Halve and double

In your book, Maths Mela (Chapter 6, The Dairy Farm), five children find 18 × 5 in five different ways, and all of them get 90. One child says: 18 × 5 = 9 × 10, so the answer is 90. Half of 18 is 9, and double 5 is 10. That is the trick of halving and doubling.

Why does it work? Think of butter packets set out in 3 rows of 18. Cut the rows in half and put the halves under each other: now there are 6 rows of 9. No packet was added or taken away, so 3 × 18 = 6 × 9 = 54. When we halve one number and double the other, the product stays the same.

This trick is best with 5 and 25. Double 5 is 10, and double 25 is 50, and double 50 is 100, and multiplying by 10 or 100 is easy. So 38 × 5 = 19 × 10 = 190, and 25 × 12 = 50 × 6 = 300. For 16 × 25, halve and double twice: 16 × 25 = 8 × 50 = 4 × 100 = 400.

If 16 × 44 = 704, then 8 × 88 = 704 too, because 16 was halved and 44 was doubled. But if only ONE number is doubled, the product doubles too: 32 × 44 is double 16 × 44, so it is 704 + 704 = 1,408.