What is the greater of the two numbers whose product is 1092 and the sum of the two numbers exceeds their difference by 42?

## Concept & Logic Translating a word problem into algebraic expressions is key here. The phrase "sum exceeds difference by" can be translated into a direct equation. Interestingly, the variables representing the "greater" number will cancel out in this translation, instantly revealing the smaller number. $$ (x + y) - (x - y) = \text{Excess Amount} $$ ## Step-by-Step Solution * **Given:** Let the two numbers be $x$ and $y$, where $x > y$. The product $x \times y = 1092$. * **Calculation / Deduction:** * The sum is $(x + y)$. The difference is $(x - y)$. * Set up the equation based on the condition "sum exceeds difference by 42": * $(x + y) - (x - y) = 42$ * Simplify the left side: * $x + y - x + y = 42$ * $2y = 42$ * $y = 21$ * We now know the smaller number is $21$. Use the product to find the greater number ($x$): * $x \times 21 = 1092$ * $x = \frac{1092}{21}$ * $x = 52$ ## Exam Strategy & Shortcut **Logical Deduction:** Realize that the difference between the sum $(A+B)$ and the difference $(A-B)$ of two numbers is ALWAYS simply twice the smaller number ($2B$). Therefore, if the difference is $42$, the smaller number is instantly $21$. Divide $1092$ by $21$ to get $52$. To quickly divide $1092$ by $21$, look at the unit digit: $1 \times \text{what} = 2$? It must end in $2$. Look at the options; $52$ is the only logical choice that works ($20 \times 50 = 1000$). ## Common Pitfall A major pitfall is trying to set up complex quadratic equations using substitution before simplifying the "sum exceeds difference" clause. Always simplify your linear conditions completely before plugging them into a product condition. ## Final Answer **Therefore, the correct answer is 52.**