101 × 99 = 9999. This is evaluated using the algebraic identity (a + b)(a − b) = a² − b². Writing 101 = (100 + 1) and 99 = (100 − 1), we get (100 + 1)(100 − 1) = 100² − 1² = 10000 − 1 = 9999.
101 × 99 = 9999.
Identity used: (a+b)(a−b) = a²−b².
101 = (100+1), 99 = (100−1) → (100²−1²) = 10000−1 = 9999.
Similarly: 83×77 = (80²−3²) = 6400−9 = 6391.
97×103 = (100²−3²) = 10000−9 = 9991.
49×51 = (50²−1²) = 2500−1 = 2499.
Method: Using identity (a + b)(a − b) = a² − b²
Step 1: Rewrite 101 and 99 101 = 100 + 1 99 = 100 − 1
Step 2: Apply the identity 101 × 99 = (100 + 1)(100 − 1) = 100² − 1² = 10000 − 1 = 9999
Answer: 101 × 99 = 9999
Verification (direct multiplication): 101 × 99 = 101 × (100 − 1) = 10100 − 101 = 9999 ✓
Identity: (a + b)(a − b) = a² − b²
Evaluate 83 × 77: 83 = 80 + 3, 77 = 80 − 3 83 × 77 = (80 + 3)(80 − 3) = 80² − 3² = 6400 − 9 = 6391
Evaluate 97 × 103: 97 = 100 − 3, 103 = 100 + 3 97 × 103 = (100 − 3)(100 + 3) = 100² − 3² = 10000 − 9 = 9991
Evaluate 49 × 51: 49 = 50 − 1, 51 = 50 + 1 49 × 51 = 50² − 1² = 2500 − 1 = 2499
Evaluate 1.8 × 2.2: 1.8 = 2 − 0.2, 2.2 = 2 + 0.2 1.8 × 2.2 = 2² − 0.2² = 4 − 0.04 = 3.96
Key insight: When two numbers are equidistant from a round number, use (a+b)(a−b) = a²−b² for quick calculation.
Using (a+b)(a−b) = a²−b²: 101×99 = (100+1)(100−1) = 100²−1² = 10000−1 = 9999.
Using (a+b)(a−b): 83×77 = (80+3)(80−3) = 80²−3² = 6400−9 = 6391.
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