The areas of two squares are in the ratio `16 : 9`. The ratio of their perimeters is :

Options

A. `16:12`
B. `12:16`
C. `9:12`
D. `9:6`

Answer

To find the ratio of the perimeters of two squares given that their areas are in the ratio \(16:9\), we can follow these steps:

### Step 1: Understand the relationship between area and side length of a square
The area \(A\) of a square is given by the formula:
\[ A = s^2 \]
where \(s\) is the length of a side of the square.

### Step 2: Let the side lengths of the two squares be \(s_1\) and \(s_2\)
Given that the areas of the two squares are in the ratio \(16:9\), we can express this as:
\[ \frac{s_1^2}{s_2^2} = \frac{16}{9} \]

### Step 3: Take the square root of both sides
Taking the square root of both sides gives us the ratio of the side lengths:
\[ \frac{s_1}{s_2} = \frac{\sqrt{16}}{\sqrt{9}} = \frac{4}{3} \]

### Step 4: Calculate the perimeters of the squares
The perimeter \(P\) of a square is given by the formula:
\[ P = 4s \]
Thus, the perimeters of the two squares can be expressed as:
\[ P_1 = 4s_1 \quad \text{and} \quad P_2 = 4s_2 \]

### Step 5: Find the ratio of the perimeters
Now, we can find the ratio of the perimeters:
\[ \frac{P_1}{P_2} = \frac{4s_1}{4s_2} = \frac{s_1}{s_2} \]
From step 3, we know that:
\[ \frac{s_1}{s_2} = \frac{4}{3} \]
Thus, the ratio of the perimeters is:
\[ \frac{P_1}{P_2} = \frac{4}{3} \]

### Final Answer
The ratio of their perimeters is \(4:3\).