Rajesh and Vimal own 20 hectares and 30 hectares of agricultural land, respectively, which are entirely covered by wheat and mustard crops. The cultivation area of wheat and mustard in the land owned by Vimal are in the ratio of 5 : 3. If the total cultivation area of wheat and mustard are in the ratio 11 : 9, then the ratio of cultivation area of wheat and mustard in the land owned by Rajesh is:

The correct option is 7 : 9.

Let us break down the solution step-by-step to understand how we arrive at this ratio.

Step 1: Determine the total land and individual holdings
Rajesh owns 20 hectares of land.
Vimal owns 30 hectares of land.
Therefore, the total agricultural land owned by both is:
Total land = 20 hectares + 30 hectares = 50 hectares.

Step 2: Find the total cultivation area of wheat and mustard
The total land of 50 hectares is entirely covered by wheat and mustard. The ratio of the total cultivation area of wheat to mustard is given as 11 : 9.
Let the total cultivation area of wheat be $11x$ and that of mustard be $9x$.
Thus, we have:
$11x+9x=50$
$20x=50$
$x=2.5$

Using the value of $x$, we can find the total area for each crop:
Total area under wheat = $11\times 2.5=27.5$ hectares.
Total area under mustard = $9\times 2.5=22.5$ hectares.

Step 3: Determine the cultivation area of wheat and mustard for Vimal
Vimal owns 30 hectares of land. The ratio of wheat to mustard cultivation in Vimal's land is 5 : 3.
Let the area under wheat for Vimal be $5y$ and under mustard be $3y$.
Thus, we have:
$5y+3y=30$
$8y=30$
$y=\frac{30}{8}=3.75$

Now, we calculate Vimal's individual crop areas:
Wheat area for Vimal = $5\times 3.75=18.75$ hectares.
Mustard area for Vimal = $3\times 3.75=11.25$ hectares.

Step 4: Determine the cultivation area of wheat and mustard for Rajesh
We can find Rajesh's cultivation area for each crop by subtracting Vimal's crop area from the total crop area:
Wheat area for Rajesh = Total wheat area − Vimal's wheat area
Wheat area for Rajesh = $27.5-18.75=8.75$ hectares.

Mustard area for Rajesh = Total mustard area − Vimal's mustard area
Mustard area for Rajesh = $22.5-11.25=11.25$ hectares.

Step 5: Calculate the ratio of wheat to mustard cultivation area for Rajesh
The ratio for Rajesh is:
Ratio = Wheat area for Rajesh : Mustard area for Rajesh
Ratio = $8.75:11.25$

To simplify, we multiply both sides of the ratio by 100 to remove decimals:
Ratio = $875:1125$

Dividing both terms by their greatest common divisor, which is 125:
$\frac{875}{125}=7$
$\frac{1125}{125}=9$

Hence, the simplified ratio is 7 : 9.