*Asked for the compound interest on £4,444 for 4,444 days at 4.5% per annum, Bidder, aged ten, gave the answer, £2,434 16s 5.25d in two minutes. When he was twelve he was asked "if a pendulum clock vibrates the distance of 9.75 inches in a second of time, how many inches will it vibrate in 7 years, 14 days, 2 hours, 1 minute, 56 seconds, each year being 365 days, 5 hours, 48 minutes, 55 seconds?" He gave the answer, 2,165,625,744.75 inches, in less than 1 minute.*

George Parker Bidder (1806-1878) was the son of a stonemason of Devonshire, England. An elder brother taught him to count, this being the only formal instruction in arithmetic he ever received. He later became one of the most prominent civil engineers of his time.

Chapter 2

Proportionately

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,_{1}2 = 152.

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.

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Exercise A

Extending The Muliplication 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

This device can also be used for other types of sum.

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**

Exercise B

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**

Exercise C

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.

Exercise D