Chapter 4 · Sections 4–5Textbook pages 74–82

Multiplying Differences വ്യത്യാസ ഗുണനം

Section 1 multiplied sums. But 26 × 17 is much easier to think of as (30 − 4)(20 − 3): numbers just below round numbers. This page finds the identity for a difference times a difference, (x − y)(u − v) = xu − xv − yu + yv, then a sum times a difference, (x + y)(u − v) = xu − xv + yu − yv, and uses them to find two numbers from the products of their neighbours.

📖 What you already know (from pages 1–5 of this guide)

(x + y)(u + v) = xu + xv + yu + yv, and (x + 1)(y + 1) = xy + x + y + 1 (page 5).
Numbers can be written near round numbers: 26 = 30 − 4, 17 = 20 − 3, 99 = 100 − 1 (page 4).
Subtracting a bracket: a − (b − c) = a − b + c; the trick 612 − 170 = 612 − 200 + 30 (page 4).
(x − y)² = (x + y)² − 4xy, and from a sum and a difference you can find the two numbers (page 3).

1. One difference times a number: 18 × 5

18 × 5 can be done as (10 + 8) × 5 = 50 + 40 = 90. But 18 is also 20 − 2, and 20 × 5 is easier than 18 × 5.

🎬 Animation: 18 × 5 = (20 − 2) × 5

The same trick: 38 × 5 = (40 − 2) × 5 = 200 − 10 = 190 and 45 × 8 = 45 × (10 − 2) = 450 − 90 = 360.

🎚️ Round up, then subtract

number times

Pick a number just below a multiple of ten (38, 47, 69, …). Multiplying the round number is easy; then take away the small extra piece.

2. A difference times a difference: 26 × 17 as a picture

Write 26 = 30 − 4 and 17 = 20 − 3. The rectangle 26 × 17 sits inside a 30 × 20 rectangle (area 600). What must we take away?

🎬 Animation: 26 × 17 inside 30 × 20 (textbook page 75)

3. In algebra: (x − y)(u − v)

Replace the numbers with letters: the big rectangle is x by u, the piece we want is (x − y) by (u − v). The textbook reads the equation off the picture and then rearranges it.

🎬 Animation: from the picture to the identity (textbook pages 76–77)

⭐ Key idea 3 (textbook page 77)

x > y, u > v ആയ ഏതു അധിസംഖ്യകളെടുത്താലും

(x − y)(u − v) = xu − xv − yu + yv

For any positive numbers with x > y and u > v. Notice the signs: the two "mixed" products xv and yu are subtracted, the product of the two small numbers yv is added back (it was subtracted twice).

Now 26 × 17 in one line: (30 − 4)(20 − 3) = 600 − 90 − 80 + 12 = 612 − 170. And 612 − 170 is easiest as 612 − 200 + 30 = 442.

🧮 Nearby round numbers calculator

first number second number

Each number is rounded up to the next ten. Blue is the product we want; red, green and amber are the pieces to subtract (the amber corner comes back once). The picture is schematic, not to scale.

Practice from the textbook (page 77)

Try each one in your notebook first, then open the working.

(i) 38 × 49

= (40 − 2)(50 − 1) = 2000 − 40 − 100 + 2 = 1862

(ii) 47 × 99

= (50 − 3)(100 − 1) = 5000 − 50 − 300 + 3 = 4653

(iii) 29 × 46

= (30 − 1)(50 − 4) = 1500 − 120 − 50 + 4 = 1334

(iv) 9½ × 19½

= (10 − ½)(20 − ½) = 200 − 5 − 10 + ¼ = 185¼

4. The "previous numbers" rule: (x − 1)(y − 1)

29 × 49 = (30 − 1)(50 − 1) = 1500 − 30 − 50 + 1 = 1421. The identity behind it:

(x − 1)(y − 1) = xy − x − y + 1 = xy − (x + y) + 1

In words: the product of the numbers one before = product − sum + 1. (Compare page 5: the product of the numbers one after = product + sum + 1.)

