The sum of the two positive numbers is $ 55 $ and their difference is $ 19 $. What are the numbers?

Question

Examoogle · Accepted Answer

The correct answer is Option C: 37, 18. Add the two numbers and divide by 2 to get the larger number, then subtract the difference from the sum's half or simply subtract the smaller number from the total sum: $$(55 + 19) / 2 = 37$$ and $$(55 - 19) / 2 = 18$$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Add the two numbers and divide by 2 to get the larger number, then subtract the difference from the sum's half or simply subtract the smaller number from the total sum: $$(55 + 19) / 2 = 37$$ and $$(55 - 19) / 2 = 18$$.

Formula Used: $$x + y = 55, \quad x - y = 19$$

Given: Sum of two positive numbers = 55, Difference between the same two numbers = 19

Steps: Set up equations: Let the two positive numbers be $x$ and $y$. Based on the problem, we have two equations: the sum is 55 ($x + y = 55$) and the difference is 19 ($x - y = 19$).. Solve for the first variable: Add both equations together to eliminate $y$: $(x + y) + (x - y) = 55 + 19$, which simplifies to $2x = 74$.. Solve for the second variable: Substitute the value of $x$ back into the first equation: $37 + y = 55$, so $y = 55 - 37 = 18$.

Answer: C is correct because solving the system of equations $x+y=55$ and $x-y=19$ yields the numbers 37 and 18.

Exam Tip: This concept of 'Sum and Difference' is a fundamental technique used in solving ages, speed-distance-time, and mixture problems across all competitive exams.

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