If the sum of two numbers is 14 and their difference is 10, find the product of these two numbers.

## Concept & Formula This problem can be solved by translating the wording directly into a system of linear equations. However, a more advanced and faster approach uses the relationship formula between the sum, difference, and product of two variables: $$(x + y)^2 - (x - y)^2 = 4xy$$ ## Step-by-Step Solution * **Given Information:** Let the two numbers be $x$ and $y$. The sum is: $x + y = 14$ The difference is: $x - y = 10$ * **Calculation Method 1: Using the Algebraic Formula** Plug the known values directly into the identity: $$(14)^2 - (10)^2 = 4xy$$ $$196 - 100 = 4xy$$ $$96 = 4xy$$ $$xy = \frac{96}{4}$$ $$xy = 24$$ * **Calculation Method 2: Solving for Variables** * Align the two simple equations: 1) $x + y = 14$ 2) $x - y = 10$ * Adding both equations vertically eliminates $y$: $$2x = 24 \implies x = 12$$ * Substitute $x = 12$ back into the first equation: $$12 + y = 14 \implies y = 2$$ * Multiply the two numbers together to find the product: $$12 \times 2 = 24$$ ## Exam Strategy & Shortcut In competitive exams, bypass algebra entirely for "sum and difference" integer questions. Use this rapid mental arithmetic rule: **Find the larger number:** Add the sum and difference, then halve it. $\frac{14 + 10}{2} = 12$. **Find the smaller number:** Subtract the difference from the sum, then halve it. $\frac{14 - 10}{2} = 2$. Multiply them: $12 \times 2 = 24$. ## Common Pitfall A common error is successfully finding the two numbers ($12$ and $2$) but then accidentally adding them back together or stopping the calculation, completely forgetting that the question specifically asked for their *product*. Always re-read the final sentence of the question before marking your option. ## Final Answer Therefore, the correct answer is 24.