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Proportion is a natural and easy concept which is fundamental to mathematics. It therefore offers some simple but very effective devices which we will be using throughout subsequent chapters. With Proportion we will also be able to extend considerably all the various formulas to come. The advantage of splitting numbers into convenient sections is also illustrated in this chapter.

MULTIPLICATION BY 4, 8, 16, 20, 40 ETC.

Doubling numbers is very easy, so in multiplying a number, by say, 4 we simply double the number twice.

Example 1 If for example we want 53×4, we double 53 to 106 and double it again to 212. So 53 × 4 = 212 Example 2 Also, for 225 × 4: twice 225 is 450, and twice 450 is 900. So 225 × 4 = 900

Of course we can double more than twice. For multiplication by 8 we would double 3 times:

Example 3 For 26 × 8 we get 52, 104, 208. So 26 × 8 = 208

And for multiplication by 16 we can double 4 times.

Example 4 For 76 × 16 we get 152, 304, 608, 1216. So 76 × 16 = 1216 In doubling 76 we double the 7 first, as discussed in the previous chapter: 14,12 = 152.

NUMBER SPLITTING

In doubling 152 above you may find it easiest to double 15 to 30 and 2 to 4, and get 304, thereby thinking of the number in two convenient parts rather than three: 152 × 2 = 15/2 × 2 = 304. This number splitting is very effective and will be in frequent use.

Example 5 Multiplying by 40, 800 etc is simply a matter of doubling the appropriate number of times and adding the appropriate number of noughts: 17 × 40? think 17, 34, 68, 680.

EXTENDING THE MULTIPLICATION TABLES

Example 6 Suppose we want 14 × 18. You may not know your 14 or 18 times tables, but you probably know that 7×9 = 63, and since 14 and 18 are double of 7 and 9 we can now simply double 63 twice: 126, 252. So 14 × 18 = 252 Example 7 Similarly for 14×16, as 7×8=56 therefore 14 × 16 = 224 (56 doubled twice) Example 8 For 14 × 7. Since 7×7=49, 14 ×7 = 98 Example 9 For 17 × 14 you can either multiply 17 by 7 and double the result, or find 16 14's and add another 14. In either case 17 × 14 = 238

MULTIPLICATION BY 5, 50, 25 ETC.

Halving numbers is also very easy, so rather than multiply by 5 we can put a 0 onto the number and halve it, because 5 is half of 10.

Example 10 So for 44 × 5 we find half of 440 which is 220 so 44 × 5 = 220 Example 11 Similarly, 68 × 5 = half of 680 = 340 Example 12 87 × 5 = half of 870 = 435 Example 13 452 × 5 = half of 4520 = 2260 Example 14 27 × 50 = half of 2700 = 1350

Since the halving of even numbers is to be preferred to the halving of odd numbers we may think of 2700 in this last example split as 2/70/0 so that 2 and 70 get halved to 1 and 35. In the example before that we think that half of 4/52/0 = 2/26/0.

For multiplication by 25 we multiply by 100 and halve twice, as 25 is half of half of 100.

Example 15 So for 82 × 25, half of 8200 is 4100, and half of 4100 is 2050 Example 16 For 181 × 25, half of 18100 is 9050 (think of 18100 as 18/10/0) half of 9050 is 4525 (split 9050 into 90/50).

We may note here the use of the Vedic formula Transpose and Apply in using division to do a multiplication sum. We can also transpose the devices shown in this chapter to obtain easy methods of division by numbers like 4, 8, 25, 35 etc. For example to divide a number by 5 we double the number and divide by 10:

Example 17 27 ÷ 5 = 54 ÷ 10 = 5.4

MULTIPLICATION BY NUMBERS THAT END IN 5, 25, 75

Example 18 Consider the sum 46 × 35. As it stands this is a 2-figure number multiplied by another 2-figure number. But 46 × 35 = 23 × 70 (by halving the first number and doubling the second), and this is effectively multiplication by 7, instead of by 35. Furthermore this has given us 23 to multiply instead of 46. So 46 × 35 = 23 × 70 = 1610 (23 × 7 is found from left to right, as described in Chapter 1). Example 19 Similarly, 66 × 15 = 33 × 30 = 990 Example 20 And 124 × 45 = 62 × 90 = 5580

Multiplication by numbers ending in 25 or 75 can be given at least two applications of this procedure:

Example 21 448 × 175 = 224 × 350 = 112 × 700 = 78400

In these examples the first number has been even. But even if the first number is odd it is still easier to multiply by twice the second number and then halve the result.

Example 22 For example, for 23 × 15 we find 23 × 30 = 690, and half of 690 is 345 Example 23 Similarly for 41 × 35: 41 × 70 = 2870 so 41 × 35 = 1435 an amusing result since the answer is a slight rearrangement of the figures in the sum.

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