The sum of two numbers is 25 and their difference is 13. Find their product.

## Concept & Formula When given the sum and difference of two numbers, you can find the individual numbers using linear equations. This is a fundamental algebraic identity. $$ x = \frac{\text{Sum} + \text{Difference}}{2} $$ $$ y = \frac{\text{Sum} - \text{Difference}}{2} $$ ## Step-by-Step Solution * **Given:** The sum of two numbers ($x + y$) is $25$. Their difference ($x - y$) is $13$. * **Calculation / Deduction:** * Add the two equations to eliminate $y$: $(x + y) + (x - y) = 25 + 13$ * $2x = 38$ * $x = 19$ * Substitute $x$ back into the sum equation: $19 + y = 25$ * $y = 6$ * Calculate the product: $x \times y = 19 \times 6 = 114$ ## Exam Strategy & Shortcut **Direct Formula:** Instead of writing out the equations, mentally compute the larger number as half the sum of $25$ and $13$ (which is $38 / 2 = 19$), and the smaller as half their difference (which is $12 / 2 = 6$). Instantly multiply $19 \times 6 = 114$. ## Common Pitfall A common mistake is correctly finding the two numbers but forgetting to read the final sentence, leading students to look for an option matching one of the numbers instead of their required product. Always double-check what the question is asking for. ## Final Answer **Therefore, the correct answer is 114.**