A rectangular field is 50 m by 40 m. It has two roads through its centre, running parallel toits sides. The width of the longer and the shorter roads are 2 m and 2.5-m-respectivelyFind the area of the roads and the area of the remaining portion of the field. 

Answer

To solve the problem step by step, we will calculate the area of the roads and the area of the remaining portion of the rectangular field.

### Step 1: Calculate the area of the rectangular field.
The area of a rectangle is given by the formula:
\[ \text{Area} = \text{Length} \times \text{Breadth} \]
For the given rectangular field:
- Length = 50 m
- Breadth = 40 m

Calculating the area:
\[ \text{Area of the field} = 50 \, \text{m} \times 40 \, \text{m} = 2000 \, \text{m}^2 \]

### Step 2: Calculate the area of the longer road.
The longer road runs parallel to the length of the rectangle and has a width of 2 m.

Using the area formula:
- Length of the road = 50 m (same as the length of the field)
- Width of the road = 2 m

Calculating the area:
\[ \text{Area of the longer road} = 50 \, \text{m} \times 2 \, \text{m} = 100 \, \text{m}^2 \]

### Step 3: Calculate the area of the shorter road.
The shorter road runs parallel to the breadth of the rectangle and has a width of 2.5 m.

Using the area formula:
- Length of the road = 40 m (same as the breadth of the field)
- Width of the road = 2.5 m

Calculating the area:
\[ \text{Area of the shorter road} = 40 \, \text{m} \times 2.5 \, \text{m} = 100 \, \text{m}^2 \]

### Step 4: Calculate the total area of the roads.
The total area of the roads is the sum of the areas of both roads:
\[ \text{Total area of the roads} = \text{Area of longer road} + \text{Area of shorter road} = 100 \, \text{m}^2 + 100 \, \text{m}^2 = 200 \, \text{m}^2 \]

### Step 5: Subtract the overlapping area.
The area where the two roads overlap (the intersection) is a rectangle formed by the widths of both roads:
- Length of the overlap = 2 m (width of the longer road)
- Width of the overlap = 2.5 m (width of the shorter road)

Calculating the area of the overlap:
\[ \text{Area of overlap} = 2 \, \text{m} \times 2.5 \, \text{m} = 5 \, \text{m}^2 \]

Now, we subtract the overlapping area from the total area of the roads:
\[ \text{Total area of the roads after removing overlap} = 200 \, \text{m}^2 - 5 \, \text{m}^2 = 195 \, \text{m}^2 \]

### Step 6: Calculate the area of the remaining portion of the field.
To find the area of the remaining portion of the field, we subtract the area of the roads from the area of the entire field:
\[ \text{Area of remaining portion} = \text{Area of field} - \text{Total area of roads} \]
\[ \text{Area of remaining portion} = 2000 \, \text{m}^2 - 195 \, \text{m}^2 = 1805 \, \text{m}^2 \]

### Final Answers:
- Area of the roads: **195 m²**
- Area of the remaining portion of the field: **1805 m²**