4. Operators □, ◊ and → are defined by:
a □ b =a − b /(a + b), a ◊ b =a + b /(a − b), a → b = ab
Find the value of
(66 □ 6) → (66 ◊ 6).
(A) −2
(B) −1
(C) 1
(D) 2
The operators □, ◊, and → are custom-defined binary operations that require step-by-step substitution to evaluate the expression (66 □ 6) → (66 ◊ 6), which simplifies to 1.
Step-by-Step Solution
- First, compute 66 □ 6 using the definition
a □ b = (a - b) / (a + b).
Substitutea = 66, b = 6:66 □ 6 = (66 - 6) / (66 + 6) = 60 / 72 = 5/6. - Next, compute 66 ◊ 6 using
a ◊ b = (a + b) / (a - b).
Substitute:66 ◊ 6 = (66 + 6) / (66 - 6) = 72 / 60 = 6/5. - Finally, apply → where
a → b = a * b.
Thus,(66 □ 6) → (66 ◊ 6) = (5/6) * (6/5) = 1.
Option Analysis
(5/6) * (6/5) yields positive 1, not negative.Introduction (SEO Optimized)
Cracking the 66 □ 6 → 66 ◊ 6 operators puzzle is essential for competitive exams like GATE. These custom operators—□ as (a-b)/(a+b), ◊ as (a+b)/(a-b), and → as ab—test precise substitution skills. This guide delivers the 66 □ 6 → 66 ◊ 6 operators solution with calculations, eliminating all wrong options for a perfect score.
Operator Definitions Breakdown
- □ simplifies differences over sums, yielding
60/72 = 5/6for 66 □ 6. - ◊ flips to sums over differences, giving
72/60 = 6/5for 66 ◊ 6. - → is simple multiplication.
Why Option C (1) Wins
The key insight: (5/6) × (6/5) cancels perfectly to 1, proving exam traps like negatives or doubles fail.
Exam Tips for Similar Problems
- Always substitute step-by-step.
- Simplify fractions early.
- Practice reciprocal patterns in custom ops.
