A buyer purchased few books of equal value for a total cost of ₹ 720. If the value of each were ₹ 2 less than the price at which the buyer originally bought, she could have purchased 4 more books than what she had bought. How many books did she originally purchase?

### Concept & Strategy The total cost is a constant product of Quantity and Price ($C = Q \times P$). When a price decrease allows for a quantity increase under a fixed budget, it creates a non-linear algebraic equation. The fastest way to solve this is to set up the equation and test the multiple-choice options rather than solving the underlying quadratic equation. $$\text{Total Cost} = \text{Number of Books} \times \text{Price per Book}$$ ### Step-by-Step Solution * **Given:** * Total Cost = ₹ $720$ * Original: Let number of books = $n$. Price per book = $\frac{720}{n}$ * Hypothetical: Number of books = $(n + 4)$. Price per book = $\frac{720}{n+4}$ * The price difference is exactly ₹ $2$. * **Equation:** * $\frac{720}{n} - \frac{720}{n+4} = 2$ * Simplify by factoring out $720$: * $720 \times \left[\frac{(n+4) - n}{n(n+4)}\right] = 2$ * $720 \times \frac{4}{n(n+4)} = 2$ * $2880 = 2n(n+4) \Rightarrow 1440 = n(n+4)$ * **Option Elimination (Calculation):** * We need to find $n$ such that $n$ multiplied by $(n+4)$ equals $1440$. * Test (a) 18: $18 \times 22 = 396$ (Too small) * Test (b) 20: $20 \times 24 = 480$ (Too small) * Test (c) 34: $34 \times 38 = 1292$ (Close) * Test (d) 36: $36 \times 40 = 1440$. (Perfect Match!) ### Exam Strategy & Shortcut Instead of writing out algebra, use factor pairs of $720$. Look at the options for the original quantity $n$: $18, 20, 34, 36$. The original quantity $n$ must divide $720$ evenly, and $(n+4)$ must also divide $720$ evenly. Test 36: Price $= 720 / 36 = 20$. If $n+4 = 40$, Price $= 720 / 40 = 18$. Difference in price $= 20 - 18 = 2$. This perfectly satisfies all conditions in under 15 seconds. ### Common Pitfall A standard trap is trying to expand and solve the quadratic equation $n^2 + 4n - 1440 = 0$ manually. Factoring large numbers under pressure is error-prone and time-consuming. Always rely on the multiple-choice options to bypass quadratic factoring. ### Final Answer **Therefore, the correct answer is 36.**