49 × 51 in Your Head? Cut One Strip and It's Easy

Step by step

1. No calculator: 49 × 51 in your head?
   Put the calculator away. Forty-nine times fifty-one, in your head. Ready?
2. A 49 × 51 garden · first a small one: 9 × 11
   Picture a garden forty-nine feet by fifty-one feet. Let's try a smaller one first: nine by eleven.
3. Way 1: cut off the top row (9 squares)
   Way one: cut off the top row. It's nine squares long.
4. Stand it up on the right → 10 × 10, one corner missing
   Stand it up along the right side. Now it's ten by ten, with one corner square missing.
5. 10 × 10 − 1 = 99 → 9 × 11 = 99
   Ten times ten is one hundred, minus one is ninety-nine. So nine times eleven is ninety-nine!
6. 49 × 51 = 50 × 50 − 1 = 2499
   The big garden works the same way: fifty by fifty, minus one corner. Twenty-five hundred minus one is twenty-four ninety-nine!
7. Way 2: (50 − 1) × (50 + 1), multiply each part
   Way two is school algebra. Forty-nine is fifty minus one, and fifty-one is fifty plus one. Multiply each part.
8. 2500 + 50 − 50 − 1 = 2500 − 1
   Fifty times fifty is twenty-five hundred. Plus fifty and minus fifty cancel out. Minus one times one is minus one. So it's twenty-five hundred minus one!
9. Big square − little square: the difference of squares
   That's the picture again: a big square, minus a little corner square. It's called the difference of squares.
10. 48 × 52 = 2500 − 4 · 47 × 53 = 2500 − 9
    It works for any two numbers the same distance from the middle. Forty-eight times fifty-two is twenty-five hundred minus four. Forty-seven times fifty-three is minus nine.
11. Middle² − distance²
    So find the middle and square it. Then take away the distance, squared.
12. On a test: (50 − 1)(50 + 1) = 50² − 1² = 2499 sq ft
    On a test, write it like this. Write forty-nine and fifty-one around the middle, fifty. Use the difference of squares. Square each part, then subtract. The garden is twenty-four ninety-nine square feet.