The sum of two number is 37 and the difference of their squares is 185, then the difference between the two numbers is:

Concept & Formula Utilize the standard algebraic identity for the difference of squares to relate the sum, difference, and squares of two numbers. $$ x^2 - y^2 = (x - y)(x + y) $$ Step-by-Step Solution * **Given:** Let the two numbers be $x$ and $y$. Sum $x + y = 37$. Difference of squares $x^2 - y^2 = 185$. * **Calculation:** Expand the difference of squares identity: $x^2 - y^2 = (x - y)(x + y)$ * Substitute the known values into the equation: $185 = (x - y)(37)$ * Solve for the difference $(x - y)$: $x - y = 185 / 37$ * Divide 185 by 37: $x - y = 5$ Exam Strategy & Shortcut **Direct Formula Application:** Recognize instantly that $(x^2 - y^2) / (x + y) = (x - y)$. Simply divide the difference of the squares by the sum of the numbers to get the difference of the numbers. $185 / 37 = 5$. This takes 5 seconds and requires no paper. Common Pitfall Students sometimes try to solve for $x$ and $y$ individually by creating a quadratic equation or guessing numbers. This wastes immense time when the identity directly yields the required difference. Final Answer **Therefore, the correct answer is 5.**