One-fourth of one-seventh of a plot of land is sold for Rs. 30,000. What is the value of four-fifteenths of the same land?

Introduction / Context:
This question tests the ability to work with fractions of a whole and proportional values. We are given the sale price of a certain fractional part of a plot of land and asked to find the value of another fractional part of the same land. Since the land has uniform value per unit area, the price is directly proportional to the fraction of land.

Given Data / Assumptions:

• One-fourth of one-seventh of the plot is sold for Rs. 30,000.
• The land is assumed to have uniform value per unit area.
• We must find the value of four-fifteenths (4/15) of the same land.
• No change in price per unit area is implied, so the rate is constant.

Concept / Approach:
If 1/4 of 1/7 of the land is worth Rs. 30,000, we can compute the total value of the entire plot by dividing by that fractional share. Once we know the total value of the full land, we can multiply by 4/15 to get the value of four-fifteenths. The key is to handle fractions carefully, noting that 1/4 of 1/7 equals 1/28 of the full land.

Step-by-Step Solution:
Step 1: The fraction 1/4 of 1/7 is (1/4) * (1/7) = 1/28.Step 2: One twenty eighth of the plot is sold for Rs. 30,000.Step 3: Let the total value of the land be V. Then (1/28) * V = 30,000.Step 4: Solve for V: V = 30,000 * 28 = 8,40,000.Step 5: We need the value of 4/15 of the land, which is (4/15) * V = (4/15) * 8,40,000.Step 6: Compute (4/15) * 8,40,000. First find 8,40,000 / 15 = 56,000, then multiply by 4 to get 2,24,000.

Verification / Alternative check:
We can check the consistency of the values. If the whole land is worth Rs. 8,40,000, then 1/28 of it is 8,40,000 / 28. Since 28 * 30,000 equals 8,40,000, the small portion worth Rs. 30,000 is indeed 1/28. For the target fraction, 4/15 of 8,40,000 should equal 2,24,000, and multiplying 56,000 by 4 confirms this. All steps align, so the computation is correct.

Why Other Options Are Wrong:
The value Rs. 1,92,000 is smaller than 2,24,000 and would correspond to a smaller fraction such as 4/17.5 of the land, not 4/15. The value Rs. 1,26,000 is much smaller and does not match any simple fraction of 8,40,000 related to the given conditions. The value Rs. 3,42,000 is larger and would correspond to approximately 4/9.8 of the land. The value Rs. 2,80,000 is also inconsistent with 4/15 of the total. Only Rs. 2,24,000 matches the exact computed value of 4/15 of the total land value.

Common Pitfalls:
A frequent mistake is to misinterpret 1/4 of 1/7 as 1/11 or 1/4 + 1/7 instead of the product 1/28. Another error occurs when students try to go directly from 30,000 to 4/15 of the land without first finding the total value. Also, some may perform fraction operations or division incorrectly, especially when dealing with large numbers. Breaking the calculation into smaller steps, such as simplifying 8,40,000 / 15 before multiplying by 4, helps to avoid arithmetic errors.

Final Answer:
The value of four-fifteenths of the land is Rs.2,24,000, so the correct option is Rs.2,24,000.