A group of boys decided to buy a few CDs whose total price was between ₹ 200 and ₹ 250. But at the time of purchase, two of the boys declined to contribute as a result of which the remaining boys had to pay Re 1 more than they had originally planned. What was the price of the CDs if the boys contributed equally and in whole number of rupees?

### Concept & Logic This is a prime example of formulating a Diophantine-style algebraic equation where the total value must fall within a specific range, and the variables are constrained to integers. We use the variables for "number of boys" and "original share" to find a structural pattern for the total price. ### Step-by-Step Solution * **Given:** Total Price $P$ is between 200 and 250. 2 boys dropped out. Remaining paid ₹ 1 extra. All shares are whole numbers. * **Calculation:** Let the original number of boys be $n$, and the original share per boy be $x$. * Total price: $$P = nx$$ * When 2 boys leave, $n - 2$ boys remain, and they pay $(x + 1)$ each: $$(n - 2)(x + 1) = P$$ * Expand the second equation: $$nx + n - 2x - 2 = P$$ * Since $nx = P$, substitute $P$ into the equation: $$P + n - 2x - 2 = P$$ $$n - 2x - 2 = 0$$ $$n = 2x + 2$$ $$n = 2(x + 1)$$ * Now, substitute this expression for $n$ back into the original total price equation $P = nx$: $$P = 2(x + 1)x = 2x(x + 1)$$ * This means the total price $P$ must be exactly twice the product of two consecutive integers ($x$ and $x+1$). Let's test integer values for $x$ to land in the 200–250 range: * If $x = 9$: $P = 2 \times 9 \times 10 = 180$ (Too low) * If $x = 10$: $P = 2 \times 10 \times 11 = 220$ (Fits the range) * If $x = 11$: $P = 2 \times 11 \times 12 = 264$ (Too high) ### Exam Strategy & Shortcut From the deduction $P = 2x(x + 1)$, we realize that $P/2$ must be the product of two consecutive integers $x(x+1)$. Look at the options for $P$: 210, 220, 230, 240. Divide them by 2: 105, 110, 115, 120. Which of these is a product of consecutive integers? $10 \times 11 = 110$. Thus, $P = 220$. ### Common Pitfall Getting overwhelmed by the lack of fixed numbers and trying to guess values randomly. Always set up the two equations, equate them, and simplify. The relationship between the variables will always reveal a predictable pattern (like consecutive integers). ### Final Answer **Therefore, the correct answer is ₹ 220.**