(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 = | 11·5 | of A |
| 7·4 |
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)