🎬 Animation: why the corner comes back

5. Finding two numbers from their neighbours' products

Textbook page 78: the product of the numbers one after two numbers is 525, and the product of the numbers one before them is 437. What are the numbers?

🎬 Animation: add, subtract, then use a Class 8 identity

💡 The pattern of the method

Two equations with the same unknown parts xy and (x + y): adding them kills (x + y) and gives xy; subtracting kills xy and gives x + y. Then (x − y)² = (x + y)² − 4xy (Class 8, page 3 of this guide) gives the difference, and sum ± difference gives the two numbers. Exercise 4 uses exactly this method.

6. A sum times a difference: (x + y)(u − v)

What if one number is just above a round number and the other just below? 14 × 59 = (10 + 4)(60 − 1). Here is the picture: a tall rectangle with a strip cut off the top.

🎬 Animation: (x + y)(u − v) (textbook page 80)

⭐ Key idea 4 (textbook page 80)

x, y, u, v എന്ന ഏതു നാലു അധിസംഖ്യകളിലും, u > v ആണെങ്കിൽ

(x + y)(u − v) = xu − xv + yu − yv

The algebra: (x + y)(u − v) = x(u − v) + y(u − v) = xu − xv + yu − yv. Each term of the sum multiplies the whole difference.

So 14 × 59 = (10 + 4)(60 − 1) = 600 − 10 + 240 − 4 = 590 + 236 = 826.

Practice from the textbook (page 81)

(i) 52 × 19

= (50 + 2)(20 − 1) = 1000 − 50 + 40 − 2 = 988

(ii) 101 × 48

= (100 + 1)(50 − 2) = 5000 − 200 + 50 − 2 = 4848

(iii) 97 × 102

= (100 − 3)(100 + 2) = 10000 + 200 − 300 − 6 = 9894

(iv) 9¾ × 20½

= (10 − ¼)(20 + ½) = 200 + 5 − 5 − ⅛ = 199⅞

7. One up, one down

8 × 5 = 40. Add 1 to the bigger number and take 1 from the smaller: 9 × 4 = 36 – smaller. The other way: 7 × 6 = 42 – bigger. Why? With the bigger number x and the smaller y:

(x + 1)(y − 1) = xy − x + y − 1 = xy − (x − y) − 1 (x − 1)(y + 1) = xy + x − y − 1 = xy + (x − y) − 1

🎚️ Move one up and one down

bigger number x smaller number y

Spreading the numbers apart (bigger up, smaller down) always lowers the product; bringing them together raises it – except for consecutive numbers (x − y = 1), where (x − 1)(y + 1) = xy + 1 − 1 = xy: no change at all. Try x = 8, y = 7.

🌍 Real life

A shop sells 99 notebooks at ₹47: 100 × 47 − 47 = ₹4653, no calculator. A customer returns one of 19 items priced ₹52: the bill is 52 × 18 = 52 × 19 − 52. A tailor trims 4 cm off a 30 cm width and 3 cm off a 20 cm length of cloth: the remaining piece is (30 − 4)(20 − 3) = 600 − 90 − 80 + 12 = 442 sq cm. Whenever something is "a little less than a round amount", the difference identities turn the sum into mental arithmetic.

8. Check yourself

🧭 Summary of Sections 4 and 5

A number times a difference: 18 × 5 = (20 − 2) × 5 = 100 − 10.
(x − y)(u − v) = xu − xv − yu + yv: 26 × 17 = (30 − 4)(20 − 3) = 600 − 90 − 80 + 12 = 442.
(x − 1)(y − 1) = xy − (x + y) + 1: product of the previous numbers = product − sum + 1.
Given (x + 1)(y + 1) and (x − 1)(y − 1): add → xy, subtract → x + y, then (x − y)² = (x + y)² − 4xy.
(x + y)(u − v) = xu − xv + yu − yv: 14 × 59 = (10 + 4)(60 − 1) = 826.
(x + 1)(y − 1) = xy − (x − y) − 1 and (x − 1)(y + 1) = xy + (x − y) − 1 (x the bigger number).

Next: Exercises 4 and 5 – all eight questions solved with animations ▶