The question you opened
A field 47 by 53
A rectangular field measures 47 metres by 53 metres.
What is its area in square metres?
2,491 square metres
Both numbers sit three away from 50. Fifty squared is 2,500 and three squared is 9, so the area is 2,500 minus 9, which is 2,491.
Forty seven and 53 are both three away from 50, one below and one above. That makes their product 50 squared minus 3 squared, so 2,500 minus 9, which is 2,491. One subtraction replaces a full multiplication. The identity is that a minus b times a plus b equals a squared minus b squared.
Any pair of numbers symmetric about a round centre qualifies. Ninety eight times 102 is 10,000 minus 4, so 9,996. Nineteen times 21 is 400 minus 1, so 399. Thirty four times 46 is 40 squared minus 6 squared, so 1,600 minus 36, which is 1,564. Finding the centre is the only real step. Add the two numbers and halve: 47 plus 53 is 100, half of which is 50.
The distance is the difference between either number and that centre. If the halving lands on something awkward, the method still works but stops being faster. It also gives a fact worth carrying: among all rectangles with a fixed perimeter, the square has the largest area. A 50 by 50 field is 2,500 square metres, and stretching it to 47 by 53 loses 9. Stretching it to 40 by 60 loses 100, and 10 by 90 loses 1,600.
The loss is always the square of how far the shape strays from even, which is why fencing a square costs the least per unit of enclosed land. The habit is to look at any multiplication and ask whether the two numbers sit either side of something round. When they do, the answer is one square and one subtraction, and both of those can be done while walking.
Technique: Difference of squares around 50