therefore to divide one fraction by another, invert the terms of the divisor, and multiply the dividend by it. 69. A complex fraction may be reduced to a simple fraction by division. For if be the complex fraction, then its value will be the 71 5吋 REDUCTION OF FRACTIONS OF CONCRETE NUMBERS. 70. A fraction of a concrete number may be reduced to the fraction of another concrete number of a higher or lower denomination, by means of the principle employed in the reduction of integers from one denomination to another (35). 2 9 EXAMPLES. 1. Reduce of a pound to the fraction of a penny. Since any integer number of pounds is reduced to pence by multiplying the number of pounds by 20, and the product by 12, so any fraction of a pound is reduced to the fraction of a penny by multiplying the numerator by 20 × 12 or 240. 3. Reduce 3 cwt. 14 lb. to the fraction of a ton; or what fraction of a ton is 3 cwt. 14 lb. ? Otherwise. Since 3 cwt. 14 lb. = 3 × 112 + 14 or 350 lb. and 1 ton or We might shorten the operation in this manner :-Since 14 lb. 14 = 112 X or of a ton. 8 20 32 4. What part or fraction of half-a-crown is 11d. Here 111d. = 45 farthings, and half-a-crown is = 30d. or 120 far things; therefore 114d. = 5 or of half-a-crown. 45 71. The value of a fraction of a concrete number is easily determined in terms of the same or lower denominations. For since of a pound 2 3 may be obtained either by dividing one pound into 3 equal parts and taking 2 of those parts, or by taking the concrete unit twice, viz., 2 pounds, and dividing it into 3 equal parts; therefore we have In every case, then, we have only to multiply the given concrete number by the numerator of the fraction, and divide the product by the denominator. 72. By means of these reductions, we may find the sum and difference of fractions of concrete numbers consisting of one or more deno of a crown, we may either reduce any two of them to the fraction of the denomination of the other, and then find the sum of these fractions, or we may find the value of each, and then take their sum. ALIQUOT FRACTIONS. 73. The value of any number of articles may always be found by Compound Multiplication when the price of one of them is given; but when the number is large, and the price of each article is only a small sum, their value may be obtained by the method of Aliquot Fractions (i. e., fractions having unity for their numerators), in a very convenient manner. process is frequently denominated Practice, and the following examples will sufficiently illustrate the mode of calculation to be adopted in all cases. EXAMPLES. The 1. Suppose it required to find the value of 115 tons, if each ton cost £3. 128. 74d. The value of 115 tons may be found either by multiplying £3. 12s. 74d. by 115, or by separating the price of 1 ton into different parts, as £3, 10s., 2s. 6d., and 14d., and finding the cost of 115 tons at each of these prices. These different sums together will evidently give the value of 115 tons, at the whole price, £3. 12s. 74d. The reason for the above separation of the price per ton into different parts will be manifest from the following process, in which £115 is the price of 115 tons at £1 per ton: S. d. £. £. S. d. £. S. 74. The separation of the price into the different parts may be made in a great variety of ways; but in all of them each of the parts must be an aliquot part of some one which precedes it, and the first is usually an aliquot part of the unit of the highest denomination in the price. Thus 10s. is an aliquot part of £1, for it is of £1; 2s. 6d. is an aliquot part of 10s., since it is of 10s.; 14 lb. is an aliquot part of 1 cwt., for it is of 1 cwt.; and so on. Consequently one number is an aliquot part of another when the former number measures the latter; that is, when the less number is a fraction of the greater, the numerator of the fraction being unity. The preceding example may be solved in a different manner, as in 75. We have seen that the fundamental operations of arithmetic can be applied to fractions with considerable facility, and that they are much more readily performed upon fractions having the same than upon those which have different denominators. Hence in all those parts of mathematics where fractions are often required, it has been customary to use only those which have a common denominator, or such as can be easily transformed to others having the same denominators. As the decimal numbers 10, 100, 1000, etc., can be operated upon with the utmost facility, so, by employing.those fractions which have decimal numbers for denominators, the highest degree of simplicity will be attained in all calculations involving fractions. A decimal fraction is one whose denominator is any of the decimal 5 27 4375 numbers 10, 100, 1000, etc. Thus 10' 100' 10000' tions. are decimal frac 76. Instead, then, of dividing the unit into so many different parts, corresponding to all the different denominators which are met with in fractions, it has been found convenient to consider the unit as divided into ten parts, each of which shall be a tenth; and each of these tenths into ten parts, of which each is a hundredth of a unit; the hundredths into ten parts, each of which is a thousandth of a unit, and so on. By continuing this division, the smallest parts may be formed, by means of which all numbers whatever may be measured to any degree of accuracy. These decimal fractions, being composed of parts of a unit which become successively ten times less, are changed one into another, in the same manner as the tens, hundreds, thousands, etc., are changed into units. For as the unit is 10 tenths, the tenth is 10 hundredths, the hundredth is 10 thousandths, etc.; therefore the tenth is ten times 10 thousandths, or 100 thousandths. Thus 2 tenths, 4 hundredths, and 9 thousandths, are equivalent to 249 thousandths, in the same way as 2 hundreds, 4 tens, and 9 units make 249 units. 77. The decimal fractions may, therefore, be written by means of figures, in the same manner as whole numbers, the tenths taking their place, of course, to the right of the units, then the hundredths to the right of the tenths, and so on, as in the following table, which may be regarded as an extension of the Numeration Table. To distinguish the figures which express the decimal parts from those which express entire units, a period is placed on the right of the units. Thus 21 234 will represent 21 units and 234 hundredths. 2 3 78. Suppose it required to represent and as decimal fractions; 1 5 fractions, it must fall between some two of them, as and 1000 1000 and consequently it cannot differ from either by a thousandth part of the unit. With respect to the other fraction, 1 1 2 5' we have = 10' which is a decimal fraction; hence we perceive that though the exact value of every fraction may not be assignable by means of decimals, still we can obtain its approximate value to any degree of accuracy that may be required. 79. The value of the decimal figures depending entirely on the place they occupy with respect to the point which separates the units from the tenths, any number of ciphers on their right may either be annexed or effaced, without altering their value. For instance, 0.7 is the same as 0.70, because the number that expresses the decimal fraction becomes ten times greater, while its parts become hundreaths, and are therefore 7 70 700 diminished ten times; thus and hence it is evident 10 100 1000' = = that annexing ciphers to the right-hand of decimals does not change their value; but if ciphers be prefixed to a decimal, its value will be diminished ten times for each cipher that is prefixed; thus― VOL. I. D |