The sum of the squares of two numbers is 3341, and the difference of their squares is 891. What are the two numbers?

Introduction:
This problem focuses on the algebraic properties of squares of numbers. You are given the sum of the squares of two numbers and the difference of their squares, and you must find the numbers themselves. The question is a good application of simultaneous equations and uses the identities involving a^2 + b^2 and a^2 - b^2 together.

Given Data / Assumptions:

• Let the two numbers be a and b, with a > b.
• a^2 + b^2 = 3341.
• a^2 - b^2 = 891.
• Both a and b are real numbers; in this context they turn out to be positive integers.

Concept / Approach:
We have two equations in terms of a^2 and b^2. Adding the equations will eliminate b^2 and give a direct expression for a^2. Subtracting appropriately will then yield b^2. Once a^2 and b^2 are known, we take square roots to obtain a and b. This is a very standard approach when dealing with sums and differences of squares.

Step-by-Step Solution:
Given: a^2 + b^2 = 3341 … (1).And a^2 - b^2 = 891 … (2).Add (1) and (2): (a^2 + b^2) + (a^2 - b^2) = 3341 + 891.2a^2 = 4232.a^2 = 4232 / 2 = 2116.a = √2116 = 46 (taking the positive root for this context).Substitute a^2 back into (1): 2116 + b^2 = 3341.b^2 = 3341 - 2116 = 1225.b = √1225 = 35.Thus the two numbers are 35 and 46.

Verification / Alternative check:
Check the sum of squares: 35^2 + 46^2 = 1225 + 2116 = 3341, correct. Check the difference of squares: 46^2 - 35^2 = 2116 - 1225 = 891, also correct. Both given conditions are satisfied, confirming the result.

Why Other Options Are Wrong:
35 and 50; 40 and 55; 45 and 60; 38 and 45: none of these pairs simultaneously satisfy both the sum 3341 and the difference 891 when you compute their squares.

Common Pitfalls:
Trying to directly solve for a and b instead of working with a^2 and b^2 first.Making arithmetic mistakes while adding 3341 and 891 or subtracting 2116 from 3341.Forgetting to consider that the numbers must match both equations simultaneously, not just one.

Final Answer:
35 and 46