In the given questions, two quantities are given. One as Quantity-I and another is Quantity-II. You have to determine relationship between these two quantities and choose the appropriate options as given below:
The side of a square garden (A) is equal to that of the breadth of a rectangular garden (B). The respective ratio between the length and the breadth of garden B is 5:4. The cost of fencing of garden A is Rs30 less than the cost of fencing of garden B. (both at the rate of Rs5 per metre)
Quantities:
I. The length of the garden B.
II. 20 metres

Options

A. Quantity- I `gt` Quantity-II
B. Quantity- I `lt` Quantity-II
C. Quantity- I `le` Quantity-II
D. Quantity- I `ge` Quantity-II

Answer

To solve the problem step by step, we will analyze the information given and derive the necessary quantities.

### Step 1: Define the Variables
Let the side of the square garden (A) be denoted as \( s \). According to the problem, the breadth of the rectangular garden (B) is equal to the side of the square garden, so:
\[ \text{Breadth of garden B} = s \]

### Step 2: Establish the Ratio
We know that the ratio of the length to the breadth of garden B is \( 5:4 \). If we let the breadth be \( 4x \), then the length will be \( 5x \). Since the breadth of garden B is equal to the side of the square garden:
\[ s = 4x \]

### Step 3: Calculate the Perimeters
- The perimeter of the square garden A is:
\[ \text{Perimeter of A} = 4s = 4(4x) = 16x \]

- The perimeter of the rectangular garden B is:
\[ \text{Perimeter of B} = 2(\text{Length} + \text{Breadth}) = 2(5x + 4x) = 2(9x) = 18x \]

### Step 4: Calculate the Cost of Fencing
The cost of fencing is given at the rate of Rs 5 per meter.
- Cost of fencing for garden A:
\[ \text{Cost of A} = 5 \times \text{Perimeter of A} = 5 \times 16x = 80x \]

- Cost of fencing for garden B:
\[ \text{Cost of B} = 5 \times \text{Perimeter of B} = 5 \times 18x = 90x \]

### Step 5: Set Up the Equation Based on the Given Condition
We know that the cost of fencing for garden A is Rs 30 less than that of garden B:
\[ 80x = 90x - 30 \]

### Step 6: Solve for \( x \)
Rearranging the equation:
\[ 80x + 30 = 90x \]
\[ 30 = 90x - 80x \]
\[ 30 = 10x \]
\[ x = 3 \]

### Step 7: Calculate the Length of Garden B
Now, we can find the length of garden B:
\[ \text{Length of garden B} = 5x = 5 \times 3 = 15 \text{ meters} \]

### Step 8: Compare Quantities
We need to compare the length of garden B (15 meters) with 20 meters (Quantity II):
- Quantity I: Length of garden B = 15 meters
- Quantity II: 20 meters

Since \( 15 < 20 \), we conclude that:
**Quantity I is less than Quantity II.**

### Final Conclusion
The answer is that Quantity I is less than Quantity II.

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