The length of a rectangle is twice its breadth. If its length is decreased by 5 cm and breadth is increased by 5 cm, the area of the rectangle is increased by 75 Sq cm. What is the length of the rectangle?

Options

A. 20 cm
B. 40 cm
C. 16 cm
D. 18 cm

Answer

To solve the problem, let’s denote the breadth of the rectangle as \( b \) cm. According to the problem, the length \( l \) of the rectangle is twice its breadth. Therefore, we can express the length as:

\[ l = 2b \]

Next, we need to analyze the changes in dimensions. The length is decreased by 5 cm and the breadth is increased by 5 cm. Thus, the new dimensions will be:

\[ \text{New Length} = l - 5 = 2b - 5 \]
\[ \text{New Breadth} = b + 5 \]

Now we can calculate the area of the rectangle before and after the changes. The area \( A \) of the rectangle before the changes is given by:

\[ A = l \times b = (2b) \times b = 2b^2 \]

The area of the rectangle after the changes is:

\[ \text{New Area} = \text{New Length} \times \text{New Breadth} = (2b - 5)(b + 5) \]

Expanding this expression using the distributive property:

\[ \text{New Area} = (2b - 5)(b + 5) = 2b^2 + 10b - 5b - 25 = 2b^2 + 5b - 25 \]

According to the problem, the new area is 75 sq cm more than the original area. Therefore, we can set up the equation:

\[ \text{New Area} = A + 75 \]

Substituting the expressions for the areas:

\[ 2b^2 + 5b - 25 = 2b^2 + 75 \]

Now, we can simplify this equation by subtracting \( 2b^2 \) from both sides:

\[ 5b - 25 = 75 \]

Next, we will isolate \( b \) by adding 25 to both sides:

\[ 5b = 100 \]

Now, divide both sides by 5 to find \( b \):

\[ b = 20 \]

Now that we have the breadth, we can find the length using the relation \( l = 2b \):

\[ l = 2 \times 20 = 40 \]

Thus, the length of the rectangle is:

\[ \boxed{40 \text{ cm}} \]