(viii) by 11 if the difference between the sum of the 1st, 3rd, 5th, ... digits and the sum of the 2nd, 4th, 6th, ... is zero or divisible by 11.
(ix) To find whether a number is divisible by 7, 11 or 13, arrange the number in groups of three figures, beginning from the end, treat each group as a separate number, and then find the difference between the sum of the 1st, 3rd, ... of these numbers and the sum of the 2nd, 4th, ... Then, if this difference is zero or is divisible by 7, 11 or 13, the original number is also so divisible; and conversely. For example, 31521 gives 521 − 31 = 490, and therefore is divisible by 7, but not by 11 or 13.
49. Casting out Nines is a process based on (vi) of the last paragraph. The remainder when a number is divided by 9 is equal to the remainder when the sum of its digits is divided by 9. Also, if the remainders when two numbers are divided by 9 are respectively a and b, the remainder when their product is divided by 9 is the same as the remainder when a·b is divided by 9. This gives a rule for testing multiplication, which is found in most text-books. It is doubtful, however, whether such a rule, giving a test which is necessarily incomplete, is of much educational value.
(v.) Relative Magnitude.
50. Fractions.—A fraction of a quantity is a submultiple, or a multiple of a submultiple, of that quantity. Thus, since 3 × 1s. 5d. = 4s. 3d., 1s. 5d. may be denoted by 1⁄3 of 4s. 3d.; and any multiple of 1s. 5d., denoted by n × 1s. 5d., may also be denoted by n/3 of 4s. 3d. We therefore use “n⁄a of A” to mean that we find a quantity X such that a × X = A, and then multiply X by n.
It must be noted (i) that this is a definition of “n/a of,” not a definition of “n/a,” and (ii) that it is not necessary that n should be less than a.
51. Subdivision of Submultiple.—By 5⁄7 of A we mean 5 times the unit, 7 times which is A. If we regard this unit as being 4 times a lesser unit, then A is 7·4 times this lesser unit, and 5⁄7 of A is 5·4 times the lesser unit. Hence 5⁄7 of A is equal to 5·4⁄7·4 of A; and, conversely, 5·4⁄7·4 of A is equal to 5⁄7 of A. Similarly each of these is equal to 5·3⁄7·3 of A. Hence the value of a fraction is not altered by substituting for the numerator and denominator the corresponding numbers in any other column of a multiple-table (§ 36). If we write 5·4⁄7·4 in the form 4·5⁄4·7 we may say that the value of a fraction is not altered by multiplying or dividing the numerator and denominator by any number.
52. Fraction of a Fraction.—To find 11⁄4 of 5⁄7 of A we must convert 5⁄7 of A into 4 times some unit. This is done by the preceding paragraph. For 5⁄7 of A = 5·4⁄7·4 of A = 4·5⁄7·4 of A; i.e. it is 4 times a unit which is itself 5 times another unit, 7·4 times, which is A. Hence, taking the former unit 11 times instead of 4 times,
|11⁄4 of 5⁄7 of A = of A.|
A fraction of a fraction is sometimes called a compound fraction.
53. Comparison, Addition and Subtraction of Fractions.—The quantities ¾ of A and 5⁄7 of A are expressed in terms of different units. To compare them, or to add or subtract them, we must express them in terms of the same unit. Thus, taking 1⁄28 of A as the unit, we have (§ 51)
|¾ of A = 21⁄28 of A; 5⁄7 of A = 20⁄28 of A.|
Hence the former is greater than the latter; their sum is 41⁄28 of A; and their difference is 1⁄28 of A.
Thus the fractions must be reduced to a common denominator. This denominator must, if the fractions are in their lowest terms (§ 54), be a multiple of each of the denominators; it is usually most convenient that it should be their L.C.M. (§ 47).
54. Fraction in its Lowest Terms.—A fraction is said to be in its lowest terms when its numerator and denominator have no common factor; or to be reduced to its lowest terms when it is replaced by such a fraction. Thus 8⁄22 of A is said to be reduced to its lowest terms when it is replaced by 4⁄11 of A. It is important always to bear in mind that 4⁄11 of A is not the same as 8⁄22 of A, though it is equal to it.
55. Diagram of Fractional Relation.—To find 10⁄24 of 14s. we have to take 10 of the units, 24 of which make up 14s. Hence the required amount will, in the multiple-table of § 36, be opposite 10 in the column in which the amount opposite 24 is 14s.; the quantity at the head of this column, representing the unit, will be found to be 7d. The elements of the multiple-table with which we are concerned are shown in the diagram in the margin. This diagram serves equally for the two statements that (i) 10⁄24 of 14s. is 5s. 10d., (ii) 24⁄10 of 5s. 10d. is 14s. The two statements are in fact merely different aspects of a single relation, considered in the next section.
56. Ratio.—If we omit the two upper compartments of the diagram in the last section, we obtain the diagram A. This diagram exhibits a relation between the two amounts 5s. 10d. and 14s. on the one hand, and the numbers 10 and 24 of the standard series on the other, which is expressed by saying that 5s. 10d. is to 14s. in the ratio of 10 to 24, or that 14s. is to 5s. 10d. in the ratio of 24 to 10. If we had taken 1s. 2d. instead of 7d. as the unit for the second column, we should have obtained the diagram B. Thus we must regard the ratio of a to b as being the same as the ratio of c to d, if the fractions a/b and c/d are equal. For this reason the ratio of a to b is sometimes written a/b, but the more correct method is to write it a:b.
If two quantities or numbers P and Q are to each other in the ratio of p to q, it is clear from the diagram that p times Q = q times P, so that Q = q/p of P.
57. Proportion.—If from any two columns in the table of § 36 we remove the numbers or quantities in any two rows, we get a diagram such as that here shown. The pair of compartments on either side may, as here, contain numerical quantities, or may contain numbers. But the two pairs of compartments will correspond to a single pair of numbers, e.g. 2 and 6, in the standard series, so that, denoting them by M, N and P, Q respectively, M will be to N in the same ratio that P is to Q. This is expressed by saying that M is to N as P to Q, the relation being written M:N :: P:Q; the four quantities are then said to be in proportion or to be proportionals.
This is the most general expression of the relative magnitude of two quantities; i.e. the relation expressed by proportion includes the relations expressed by multiple, submultiple, fraction and ratio.
If M and N are respectively m and n times a unit, and P and Q are respectively p and q times a unit, then the quantities are in proportion if mq = np; and conversely.
IV. Laws of Arithmetic
58. Laws of Arithmetic.—The arithmetical processes which we have considered in reference to positive integral numbers are subject to the following laws:—
(i) Equalities and Inequalities.—The following are sometimes called Axioms (§ 29), but their truth should be proved, even if at an early stage it is assumed. The symbols “>” and “<” mean respectively “is greater than” and “is less than.” The numbers represented by a, b, c, x and m are all supposed to be positive.