The perimeter of a rectangular garden is 420 cm. If its length is increased by 20% and breadth is decreased by 40%, we get the same perimeter. Then the length and breadth of the new rectangular garden, respectively are

Options

A. 115 cm and 95 cm,
B. 168 cm and 42 cm
C. 210 cm and 210 cm
D. 95 cm and 115 cm

Answer

To solve the problem step by step, we need to find the length and breadth of the rectangular garden after the changes in dimensions.

### Step 1: Understand the Perimeter Formula
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2 \times (L + B) \]
where \( L \) is the length and \( B \) is the breadth.

### Step 2: Set Up the Equation for the Original Garden
Given that the perimeter of the rectangular garden is 420 cm, we can write:
\[ 2 \times (L + B) = 420 \]
Dividing both sides by 2 gives:
\[ L + B = 210 \quad \text{(Equation 1)} \]

### Step 3: Define the New Dimensions
According to the problem:
- The length is increased by 20%, so the new length \( L' \) is:
\[ L' = L + 0.2L = 1.2L \]
- The breadth is decreased by 40%, so the new breadth \( B' \) is:
\[ B' = B - 0.4B = 0.6B \]

### Step 4: Set Up the Equation for the New Garden
The new perimeter remains the same (420 cm), so we can write:
\[ 2 \times (L' + B') = 420 \]
Substituting \( L' \) and \( B' \):
\[ 2 \times (1.2L + 0.6B) = 420 \]
Dividing both sides by 2 gives:
\[ 1.2L + 0.6B = 210 \quad \text{(Equation 2)} \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( L + B = 210 \) (Equation 1)
2. \( 1.2L + 0.6B = 210 \) (Equation 2)

We can solve these equations simultaneously. From Equation 1, we can express \( B \) in terms of \( L \):
\[ B = 210 - L \]

### Step 6: Substitute \( B \) in Equation 2
Substituting \( B \) in Equation 2:
\[ 1.2L + 0.6(210 - L) = 210 \]
Expanding this gives:
\[ 1.2L + 126 - 0.6L = 210 \]
Combining like terms:
\[ 0.6L + 126 = 210 \]
Subtracting 126 from both sides:
\[ 0.6L = 84 \]
Dividing by 0.6:
\[ L = \frac{84}{0.6} = 140 \]

### Step 7: Find the Breadth
Now substituting \( L \) back into Equation 1 to find \( B \):
\[ B = 210 - 140 = 70 \]

### Step 8: Find the New Dimensions
Now we can find the new dimensions:
- New Length \( L' = 1.2L = 1.2 \times 140 = 168 \)
- New Breadth \( B' = 0.6B = 0.6 \times 70 = 42 \)

### Final Answer
The length and breadth of the new rectangular garden are:
\[ \text{Length} = 168 \text{ cm}, \quad \text{Breadth} = 42 \text{ cm} \]