Multiplying Sums തുകകളുടെ ഗുണനം
You already multiply numbers like 26 × 15 every day. This section looks inside that multiplication, using the area of a rectangle, and finds a rule that works for all numbers: (x + y)(u + v) = xu + xv + yu + yv. Then it turns the rule into mental-maths shortcuts like 31 × 51 = 1581 in your head.
- 26 = 20 + 6: a number is a sum of its place values.
- Area of a rectangle = length × breadth, and a rectangle cut into pieces has area = sum of the pieces.
- a(b + c) = ab + ac (the distributive law); letters stand for any numbers.
- 6½ = 6 + ½, and ½ × ½ = ¼.
1. One sum times a number: 26 × 5
What is the area of a rectangle 26 cm long and 5 cm wide? Instead of multiplying directly, write 26 = 20 + 6 and cut the rectangle.
2. A sum times a sum: 26 × 15 in four pieces
Now take 26 cm × 15 cm. Write both numbers as sums: 26 = 20 + 6 and 15 = 10 + 5. Cutting both ways gives four rectangles.
When you multiply 26 × 15 the usual way, the two rows you write are exactly these four areas, added in a different order:
Each row is a sum times a number; stacking the rows adds all four pieces: 30 + 100 + 60 + 200 = 390.
3. It works for fractions too
The numbers in the sums need not be whole numbers. Mixed numbers are sums as well:
6½ × 8⅓ = (6 + ½)(8 + ⅓) = (6 × 8) + (6 × ⅓) + (½ × 8) + (½ × ⅓) = 48 + 2 + 4 + ⅙ = 54⅙തുകയെ തുക കൊണ്ടു ഗുണിക്കാൻ, ആദ്യത്തെ തുകയിലെ ഓരോ സംഖ്യയെയും രണ്ടാമത്തെ തുകയിലെ ഓരോ സംഖ്യകൊണ്ടും ഗുണിച്ച്, ഗുണനഫലങ്ങളെല്ലാം കൂട്ടണം.
To multiply a sum by a sum, multiply each number of the first sum by each number of the second sum, and add all the products.
4. In algebra: (x + y)(u + v)
Call the first sum x + y and the second u + v. The four products are xu, xv, yu, yv.
x, y, u, v എന്ന ഏതു നാല് അധിസംഖ്യകളെടുത്താലും
(x + y)(u + v) = xu + xv + yu + yvFor any four positive numbers x, y, u, v. (It is in fact true for all numbers, as you will see in later classes.)
Whatever the four numbers, the total area equals the sum of the four pieces. The textbook colours: xv (red, top-left), yv (yellow, top-right), xu (cream, bottom-left), yu (orange, bottom-right).
5. Mental maths: 31 × 51 and the "next numbers" rule
30 × 50 = 1500 is easy. Then 31 × 51 = (30 + 1)(50 + 1) = 1500 + 30 + 50 + 1 = 1581. The identity behind this:
(x + 1)(y + 1) = xy + x + y + 1In words: if you know the product of two numbers, the product of the next two numbers is product + sum + 1.
Three examples from the textbook
| Product | Known product | + sum | + 1 | Answer |
|---|---|---|---|---|
| 21 × 71 | 20 × 70 = 1400 | + 20 + 70 | + 1 | 1491 |
| 81 × 91 | 80 × 90 = 7200 | + 80 + 90 | + 1 | 7371 |
| 201 × 401 | 200 × 400 = 80000 | + 200 + 400 | + 1 | 80601 |
The slider gives round numbers x and y. Compute (x + 1)(y + 1) in your head first, then read the working.
(x + 2)(y + 2) = xy + 2x + 2y + 4 = xy + 2(x + y) + 4. So 22 × 72 = 1400 + 2 × 90 + 4 = 1584. In general (x + a)(y + a) = xy + a(x + y) + a².
6. Halves: (x + ½)(y + ½)
The same identity with ½ in place of 1:
(x + ½)(y + ½) = xy + ½x + ½y + ¼ = xy + ½(x + y) + ¼- 6½ × 8½ = 48 + ½(6 + 8) + ¼ = 48 + 7 + ¼ = 55¼
- 10½ × 5½ = 50 + ½(10 + 5) + ¼ = 50 + 7½ + ¼ = 57¾
A shopkeeper selling 31 items at ₹51 each does not reach for a calculator: 30 × 50 = 1500, plus 30, plus 50, plus 1 – ₹1581. A tailor needing 6½ m of cloth at ₹8½ per metre thinks 48 + 7 + ¼ = ₹55.25. Engineers estimating the area of a 20.5 m × 10.5 m hall do the same: 200 + ½(31) + ¼ = 215.25 sq m. Breaking numbers into convenient sums is how fast mental arithmetic works everywhere.
7. Check yourself
- A rectangle (x + y) by (u + v) splits into four rectangles: (x + y)(u + v) = xu + xv + yu + yv.
- To multiply a sum by a sum: every term of the first times every term of the second, then add.
- (x + 1)(y + 1) = xy + x + y + 1: product of the next numbers = product + sum + 1 (31 × 51 = 1500 + 80 + 1).
- (x + 2)(y + 2) = xy + 2(x + y) + 4; (x + ½)(y + ½) = xy + ½(x + y) + ¼.
- The column method of multiplication is this same picture, two rows at a time.
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