A rectangular grassy plot measures 90 m by 65 m. Two pathways, each `1.5 m` wide are running through the middle of the plot, parallel to its length and breadth. Find the area of the grassy portion.

Answer

To find the area of the grassy portion of the rectangular plot with pathways, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the Area of the Rectangular Plot:**
The area of a rectangle is calculated using the formula:
\[ \text{Area} = \text{Length} \times \text{Breadth} \]
For our plot:
- Length = 90 m
- Breadth = 65 m

So, the area of the plot is:
\[ \text{Area} = 90 \, \text{m} \times 65 \, \text{m} = 5850 \, \text{m}^2 \]

2. **Calculate the Area of the Pathways:**
There are two pathways, each 1.5 m wide, running through the middle of the plot. One pathway runs parallel to the length and the other runs parallel to the breadth.

- **Area of the pathway parallel to the length:**
\[ \text{Area} = \text{Length} \times \text{Width} = 90 \, \text{m} \times 1.5 \, \text{m} = 135 \, \text{m}^2 \]

- **Area of the pathway parallel to the breadth:**
\[ \text{Area} = \text{Breadth} \times \text{Width} = 65 \, \text{m} \times 1.5 \, \text{m} = 97.5 \, \text{m}^2 \]

- **Total area of the pathways before adjusting for overlap:**
\[ \text{Total Area of Pathways} = 135 \, \text{m}^2 + 97.5 \, \text{m}^2 = 232.5 \, \text{m}^2 \]

3. **Adjust for Overlap:**
The area where the two pathways overlap (the intersection) is a square of side 1.5 m:
\[ \text{Area of Overlap} = 1.5 \, \text{m} \times 1.5 \, \text{m} = 2.25 \, \text{m}^2 \]

Therefore, the actual area of the pathways is:
\[ \text{Actual Area of Pathways} = 232.5 \, \text{m}^2 - 2.25 \, \text{m}^2 = 230.25 \, \text{m}^2 \]

4. **Calculate the Area of the Grassy Portion:**
Finally, we find the area of the grassy portion by subtracting the area of the pathways from the area of the plot:
\[ \text{Area of Grassy Portion} = \text{Area of Plot} - \text{Actual Area of Pathways} \]
\[ \text{Area of Grassy Portion} = 5850 \, \text{m}^2 - 230.25 \, \text{m}^2 = 5619.75 \, \text{m}^2 \]

### Final Answer:
The area of the grassy portion is \( 5619.75 \, \text{m}^2 \).

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