The ratio of the length and breadth of a rectangle is 3:1. If the perimeter is 64 cm, find the dimensions of the rectangle.

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the Problem
We need to find the dimensions (length and breadth) of a rectangle given that the ratio of length to breadth is 3:1 and the perimeter is 64 cm.

### Step 2: Write the Perimeter Formula
The formula for the perimeter (P) of a rectangle is:
\[ P = 2 \times (L + B) \]
where \( L \) is the length and \( B \) is the breadth.

### Step 3: Set Up the Equation for Perimeter
Since the perimeter is given as 64 cm, we can write:
\[ 64 = 2 \times (L + B) \]

### Step 4: Simplify the Equation
Dividing both sides of the equation by 2 gives:
\[ L + B = 32 \]
This will be our **Equation 1**.

### Step 5: Use the Ratio of Length to Breadth
We know that the ratio of length to breadth is 3:1. This can be expressed as:
\[ \frac{L}{B} = \frac{3}{1} \]
From this, we can express length in terms of breadth:
\[ L = 3B \]
This will be our **Equation 2**.

### Step 6: Substitute Equation 2 into Equation 1
Now, we will substitute the expression for \( L \) from Equation 2 into Equation 1:
\[ 3B + B = 32 \]

### Step 7: Combine Like Terms
Combining the terms gives:
\[ 4B = 32 \]

### Step 8: Solve for Breadth
Now, divide both sides by 4 to find \( B \):
\[ B = \frac{32}{4} = 8 \, \text{cm} \]

### Step 9: Find the Length
Now that we have the breadth, we can find the length using Equation 2:
\[ L = 3B = 3 \times 8 = 24 \, \text{cm} \]

### Conclusion
The dimensions of the rectangle are:
- Length \( L = 24 \, \text{cm} \)
- Breadth \( B = 8 \, \text{cm} \)

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