multiple of a small number is simpler than that of a small multiple of a large number, but the calculation of the latter is easier. It is therefore convenient, in finding the product of two numbers, to take the smaller as the multiplier.
To find 3 times 427, we apply the distributive law (§ 58 (vi)) that 3.427 = 3(400 + 20 + 7) = 3.400 + 3.20 + 3.7. This, if we regard 3.427 as 427 + 427 + 427, is a direct consequence of the commutative law for addition (§ 58 (iii)), which enables us to add separately the hundreds, the tens and the ones. To find 3.400, we treat 100 as the unit (as in addition), so that 3.400 = 3.4.100 = 12.100 = 1200; and similarly for 3.20. These are examples of the associative law for multiplication (§ 58 (iv)).
100. Special Cases.—The following are some special rules:—
(i) To multiply by 5, multiply by 10 and divide by 2. (And conversely, to divide by 5, we multiply by 2 and divide by 10.)
(ii) In multiplying by 2, from the left, add 1 if the next figure of the multiplicand is 5, 6, 7, 8 or 9.
(iii) In multiplying by 3, from the left, add 1 when the next figures are not less than 33 . . . 334 and not greater than 66 . . . 666, and 2 when they are 66 . . . 667 and upwards.
(iv) To multiply by 7, 8, 9, 11 or 12, treat the multiplier as 10 − 3, 10 − 2, 10 − 1, 10 + 1 or 10 + 2; and similarly for 13, 17, 18, 19, &c.
(v) To multiply by 4 or 6, we can either multiply from the left by 2 and then by 2 or 3, or multiply from the right by 4 or 6; or we can treat the multiplier as 5 − 1 or 5 + 1.
101. Multiplication by a Large Number.—When both the numbers are large, we split up one of them, preferably the multiplier, into separate portions. Thus 231.4273 = (200 + 30 + 1).4273 = 200.4273 + 30.4273 + 1.4273. This gives the partial products, the sum of which is the complete products. The process is shown fully in A below,—
and more concisely in B. To multiply 4273 by 200, we use the commutative law, which gives 200.4273 = 2 × 100 × 4273 = 2 × 4273 × 100 = 8546 × 100 = 854600; and similarly for 30.4273. In B the terminal 0’s of the partial products are omitted. It is usually convenient to make out a preliminary table of multiples up to 10 times; the table being checked at 5 times (§ 100) and at 10 times.
The main difficulty is in the correct placing of the curtailed partial products. The first step is to regard the product of two numbers as containing as many digits as the two numbers put together. The table of multiples will them be as in C. The next step is to arrange the multiplier and the multiplicand above the partial products. For elementary work the multiplicand may come immediately after the multiplier, as in D; the last figure of each partial product then comes immediately under the corresponding figure of the multiplier. A better method, which leads up to the multiplication of decimals and of approximate values of numbers, is to place the first figure of the multiplier under the first figure of the multiplicand, as in E; the first figure of each partial product will then come under the corresponding figure of the multiplier.
102. Contracted Multiplication.—The partial products are sometimes omitted; the process saves time in writing, but is not easy. The principle is that, e.g. (a.102 + b.10 + c)(p.102 + q.10 + r ) = ap.104 + (aq + bp) 103 + (ar + bq + cp) 102 + (br + cq) 10 + cr. Hence the digits are multiplied in pairs, and grouped according to the power of 10 which each product contains. A method of performing the process is shown here for the case of 162.427. The principle is that 162.427 = 100.427 + 60.427 + 2.427 = 1.42700 + 6.4270 + 2.427; but, instead of writing down the separate products, we (in effect) write 42700, 4270, and 427 in separate rows, with the multipliers 1, 6, 2 in the margin, and then multiply each number in each column by the corresponding multiplier in the margin, making allowance for any figures to be “carried.” Thus the second figure (from the right) is given by 1 + 2.2 + 6.7 = 47, the 1 being carried.
103. Aliquot Parts.—For multiplication by a proper fraction or a decimal, it is sometimes convenient, especially when we are dealing with mixed quantities, to convert the multiplier into the sum or difference of a number of fractions, each of which has 1 as its numerator. Such fractions are called aliquot parts (from Lat. aliquot, some, several). This can usually be done in a good many ways. Thus 56 = 1 − 16, and also = 12 + 13; and 15% = ·15 = 110 + 120 = 16 − 160 = 18 + 140. The fractions should generally be chosen so that each part of the product may be obtained from an earlier part by a comparatively simple division. Thus 12 + 120 − 160 is a simpler expression for 815 than 12 + 130.
The process may sometimes by applied two or three times in succession; thus 815 = 45·23 = (1 − 15)(1 − 13), and 3340 = 34·1110 = (1 − 14)(1 + 110).
104. Practice.—The above is a particular case of the method called practice, but the nomenclature of the method is confusing. There are two kinds of practice, simple practice and compound practice, but the latter is the simpler of the two. To find the cost of 2 ℔ 8 oz. of butter at 1s. 2d. a ℔, we multiply 1s. 2d. by 2816 = 212. This straightforward process is called “compound” practice. “Simple” practice involves an application of the commutative law. To find the cost of n articles at £a, bs, cd. each, we express £a, bs, cd. in the form £(a + f ), where f is a fraction (or the sum of several fractions); we then say that the cost, being n × £(a + f ), is equal to (a + f ) × £n, and apply the method of compound practice, i.e. the method of aliquot parts.
105. Multiplication of a Mixed Number.—When a mixed quantity or a mixed number has to be multiplied by a large number, it is sometimes convenient to express the former in terms of one only of its denominations. Thus, to multiply £7, 13s. 6d. by 469, we may express the former in any of the ways £7·675, 30740 of £1, 15312s., 153·5s., 307 sixpences, or 1842 pence. Expression in £ and decimals of £1 is usually recommended, but it depends on circumstances whether some other method may not be simpler.
A sum of money cannot be expressed exactly as a decimal of £1 unless it is a multiple of 34d. A rule for approximate conversion is that 1s. = ·05 of £1, and that 212d.= ·01 of £1. For accurate conversion we write ·1£ for each 2s., and ·001£ for each farthing beyond 2s., their number being first increased by one twenty-fourth.
106. Division.—Of the two kinds of division, although the idea of partition is perhaps the more elementary, the process of measuring is the easier to perform, since it is equivalent to a series of subtractions. Starting from the dividend, we in theory keep on subtracting the unit, and count the number of subtractions that have to be performed until nothing is left. In actual practice, of course, we subtract large multiples at a time. Thus, to divide 987063 by 427, we reverse the procedure of § 101, but with intermediate stages. We first construct the multiple-table C, and then subtract successively 200 times, 30 times and 1 times; these numbers being the partial quotients. The theory of the process is shown fully in F. Treating x as the unknown quotient corresponding to the original dividend,
