Question

If the perimeters of two squares are in the ratio $1:3,$ then the ratio of their areas is:

• A. $3:1$
• B. $1:9$
• C. $9:1$
• D. $1:3$

Answer 1

The correct option is B $1:9$

Given, the ratio of the perimeters of two squares is $1:3.$

Let the perimeter of the first square be $x,$ then the perimeter of the second square will be $3x.$
We know that, $\text{Perimeter of a square = 4}\times \text{Side length}$
$\Rightarrow \text{Side length =}\frac{\text{Perimeter of the square}}{4}$

Step 1: Side length of the first square $=\frac{\text{Perimeter of the first square}}{4}$
$=\frac{x}{4}$
Area of the first square $=({\text{side length)}}^2$
$={\left(\frac{x}{4}\right)}^2$
$=\frac{x^2}{4^2}$
$=\frac{x^2}{16}$

Step 2: Side length of the second square $=\frac{\text{Perimeter of the second square}}{4}$
$=\frac{3x}{4}$
Area of the second square $=({\text{side length)}}^2$
$={\left(\frac{3x}{4}\right)}^2$
$=\frac{(3x{)}^2}{4^2}$
$=\frac{9x^2}{16}$

Step 3: Ratio of the areas
$=\text{Area of the first square : Area of the second square}$
$=\frac{x^2}{16}:\frac{9x^2}{16}$
Multiply both sides by 16
$=\frac{x^2}{16}\times 16:\frac{9x^2}{16}\times 16$
$=x^2:9x^2$
Divide both sides by $x^2$
$=\frac{x^2}{x^2}:\frac{9x^2}{x^2}$
$=1:9$

Hence, option (b.) is the correct choice.