Difficulty: Hard
Correct Answer: 100
Explanation:
Introduction:
This problem tests recognizing and applying a standard algebraic identity involving a^3 + b^3 + c^3 - 3abc. When combined with the denominator a^2 + b^2 + c^2 - ab - bc - ca, the ratio often simplifies dramatically. Instead of expanding large cubes and squares, you use the identity to reduce the expression to a + b + c (provided a + b + c is not zero). This saves huge computation and is a classic simplification trick in aptitude and algebra tests.
Given Data / Assumptions:
Concept / Approach:
Apply the identity directly. The whole fraction becomes (a + b + c)(denominator) / (denominator) = a + b + c, as long as the denominator is not zero (equivalently, as long as a, b, c are not all equal and a + b + c is not forcing cancellation). Here the numbers are distinct, so the simplification is valid.
Step-by-Step Solution:
Let a = 37, b = 35, c = 28
Use identity: a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)
So the given fraction = (a + b + c)(denominator) / (denominator)
This simplifies to a + b + c
a + b + c = 37 + 35 + 28 = 100
Verification / Alternative check:
A quick sanity check: since the expression is built exactly in the identity form, the simplification must be linear (not huge). Getting 100 matches the sum 37+35+28, confirming correct use of the identity.
Why Other Options Are Wrong:
0 or 1 would occur only if a + b + c were 0 or 1, which it is not. 4 and 64 are random outcomes from partial cancellation or incorrect identity use. Only 100 matches the identity result.
Common Pitfalls:
Trying to expand cubes and squares (leading to arithmetic errors), forgetting the exact denominator structure, or misremembering the identity as (a+b+c)^3 - 3abc.
Final Answer:
The expression simplifies to 100.
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