Doubling and halving

In your book, Maths Mela (Chapter 13, The Transport Museum), we use one answer we know to find new ones. We know 11 × 100 = 1100. If we double the number of groups, 22 × 100 = 2200. If instead we double the size of each group, 11 × 200 = 2200 too. And if we double both, 22 × 200 = 4400.

Halving works the same way. 14 × 500 = 7000. Half as many groups: 7 × 500 = 3500. Now make each group half as big as well: 7 × 250 = 1750. We know 80 × 100 = 8000, so 80 × 50 = 4000 (50 is half of 100), and 40 × 50 = 2000 (40 is half of 80).

Here is a surprise. If we double one number and halve the other, the answer does not change. 44 × 10 = 440, and 22 × 20 = 440 too: 22 is half of 44, and 20 is double 10. In the same way, 16 × 100 = 1600, and 4 × 400 = 1600: 4 is a quarter of 16, and 400 is four times 100.