The sum and product of two numbers are $12$ and $35$ respectively. The sum of their reciprocals will be

### Concept & Formula The sum of the reciprocals of two numbers can be found directly if their sum and product are known, without needing to calculate the individual numbers. For any two numbers $x$ and $y$, the sum of their reciprocals is: $$ \frac{1}{x} + \frac{1}{y} = \frac{x + y}{xy} $$ ### Step-by-Step Solution * **Given:** The sum of the two numbers $(x + y) = 12$ The product of the two numbers $(xy) = 35$ * **Calculation:** We need to find the sum of their reciprocals: $\frac{1}{x} + \frac{1}{y}$. Using the algebraic identity established in the concept: $$ \frac{1}{x} + \frac{1}{y} = \frac{x + y}{xy} $$ Substitute the given values directly into the formula: Sum of reciprocals $= \frac{12}{35}$ ### Exam Strategy & Shortcut **Direct Substitution** is the ultimate shortcut here. Whenever you see "sum and product are $S$ and $P$, find the sum of reciprocals", instantly write down $\frac{S}{P}$. You do not need to figure out that the numbers are $7$ and $5$. The answer is just $\frac{12}{35}$ immediately. ### Common Pitfall Students often waste time trying to solve the quadratic equation to find the individual numbers (finding $7$ and $5$), and then adding their reciprocals $\frac{1}{7} + \frac{1}{5} = \frac{5+7}{35} = \frac{12}{35}$. While this gets the right answer, it consumes precious time unnecessarily. ### Final Answer **Therefore, the correct answer is 12/35.**