b - 16\]
\[ \Rightarrow l = 2b - 16\] (1)
 Also the perimeter is given to be \[112 m\].
We know that the formula of the perimeter of the rectangle is \[P = 2\left( {l + b} \right)\].
 Now, substitute the given value of perimeter in the above formula.
\[ \Rightarrow 112 = 2\left( {l + b} \right)\]
Or
$ \Rightarrow l + b = 56$ (2)
Now, substitute the value of $l$ from equation $\left( 1 \right)$ into equation $\left( 2 \right)$ .
$ \Rightarrow \left( {2b - 16} \right) + b = 56$
$\Rightarrow$ $2b + b = 56 + 16$
$\Rightarrow$ $3b = 72$
 Dividing both sides by \[3\].
\[b = 24\]
 Now, substitute \[b = 24\] in the equation $\left( 1 \right)$ to get the value of length.
\[ \Rightarrow l = 2\left( {24} \right) - 16\]
 Simplify it further,
\[ \Rightarrow l = 48 - 16\]
$\Rightarrow $ $l = 32$
Therefore, the length and breadth of rectangle is $32\;{\text{m}}$ and $24\;{\text{m}}$ respectively.

Note: In these types of questions, form a linear equation from the given condition. Use the formula for given values such as area or perimeter, just to create another linear equation. Once the equations are obtained, simplify to get the value of the unknown variable.
In case, the area of the rectangle is given then use the formula $A = l \times b$ ,where $A$ denotes the area of the rectangle.