Find the whole number which when increased by $20$ is equal to $69$ times the reciprocal of the number.

### Concept & Formula This requires formulating a simple quadratic equation based on the text. Let the whole number be $x$. ### Step-by-Step Solution * **Given:** The number increased by $20$ is $x + 20$. This is equal to $69$ times the reciprocal of the number, which is $\frac{69}{x}$. * **Calculation:** Equate the two expressions: $$x + 20 = \frac{69}{x}$$ Multiply the entire equation by $x$ to clear the denominator: $$x^2 + 20x = 69$$ Rearrange into standard quadratic form: $$x^2 + 20x - 69 = 0$$ We need two numbers that multiply to $-69$ and add to $20$. These are $23$ and $-3$. $$x^2 + 23x - 3x - 69 = 0$$ $$x(x + 23) - 3(x + 23) = 0$$ $$(x - 3)(x + 23) = 0$$ Therefore, $x = 3$ or $x = -23$. Since the problem specifically asks for a "whole number", we discard the negative value. $x = 3$. ### Exam Strategy & Shortcut **Option Elimination** is perfect here. The question asks for a "whole number", so we can immediately ignore Option (a) $2.5$. Test Option (b) which is $3$: Increased by $20 \rightarrow 3 + 20 = 23$ Does this equal $69$ times its reciprocal? $69 \times \frac{1}{3} = 23$. Since $23 = 23$, we immediately know Option (b) is correct without solving the quadratic. ### Common Pitfall Students might waste time trying to factorize $x^2 + 20x - 69 = 0$ if they aren't comfortable with their multiplication tables (recognizing that $23 \times 3 = 69$). In competitive exams, verifying options is often much safer. ### Final Answer **Therefore, the correct answer is 3.**