Summary, Formula Sheet & Quick Quiz പുനരവലോകനം
All the identities of the chapter with their pictures, the number facts they prove, the recipe for "find the numbers" problems, the mistakes to avoid, flashcards, and a scored quiz to find out what needs another look.
1. The four identities
Sum × sum: every term times every term, all added. 26 × 15 = (20 + 6)(10 + 5) = 200 + 100 + 60 + 30.
Difference × difference: remove two strips from the big rectangle, add the corner back. 26 × 17 = (30 − 4)(20 − 3) = 600 − 90 − 80 + 12 = 442.
Sum × difference: 14 × 59 = (10 + 4)(60 − 1) = 600 − 10 + 240 − 4 = 826.
(x − 1)(y − 1) = xy − (x + y) + 1
(x + 1)(y − 1) = xy − (x − y) − 1
(x − 1)(y + 1) = xy + (x − y) − 1
Next numbers: product + sum + 1 (31 × 51 = 1500 + 80 + 1). Previous numbers: product − sum + 1. One up, one down (x the bigger): product ∓ difference − 1.
(x + ½)(y + ½) = xy + ½(x + y) + ¼
(x + 2)(y + 2) = xy + 2(x + y) + 4
6½ × 8½ = 48 + 7 + ¼ = 55¼. Two odd numbers with product 621 and sum 50: next odd pair 621 + 100 + 4 = 725.
2. Number facts the identities prove
- Odd × odd = odd.
(2m + 1)(2n + 1) = 2(2mn + m + n) + 1. - Remainder 1 × remainder 1 (÷ 3) leaves remainder 1.
(3m + 1)(3n + 1) = 3(3mn + m + n) + 1. Likewise remainder 1 × remainder 2 leaves 2:(3m + 1)(3n + 2) = 3(3mn + 2m + n) + 2. - Four consecutive numbers: middles' product − ends' product = 2.
(x + 1)(x + 2) − x(x + 3) = 2. Six consecutive numbers: difference 6. - Calendar 2 × 2 square: diagonal products differ by 7.
(x + 1)(x + 7) − x(x + 8) = 7. In a grid with 5 columns the difference is 5. - Multiplication-table squares: diagonal sums differ by 1 (2 × 2), 4 (3 × 3), 9 (4 × 4). Cells
mn, mn + n, mn + m, mn + m + n + 1. - Digit reversal 12 × 63 = 21 × 36 happens exactly when
mp = nq(tens × tens = units × units), because the two products differ by99(mp − nq).
3. Recipe: finding two numbers from their neighbours' products
- Write the two facts as identities: e.g. (x + 1)(y + 1) = xy + (x + y) + 1 and (x − 1)(y − 1) = xy − (x + y) + 1.
- Add them: 2xy + 2 = (sum of the two products) ⇒ xy.
- Subtract them: 2(x + y) = (difference of the two products) ⇒ x + y. (For one-up-one-down products, subtracting gives x − y instead.)
- Use the Class 8 bridge: (x − y)² = (x + y)² − 4xy (or (x + y)² = (x − y)² + 4xy) to get the other one.
- Finish: x = ½(sum + difference), y = ½(sum − difference).
Textbook example: 525 and 437 ⇒ xy = 480, x + y = 44, (x − y)² = 1936 − 1920 = 16 ⇒ x − y = 4 ⇒ 24 and 20.
Try 861 and 741 (Exercise 4), 1271 and 1131, or 1056 and 930. If the result is not a whole number, no such pair of natural numbers exists.
4. Formula sheet
| Situation | Formula / fact | Where |
|---|---|---|
| Number as a sum / difference | 26 = 20 + 6 = 30 − 4; 99 = 100 − 1; 6½ = 6 + ½; 9½ = 10 − ½ | p. 1, p. 4 |
| Sum × sum | (x + y)(u + v) = xu + xv + yu + yv | p. 5 |
| Difference × difference | (x − y)(u − v) = xu − xv − yu + yv | p. 9 |
| Sum × difference | (x + y)(u − v) = xu − xv + yu − yv | p. 9 |
| Next / previous numbers | (x + 1)(y + 1) = xy + (x + y) + 1; (x − 1)(y − 1) = xy − (x + y) + 1 | p. 5, p. 9 |
| One up, one down (x > y) | (x + 1)(y − 1) = xy − (x − y) − 1; (x − 1)(y + 1) = xy + (x − y) − 1 | p. 9 |
| Same amount added | (x + a)(y + a) = xy + a(x + y) + a²; a = ½ gives xy + ½(x + y) + ¼ | p. 5 |
| Class 8 identities | (x + y)² = x² + 2xy + y²; (x − y)² = x² − 2xy + y²; (x + y)(x − y) = x² − y² | p. 3 |
| Sum, difference, product bridges | (x − y)² = (x + y)² − 4xy; (x + y)² = (x − y)² + 4xy; x = ½(s + d), y = ½(s − d) | p. 3 |
| General forms | even 2n; odd 2n + 1; remainder r on ÷ k: kn + r; two-digit 10m + n; consecutive x, x + 1, x + 2; calendar below x: x + 7 | p. 2, p. 7 |
| Path of width w around x × y | (x + 2w)(y + 2w) − xy = 2w(x + y) + 4w² | p. 11 |
| Squares ending in 5 | (10a + 5)² = 100a(a + 1) + 25: 35² = 1225 | p. 11 |
5. Common mistakes
- Forgetting the cross terms. (20 + 6)(10 + 5) is NOT 20 × 10 + 6 × 5. There are four products, not two.
- Sign slips in (x − y)(u − v). The last term is plus yv: (30 − 4)(20 − 3) = 600 − 90 − 80 + 12. "Minus times minus is plus" – the corner was removed twice, so it is added back.
- Previous numbers use − sum + 1. (x − 1)(y − 1) = xy − (x + y) + 1, not xy − (x + y) − 1. 29 × 49 = 1500 − 80 + 1 = 1421.
- Which number is bigger? In (x + 1)(y − 1) = xy − (x − y) − 1 the x is the number that went up. Decide first which one you call x.
- Rounding the wrong way. 47 is 50 − 3, not 40 + 7, if you want small corrections; both are right, but one is far easier.
- Mixed numbers. 6½ × 8½ is not 6 × 8 + ½ × ½. The middle terms ½ × 8 and 6 × ½ are needed: 48 + 4 + 3 + ¼ = 55¼.
- "It worked for three examples" is not a proof. Use a general form (2n + 1, x, x + 1, …) and the identity to prove a pattern for all numbers.
6. Flashcards പദങ്ങൾ
Click a card to flip it.
7. Scored quiz (10 questions)
Answer every question, then press Submit. Each wrong answer tells you which page to revisit.
8. Your study checklist
Pages you have marked as completed are ticked. Aim to tick them all before the model test.
If you scored 8 or more in the quiz and the checklist is complete, go to the Model Test Paper. Attempt it in 1½ hours without looking at the solutions.
Next: Model Test Paper ▶