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
- 7 : 9
- 3 : 7
- 1 : 1
- 4 : 3
The Correct Option is A
Solution and Explanation
Let the area of wheat and mustard cultivated by Vimal be represented as $W_v$ and $M_v$, respectively. We are given that the ratio of wheat to mustard in Vimal's land is 5 : 3. Therefore, we can express this as:
\[ \frac{W_v}{M_v} = \frac{5}{3} \quad \text{or} \quad W_v = \frac{5}{3} M_v \]
We also know that the total area of Vimal's land is 30 hectares, so:
\[ W_v + M_v = 30 \]
Substitute $W_v = \frac{5}{3} M_v$ into the equation:
\[ \frac{5}{3} M_v + M_v = 30 \]
Simplify:
\[ \frac{8}{3} M_v = 30 \implies M_v = 30 \times \frac{3}{8} = 11.25 \]
Now, substitute $M_v = 11.25$ back into $W_v = \frac{5}{3} M_v$:
\[ W_v = \frac{5}{3} \times 11.25 = 18.75 \]
So, Vimal's land has $W_v = 18.75$ hectares of wheat and $M_v = 11.25$ hectares of mustard.
Next, let's consider Rajesh's land, where the total area of wheat and mustard is divided. The total area of Rajesh's land is 20 hectares, so:
\[ W_r + M_r = 20 \]
We are also told that the overall ratio of wheat to mustard across both Rajesh's and Vimal's lands is 11 : 9, i.e.,
\[ \frac{W_v + W_r}{M_v + M_r} = \frac{11}{9} \]
Substitute the values of $W_v = 18.75$ and $M_v = 11.25$ into the equation:
\[ \frac{18.75 + W_r}{11.25 + M_r} = \frac{11}{9} \]
Cross-multiply to solve for $W_r$ and $M_r$:
\[ 9(18.75 + W_r) = 11(11.25 + M_r) \]
Simplifying:
\[ 9W_r + 168.75 = 11M_r + 123.75 \]
\[ 9W_r - 11M_r = -45 \]
We also have the equation $W_r + M_r = 20$. Now, solve this system of equations. From $W_r + M_r = 20$, express $W_r$ as:
\[ W_r = 20 - M_r \]
Substitute into the equation $9W_r - 11M_r = -45$:
\[ 9(20 - M_r) - 11M_r = -45 \]
Simplify:
\[ 180 - 9M_r - 11M_r = -45 \]
\[ 180 - 20M_r = -45 \implies -20M_r = -225 \implies M_r = 11.25 \]
Substitute $M_r = 11.25$ into $W_r + M_r = 20$:
\[ W_r = 20 - 11.25 = 8.75 \]
Finally, the ratio of the areas of wheat to mustard in Rajesh's land is:
\[ \frac{W_r}{M_r} = \frac{8.75}{11.25} = \frac{7}{9} \]
Thus, the correct answer is Option (1).
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