Understanding Rectangle Area Changes and Ratios

This problem asks us to find the ratio of the area of an original rectangle to the area of a new rectangle created by changing its dimensions. We need to calculate the area in both cases and then find their ratio.

Original Rectangle Dimensions and Area

Let's assume the original rectangle has:

• Length = $\text{l}$
• Breadth = $\text{b}$

The area of the original rectangle is given by the formula: Area = Length $\times$ Breadth.

So, the Original Area = $\text{l} \times \text{b}$. We can denote this as $\text{A}_{original}$.

$\text{A}_{original} = \text{lb}$

New Rectangle Dimensions and Area

According to the problem, the dimensions of the new rectangle are changed:

• The length is increased to three times its original length.
• The breadth is decreased to half its original breadth.

So, the new dimensions are:

• New Length = $3 \times \text{l} = 3\text{l}$
• New Breadth = $\frac{\text{b}}{2}$

The area of the new rectangle is New Length $\times$ New Breadth.

New Area = $(3\text{l}) \times \left(\frac{\text{b}}{2}\right)$

New Area = $\frac{3\text{lb}}{2}$

We can denote this as $\text{A}_{new}$.

$\text{A}_{new} = \frac{3\text{lb}}{2}$

Calculating the Area Ratio

We need to find the ratio of the area of the given (original) rectangle to the area of the new rectangle. This ratio is $\text{A}_{original} : \text{A}_{new}$.

Ratio = $\frac{\text{A}_{original}}{\text{A}_{new}} = \frac{\text{lb}}{\frac{3\text{lb}}{2}}$

To simplify this fraction, we multiply the numerator by the reciprocal of the denominator:

Ratio = $\text{lb} \times \frac{2}{3\text{lb}}$

We can cancel out the term $\text{lb}$ from both the numerator and the denominator, provided $\text{l} \neq 0$ and $\text{b} \neq 0$, which is true for a rectangle.

Ratio = $\frac{2}{3}$

So, the ratio of the area of the given rectangle to the area of the new rectangle is $2 : 3$.

Revision Table: Rectangle Area Ratio

Additional Information: Area of Rectangles and Proportions

The area of a rectangle directly depends on its length and breadth. If you change the dimensions, the area changes proportionally. In this problem, we scaled the length by a factor of 3 and the breadth by a factor of $1/2$. The overall scaling factor for the area is the product of the scaling factors for length and breadth.

• Length scaling factor = 3
• Breadth scaling factor = $\frac{1}{2}$
• Area scaling factor = Length scaling factor $\times$ Breadth scaling factor = $3 \times \frac{1}{2} = \frac{3}{2}$

This means the new area is $\frac{3}{2}$ times the original area:

$\text{A}_{new} = \frac{3}{2} \times \text{A}_{original}$

The ratio $\text{A}_{original} : \text{A}_{new}$ is $\text{A}_{original} : \left(\frac{3}{2} \times \text{A}_{original}\right)$. Dividing both sides by $\text{A}_{original}$ (assuming $\text{A}_{original} \neq 0$), we get $1 : \frac{3}{2}$. Multiplying both sides by 2 to remove the fraction gives the ratio $2 : 3$. This confirms our calculation.