If as our unit we take X = 1⁄100 of A = 1% of A, the above quantity might equally be written 2376 X = 2376⁄100 of A; i.e. 23.76 × A is equal to 2376% of A.
71. Approximate Expression by Percentage.—When a fraction cannot be expressed by an integral percentage, it can be so expressed approximately, by taking the nearest integer to the numerator of an equal fraction having 100 for its denominator. Thus 1⁄7 = 142⁄7 / 100, so that 1⁄7 is approximately equal to 14%; and 2⁄7 = (284⁄7)/100, which is approximately equal to 29%. The difference between this approximate percentage and the true value is less than ½%, i.e. is less than 1⁄200.
If the numerator of the fraction consists of an integer and ½—e.g. in the case of 3⁄8 = (37½)/100—it is uncertain whether we should take the next lowest or the next highest integer. It is best in such cases to retain the ½; thus we can write 3⁄8 = 37½% = .37½.
72. Addition and Subtraction of Percentages.—The sum or difference of two percentages is expressed by the sum or difference of the numbers expressing the two percentages.
73. Percentage of a Percentage.—Since 37% of 1 is expressed by 0.37, 37% of 1% (i.e. of 0.01) might similarly be expressed by 0.00.37. The second point, however, is omitted, so that we write it 0.0037 or .0037, this expression meaning 37⁄100 of 1⁄100 = 37⁄10000.
On the same principle, since 37% of 45% is equal to 37⁄100 of 45⁄100 = 1665⁄10000 = 16⁄100 + (65⁄100 of 1⁄100), we can express it by .1665; and 3% of 2% can be expressed by .0006. Hence, to find a percentage of a percentage, we multiply the two numbers, put 0’s in front if necessary to make up four figures (not counting fractions), and prefix the point.
74. Decimal Fractions.—The percentage-notation can be extended to any fraction which has any power of 10 for its denominator. Thus 153⁄1000 can be written .153 and 15300⁄100000 can be written .15300. These two fractions are equal to each other, and also to .1530. A fraction written in this way is called a decimal fraction; or we might define a decimal fraction as a fraction having a power of 10 for its denominator, there being a special notation for writing such fractions.
A mixed number, the fractional part of which is a decimal fraction, is expressed by writing the integral part in front of the point, which is called the decimal point. Thus 271530⁄10000} can be written 27.1530. This number, expressed in terms of the fraction 1⁄10000 or .0001, would be 271530. Hence the successive figures after the decimal point have the same relation to each other and to the figures before the point as if the point did not exist. The point merely indicates the denomination in which the number is expressed: the above number, expressed in terms of 1⁄10, would be 271.530, but expressed in terms of 100 it would be .271530.
Fractions other than decimal fractions are usually called vulgar fractions.
75. Decimal Numbers.—Instead of regarding the .153 in 27.153 as meaning 153⁄1000, we may regard the different figures in the expression as denoting numbers in the successive orders of submultiples of 1 on a denary scale. Thus, on the grouping system, 27.153 will mean 2·10 + 7 + 1/10 + 5/102 + 3/103, while on the counting system it will mean the result of counting through the tens to 2, then through the ones to 7, then through tenths to 1, and so on. A number made up in this way may be called a decimal number, or, more briefly, a decimal. It will be seen that the definition includes integral numbers.