Doubling and halving together
Last week we discussed ways of halving or dividing a number by two. The week before that we dealt on doubling or multiplying a number by two.
We will now apply the two techniques together in a Sutra called Proportionately to simplify multiplication.
For example, many could not easily solve 18 x 4 mentally. However, if we halve 18, we get 9 and if we double 4, we get 8, that is, 18 x 4 = 9 x 8. This is easily solved as 72.
The easy way to multiply by 5 is illustrated by this technique. 26 x 5 = 13 x 10 = 130. (We will discuss more of multiplication by 5 in a future post).
Doubling and halving together works well when one factor is even, and one factor ends in 5:
24 x 35 = 12 x 70 = 840
44 x 75 = 22 x 150 = 11 x 300 = 3300
28 x 275 = 14 x 550 = 7 x 1100 = 7700
64 x 625 = 32 x 1250 = 16 x 2500 = 8 x 5000 = 4 x 10000 = 40,000
They can also lead to other multiplication techniques.
49 x 196 = 98 x 98 = 9604
23 x 92 = 46 x 46 = 2116
The doubling and halving can have other variations:
8 x 16 = 8 x (8 x 2) = (8 x 8) x 2 = 64 x 2 = 128
18 x 14 = (9 x 2) x (7 x 2) = (9 x 7) x 2 x 2 = 63 x 2 x 2 = 126 x 2 = 252
Try using halving and/or doubling to multiply these numbers mentally:
- 4 x 14 =
- 24 x 6 =
- 18 x 45 =
- 6 x 85 =
- 14 x 16 =
- 208 x 54 =
- 408 x 26 =
- 46 x 194 =
- 25 x 48 =
- 225 x 72 =
