Let $W_v$ and $M_v$ represent the areas of wheat and mustard cultivated by Vimal, respectively. The ratio of wheat to mustard on Vimal's land is 5:3, which can be written as:
\[ \frac{W_v}{M_v} = \frac{5}{3} \quad \text{or} \quad W_v = \frac{5}{3} M_v \]
Vimal's total land area is 30 hectares:
\[ W_v + M_v = 30 \]
Substitute $W_v = \frac{5}{3} M_v$ into the total area equation:
\[ \frac{5}{3} M_v + M_v = 30 \]
Combine terms:
\[ \frac{8}{3} M_v = 30 \implies M_v = 30 \times \frac{3}{8} = 11.25 \]
Now, find $W_v$ using $M_v = 11.25$:
\[ W_v = \frac{5}{3} \times 11.25 = 18.75 \]
Vimal cultivates 18.75 hectares of wheat and 11.25 hectares of mustard.
Now consider Rajesh's land. The total area is 20 hectares:
\[ W_r + M_r = 20 \]
The combined ratio of wheat to mustard across both Vimal's and Rajesh's lands is 11:9:
\[ \frac{W_v + W_r}{M_v + M_r} = \frac{11}{9} \]
Substitute Vimal's areas ($W_v = 18.75$ and $M_v = 11.25$):
\[ \frac{18.75 + W_r}{11.25 + M_r} = \frac{11}{9} \]
Cross-multiply:
\[ 9(18.75 + W_r) = 11(11.25 + M_r) \]
Simplify:
\[ 168.75 + 9W_r = 123.75 + 11M_r \]
Rearrange the terms:
\[ 9W_r - 11M_r = -45 \]
We have a system of two equations: $W_r + M_r = 20$ and $9W_r - 11M_r = -45$. From the first equation, express $W_r$ as $W_r = 20 - M_r$. Substitute this into the second equation:
\[ 9(20 - M_r) - 11M_r = -45 \]
Simplify:
\[ 180 - 9M_r - 11M_r = -45 \]
\[ 180 - 20M_r = -45 \implies -20M_r = -225 \implies M_r = 11.25 \]
Now find $W_r$ using $W_r + M_r = 20$:
\[ W_r = 20 - 11.25 = 8.75 \]
The ratio of wheat to mustard on Rajesh's land is:
\[ \frac{W_r}{M_r} = \frac{8.75}{11.25} = \frac{7}{9} \]
Thus, the correct answer is Option (1).
| Mutual fund A | Mutual fund B | Mutual fund C | |
| Person 1 | ₹10,000 | ₹20,000 | ₹20,000 |
| Person 2 | ₹20,000 | ₹15,000 | ₹15,000 |
List I | List II | ||
| A. | Duplicate of ratio 2: 7 | I. | 25:30 |
| B. | Compound ratio of 2: 7, 5:3 and 4:7 | II. | 4:49 |
| C. | Ratio of 2: 7 is same as | III. | 40:147 |
| D. | Ratio of 5: 6 is same as | IV. | 4:14 |