Difficulty: Easy
Correct Answer: 174840
Explanation:
Introduction / Context:
This problem tests algebraic simplification and smart substitution. The expression x^2 − y^2 + 10xz + 10yz looks large, but it becomes very easy once you notice that x and y are equal. When x = y, the difference of squares x^2 − y^2 becomes zero immediately, eliminating a big part of the expression. The remaining terms 10xz and 10yz are also easy because z is a known constant and x and y are the same value. These questions are common in aptitude sections because they reward observation (spotting x = y) and clean arithmetic rather than long calculations. Another key idea is factoring: 10xz + 10yz can be factored as 10z(x + y), which often reduces work further. The final result must be an exact integer, so careful multiplication is the only remaining task.
Given Data / Assumptions:
Concept / Approach:
Use the fact that if x = y then x^2 − y^2 = 0. Then combine the remaining terms:
10xz + 10yz = 10z(x + y) or, since x = y, it becomes 20xz. Finally compute the product exactly.
Step-by-Step Solution:
1) Substitute x = 93 and y = 93 into the first part:
x^2 − y^2 = 93^2 − 93^2 = 0
2) Now evaluate the remaining terms:
10xz + 10yz = 10*93*94 + 10*93*94
3) Combine identical terms:
= 20*93*94
4) Multiply 93*94:
93*94 = 8742
5) Multiply by 20:
20*8742 = 174840
Verification / Alternative check:
Factor-check method: 10xz + 10yz = 10z(x + y). Since x + y = 186 and z = 94, we get 10*94*186 = 940*186 = 174840. Same answer, confirming correctness.
Why Other Options Are Wrong:
• 147840 and 184740: typically come from incorrect multiplication of 93*94.
• 104784: often happens if you mistakenly compute only 10xz once and forget 10yz.
• 194840: indicates a wrong intermediate product or an addition mistake.
Common Pitfalls:
• Forgetting that x^2 − y^2 becomes 0 when x = y.
• Missing one of the two identical terms 10xz and 10yz.
• Arithmetic slips in 93*94 or multiplying by 20.
Final Answer:
174840
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