Real-World Applications പ്രയോഗങ്ങൾ
The identities of this chapter are the secret behind fast mental arithmetic, bill checking, garden paths, discounts and a whole collection of number tricks you can show your friends. Every tool below is one of the four identities wearing different clothes.
- (x + y)(u + v) = xu + xv + yu + yv – break numbers into convenient sums (page 5).
- (x − y)(u − v) = xu − xv − yu + yv – use nearby round numbers that are too big (page 9).
- (x + y)(u − v) = xu − xv + yu − yv – one number rounded down, the other rounded up (page 9).
- General forms 2n + 1, 10m + n, x, x + 1, x + 7 – to prove that a trick always works (page 7).
1. Shop-counter mental maths
At a counter nobody writes 48 × 25 in columns. They think "50 × 25 = 1250, take away 2 × 25": 1200. That is (50 − 2) × 25 = 50 × 25 − 2 × 25. With two awkward numbers, round both and use the matching identity.
Each number is written as the nearest multiple of 10 plus or minus a small number, and the four products are added or subtracted according to the signs. Try 31 × 51, 99 × 37, 26 × 17, 102 × 48, 9 × 99.
2. One more item, one less item
You bought 24 pens at ₹13 each (₹312). One more pen? Just add ₹13: (x + 1)y = xy + y. One less? Subtract ₹13. If the shop also changes the price by ₹1, both change: (x + 1)(y + 1) = xy + x + y + 1 – add the old quantity, the old price, and one more rupee.
3. A path around a garden
A rectangular garden x by y gets a path of width w all round it. The outer rectangle is (x + 2w) by (y + 2w). The area of the path is the outer area minus the garden:
(x + 2w)(y + 2w) − xy = 2wx + 2wy + 4w² = 2w(x + y) + 4w²Two long strips, two short strips and the four corner squares – you can see every term in the picture.
4. Trimming a field or a cloth
A field x by y loses a strip of width a from its length and b from its breadth (for a road, a drain, a hem). The remaining area is (x − a)(y − b) = xy − bx − ay + ab. Notice the + ab: the corner piece was subtracted twice, so it has to be added back once.
5. Up 10 %, then down 10 %: you lose
A price goes up by 10 % and then comes down by 10 %. Back where it started? No. Multiplying by (1 + a) and then by (1 − a) gives (1 + a)(1 − a) = 1 − a². With a = 0.1 that is 0.99: you lose 1 %. The bigger the swing, the bigger the loss, and the order does not matter.
6. Calendar magic for friends
Ask a friend to draw any 2 × 2 square on a calendar page. Multiply the diagonals: the two products always differ by 7. For any 3 × 3 square, the nine dates add up to 9 × the centre. Click a date below to place the square, and watch the algebra.
Why it works: in a 2 × 2 square x, x + 1, x + 7, x + 8 the diagonal products are x(x + 8) = x² + 8x and (x + 1)(x + 7) = x² + 8x + 7. In a 3 × 3 square with centre c the dates are c − 8, c − 7, c − 6, c − 1, c, c + 1, c + 6, c + 7, c + 8; the plus and minus parts cancel, leaving 9c.
7. The digit-reversal puzzle
12 × 63 = 756 and 21 × 36 = 756. Reverse both numbers and the product does not change! Not for every pair: it works exactly when (tens digit × tens digit) = (units digit × units digit), that is mp = nq for 10m + n and 10p + q. Type two two-digit numbers and test them.
8. Squaring a number ending in 5 – instantly
35² = 1225, 65² = 4225, 85² = 7225. The rule: take the tens digit a, multiply by the next number a + 1, and write 25 after it. 3 × 4 = 12 → 1225. Why? (10a + 5)² = 100a² + 100a + 25 = 100a(a + 1) + 25.
9. The vertical-cross speed method
The textbook's trick for 36 × 28: 3 × 2 = 6 → 600; (3 × 8) + (6 × 2) = 36 → 360; 6 × 8 = 48; total 1008. It is the identity (10m + n)(10p + q) = 100mp + 10(mq + np) + nq written as three rows. People who practise it multiply two-digit numbers faster than they can type them.
10. Checking a bill by rough products
Before paying ₹1,581 for 31 items at ₹51, estimate: 30 × 50 = 1500, plus roughly 30 + 50. So the bill should be a little above 1580 – it is. The identity tells you the size of the correction: (x + y)(u + v) = xu + (xv + yu) + yv, and the middle terms are the ones that matter.
11. Try these word problems
- Round each number to a convenient one, write it as a sum or a difference, and apply the matching identity – that is all mental multiplication is.
- Paths, frames and trimmed fields are the four-piece rectangle with pieces added or removed: watch the sign of the corner term.
- +a % then −a % multiplies by 1 − a²: a small loss every time.
- Number tricks (calendar, digit reversal, squares ending in 5, the cross method) are not magic – a general form like x, x + 1, x + 7 or 10m + n proves them.