The area of a rectangular plot of land is 817`4/5` sq. m. If its breadth is 21`3/4` m, find its length.

Answer

To find the length of the rectangular plot of land given its area and breadth, we can follow these steps:

### Step 1: Convert the mixed numbers to improper fractions.
The area is given as \( 817 \frac{4}{5} \) sq. m and the breadth as \( 21 \frac{3}{4} \) m.

**Conversion:**
- For the area:
\[ 817 \frac{4}{5} = 817 + \frac{4}{5} = \frac{817 \times 5 + 4}{5} = \frac{4085 + 4}{5} = \frac{4089}{5} \text{ sq. m} \]

- For the breadth:
\[ 21 \frac{3}{4} = 21 + \frac{3}{4} = \frac{21 \times 4 + 3}{4} = \frac{84 + 3}{4} = \frac{87}{4} \text{ m} \]

### Step 2: Use the formula for the area of a rectangle.
The area of a rectangle is given by the formula:
\[ \text{Area} = \text{Length} \times \text{Breadth} \]
Let the length be \( L \). Then we can write:
\[ \frac{4089}{5} = L \times \frac{87}{4} \]

### Step 3: Solve for the length \( L \).
Rearranging the equation to solve for \( L \):
\[ L = \frac{\frac{4089}{5}}{\frac{87}{4}} = \frac{4089}{5} \times \frac{4}{87} \]

### Step 4: Simplify the expression.
Calculating \( L \):
\[ L = \frac{4089 \times 4}{5 \times 87} \]

### Step 5: Calculate the numerator and denominator.
Calculating the numerator:
\[ 4089 \times 4 = 16356 \]

Calculating the denominator:
\[ 5 \times 87 = 435 \]

So we have:
\[ L = \frac{16356}{435} \]

### Step 6: Simplify the fraction.
To simplify \( \frac{16356}{435} \), we can divide both the numerator and the denominator by their greatest common divisor (GCD).

Calculating:
\[ L = \frac{16356 \div 87}{435 \div 87} = \frac{188}{5} \]

### Step 7: Convert back to a mixed number.
To convert \( \frac{188}{5} \) to a mixed number:
\[ 188 \div 5 = 37 \quad \text{(whole number)} \]
\[ 188 - (37 \times 5) = 3 \quad \text{(remainder)} \]
Thus, we have:
\[ L = 37 \frac{3}{5} \text{ m} \]

### Final Answer:
The length of the rectangular plot is \( 37 \frac{3}{5} \) meters.

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