The price of commodity X increases by 40 paise every year, while the price of commodity Y increases by 15 paise every year. In 2001, the price of X was Rs. 4.20 and the price of Y was Rs. 6.30. In which year will commodity X cost exactly 40 paise more than commodity Y?

Introduction / Context:
This question involves linear growth of prices over time. It tests your ability to model year on year increments as simple arithmetic progressions and then solve for the time when a given difference between two prices is achieved. Such questions are common in topics related to simple interest, progressions, and basic algebra.

Given Data / Assumptions:

• In 2001, price of commodity X = Rs. 4.20.
• In 2001, price of commodity Y = Rs. 6.30.
• Price of X increases by 40 paise (Rs. 0.40) every year.
• Price of Y increases by 15 paise (Rs. 0.15) every year.
• We must find the year when price of X is exactly 40 paise more than price of Y.

Concept / Approach:
We treat the price of each commodity as a linear function of the number of years after 2001. Let n be the number of years after 2001. Then we express the prices of X and Y in terms of n using their annual increases. We then impose the condition that X is 0.40 rupee more than Y and solve the resulting linear equation for n, finally converting n into the actual calendar year.

Step-by-Step Solution:
Let n be the number of years after 2001.Price of X after n years: X(n) = 4.20 + 0.40n.Price of Y after n years: Y(n) = 6.30 + 0.15n.We want X(n) to be 0.40 higher than Y(n): X(n) = Y(n) + 0.40.Substitute the expressions: 4.20 + 0.40n = 6.30 + 0.15n + 0.40.Simplify the right side: 6.30 + 0.40 = 6.70, so 4.20 + 0.40n = 6.70 + 0.15n.Rearrange: 0.40n - 0.15n = 6.70 - 4.20.This gives 0.25n = 2.50.Solve for n: n = 2.50 / 0.25 = 10.The required year is 2001 + 10 = 2011.

Verification / Alternative check:
Compute prices in 2011 directly. After 10 years, price of X = 4.20 + 10 × 0.40 = 4.20 + 4.00 = Rs. 8.20. Price of Y in 2011 = 6.30 + 10 × 0.15 = 6.30 + 1.50 = Rs. 7.80. The difference is 8.20 - 7.80 = Rs. 0.40, which matches the requirement that X be 40 paise more than Y. This confirms that 2011 is the correct year.

Why Other Options Are Wrong:
If you try 2010 (n = 9), prices become X = 4.20 + 3.60 = 7.80 and Y = 6.30 + 1.35 = 7.65, difference = 0.15. For 2012 (n = 11), the difference is 0.65. For 2013, the difference becomes even larger. None of these match the required 0.40 difference. Therefore, only the year 2011 fits the condition exactly.

Common Pitfalls:
One common mistake is to set X equal to Y instead of X equal to Y plus 0.40. Another error is mixing rupees and paise improperly, for example by working in paise for one quantity and rupees for another. Some students might also misinterpret 2001 as year zero and then add or subtract incorrectly. Keeping all amounts in rupees and defining n clearly as the number of years after 2001 helps maintain accuracy.

Final Answer:
Commodity X will cost exactly 40 paise more than commodity Y in the year 2011.