The price of commodity X increases by 40 paise every year, while the price of commodity Y increases by 15 paise every year. If in 2004, the price of commodity X was ₹ 4.20 and that of Y was ₹ 6.30, in which year commodity X will cost 40 paise more than the commodity Y?

## Concept & Logic This is a relative speed or linear equations problem. The price gap between the two commodities shrinks every year because Commodity X increases faster than Commodity Y. We need to find the number of years it takes for X to catch up and surpass Y by a specific margin. $$ \text{Target Year} = \text{Base Year} + \frac{\text{Initial Gap} + \text{Target Gap}}{\text{Relative Increase Rate}} $$ ## Step-by-Step Solution * **Given:** Initial Price of X in 2004 = ₹ 4.20 Initial Price of Y in 2004 = ₹ 6.30 Annual increase of X = 0.40 ₹ (40 paise) Annual increase of Y = 0.15 ₹ (15 paise) Target difference = X is 40 paise (0.40 ₹) more than Y. * **Calculation:** Let $n$ be the number of years after 2004. Price of X after $n$ years = $4.20 + 0.40n$ Price of Y after $n$ years = $6.30 + 0.15n$ Set up the equation based on the condition that X costs 0.40 ₹ more than Y: $(4.20 + 0.40n) - (6.30 + 0.15n) = 0.40$ Simplify the equation: $0.25n - 2.10 = 0.40$ $0.25n = 2.50$ Solve for $n$: $n = \frac{2.50}{0.25}$ $n = 10 \text{ years}$ Add 10 years to the base year: $2004 + 10 = 2014$ ## Exam Strategy & Shortcut Use the concept of relative speed. Commodity X catches up to Commodity Y at a relative rate of $40 - 15 = 25$ paise per year. Current price gap = $6.30 - 4.20 = 2.10$ ₹ (or 210 paise). Target state: X is 40 paise ahead. Total gap to cover = $210 \text{ paise (to tie)} + 40 \text{ paise (to lead)} = 250 \text{ paise}$. Time required = $\frac{250}{25} = 10 \text{ years}$. $2004 + 10 = 2014$. This skips building algebraic equations entirely. ## Common Pitfall Students often solve for when the prices become equal ($210 / 25 = 8.4$ years) and forget to account for the additional 40 paise target margin required by the question. ## Final Answer **Therefore, the correct answer is 2014.**