A man purchased 40 fruits - apples and oranges - for ₹ 17. Had he purchased as many oranges as apples and as many apples as oranges, he would have paid ₹ 15. Find the cost of one pair of an apple and an orange.

### Concept & Strategy This problem creates a symmetrical system of linear equations. When the coefficients (quantities) of two variables (prices) are perfectly swapped across two equations, simply adding the two equations together yields the sum of the variables directly, bypassing the need to solve for each variable individually. $$\text{If } ax + by = C_1 \text{ and } bx + ay = C_2 \text{, then } (a+b)(x+y) = C_1 + C_2$$ ### Step-by-Step Solution * **Given:** * Let $a$ be the number of apples and $o$ be the number of oranges. * Total fruits: $a + o = 40$ * Let $p_a$ be the price of one apple and $p_o$ be the price of one orange. * **Formulate Equations:** * Actual purchase: $a(p_a) + o(p_o) = 17$ (Eq. 1) * Swapped purchase: $o(p_a) + a(p_o) = 15$ (Eq. 2) * **Calculation:** * Add Equation 1 and Equation 2 together: * $a(p_a) + o(p_a) + o(p_o) + a(p_o) = 17 + 15$ * Factor out the prices: * $p_a(a + o) + p_o(a + o) = 32$ * Factor out the total quantity $(a + o)$: * $(a + o)(p_a + p_o) = 32$ * Substitute the known total quantity ($a + o = 40$): * $40(p_a + p_o) = 32$ * $p_a + p_o = \frac{32}{40} = 0.8$ rupees. * **Conversion:** * The question asks for the cost of "one pair" (one apple + one orange), which is exactly $p_a + p_o$. * $0.8 \text{ rupees} = 80 \text{ paise}$. ### Exam Strategy & Shortcut Recognize the "swap" pattern instantly. When a prompt says "had he purchased as many X as Y and Y as X," immediately add the two total costs ($17 + 15 = 32$) and divide by the total number of items ($40$). Cost of 1 pair = $32 / 40 = 0.8$ INR = $80$ paise. You can solve this entirely in your head in 10 seconds. ### Common Pitfall The biggest trap is trying to find the exact number of apples and oranges ($a$ and $o$), or the individual price of an apple ($p_a$) and an orange ($p_o$). Because there are 4 unknowns and only 3 distinct equations, the individual values cannot be determined. Attempting to solve for them will waste massive amounts of time. Always focus on the exact expression requested (the sum). ### Final Answer **Therefore, the correct answer is 80 paise.**