Exercise 1 – Animated Solutions തുകകളുടെ ഗുണനം
Three questions, all built on one idea from Section 1: a product of two sums is the sum of four products, and in particular (x + 1)(y + 1) = xy + x + y + 1. For each question, read it, try it in your notebook (use the hint if you are stuck), then press Next ▶ to watch the solution grow step by step.
- Key idea 2: (x + y)(u + v) = xu + xv + yu + yv – the four pieces of a rectangle (page 5 of this guide).
- Next numbers: (x + 1)(y + 1) = xy + x + y + 1, that is product + sum + 1.
- Two more: (x + 2)(y + 2) = xy + 2(x + y) + 4. Halves: (x + ½)(y + ½) = xy + ½(x + y) + ¼.
- Odd numbers (ഒറ്റസംഖ്യകൾ) go up in steps of 2: after 23 comes 25, so "the next odd number" is x + 2 (page 2 of this guide).
Question 1 · Five products in your head
Q1Do the following calculations mentally and find the answers: (i) 71 × 91 (ii) 42 × 62 (iii) 10½ × 6½ (iv) 9.5 × 3.5 (v) 10¼ × 6¼
(ചുവടെയുള്ള കണക്കുകൾ മനസ്സിൽത്തന്നെ ചെയ്ത് ഉത്തരം കണ്ടുപിടിക്കുക.)
Write each number as a round number plus a small extra: 71 = 70 + 1, 91 = 90 + 1. Then the product is (round × round) + (round × extra) + (extra × round) + (extra × extra). For (iv) remember that 9.5 = 9 + ½ and 3.5 = 3 + ½.
(i) and (ii) use product + a × sum + a² with a = 1 and a = 2: 6300 + 160 + 1 and 2400 + 200 + 4. (iii) and (iv) use a = ½: 60 + 8 + ¼ and 27 + 6 + ¼. (v) uses a = ¼: 60 + 4 + 1/16. One identity, five problems.
Question 2 · The product of the next numbers
Q2The product of two natural numbers is 1400 and their sum is 81. What is the product of the two natural numbers next to each of them?
Call the numbers x and y. You know xy and x + y. The next numbers are x + 1 and y + 1, and (x + 1)(y + 1) = xy + x + y + 1. You never need x and y themselves.
A shop sells x boxes at ₹y each for ₹1400 in total, and x + y = 81. Next week it sells one more box at ₹1 more each: the takings are 1400 + 81 + 1 = ₹1482, without ever knowing x and y. Identities let you reason with quantities you do not know exactly – that is the whole power of algebra.
Question 3 · The next odd numbers
Q3The product of two odd numbers is 621 and their sum is 50. What is the product of the odd numbers next to each of them?
The odd number after x is not x + 1 (that is even) but x + 2. So you need (x + 2)(y + 2) = xy + 2x + 2y + 4 = xy + 2(x + y) + 4.
Using (x + 1)(y + 1) here gives 621 + 50 + 1 = 672 = 24 × 28 – the product of the next even numbers, not the next odd ones. Read "next odd number" as "+ 2".
Make your own problem
Tell a friend only the product and the sum, and ask for the product of the numbers "a more". The identity (x + a)(y + a) = xy + a(x + y) + a² gives the answer directly. Step 1 is Question 2; step 2 (for odd numbers) is Question 3.
Check yourself
- Mental products: split each number into a round part and a small extra, then add the four pieces.
- (x + 1)(y + 1) = xy + x + y + 1 – next numbers: product + sum + 1.
- (x + 2)(y + 2) = xy + 2(x + y) + 4 – next odd (or even) numbers.
- (x + ½)(y + ½) = xy + ½(x + y) + ¼ and (x + ¼)(y + ¼) = xy + ¼(x + y) + 1/16 – mixed numbers.