38. It will often happen in practice that fractions present themselves which require the application, not of one single rule only, as of Addition, or Subtraction, or Multiplication, &c., but of several rules in one operation. Thus,

Ex. 1. Required to find the single fraction which is equivalent to

71315 21

100

3

DECIMAL FRACTIONS.

39. In order to lessen the trouble which in many cases attends the use of vulgar fractions, decimal fractions have been introduced, which differ from the former in this respect, that their denominators are always 10 or some power of 10, as 100, 1000, 10000, &c. and instead of writing the denominator under the numerator, it is expressed by pointing off from the right of the numerator as many figures as there are cyphers in the denominator;

40. COR. 1.

The value of each figure in a decimal decreases from the left to the right in a tenfold proportion, that is, each figure is ten times as great as if it were removed one place to the right, as in whole numbers; thus

and the decimal 127 is one tenth, two hundredths, and seven

thousandths, of an unit.

41. COR. 2. Adding cyphers to the right of a decimal does not alter its value; thus,

the numerator and denominator having been multiplied by the same number. (Art. 12.)

42. COR. 3. Decimals may be reduced to a common denominator by adding cyphers to the right, where it is necessary, till the number of decimal places is the same in all.

Ex. 5, 01, and 311 reduced to a common denominator, are 500, 010 and 311;

43. COR. 4. Hence in all complicated numerical reductions decimal fractions possess great advantages over vulgar fractions. For, 1st, the denominators of the former being always 10 or some power of 10, their reduction to a common denominator is easily effected, and consequently all operations requiring that previous reduction are facilitated: 2nd, the numerators and denominators of decimal fractions being usually written in one line, and the value of each figure decreasing in a tenfold proportion, from left to right, as in whole numbers, the common rules of Arithmetic are immediately applicable to such fractions, care only being taken, by means of rules for that purpose, to mark off correctly the decimal result.

As decimals are only fractions of a particular description, their operations must depend upon the principles already laid down.

ADDITION OF DECIMALS.

44. To find the sum of any number of decimals place the figures in such a manner that those of the same denomination may stand under each other; add them together as in whole numbers, and place the decimal point in the sum under the other points.

Ex. Add together 79, 51-43 and 0118.

These, when reduced to a common denominator, are 7.9000 51-4300 and 0118; and proceeding according to the rule,

7.9000 51.4300

'0118

59.3418 the sum required. (Art. 31).

In the operation the cyphers may be omitted, if the several decimal points stand exactly under each other, thus,

7.9

51.43

0118

59.3418

SUBTRACTION.

45. To find the difference of two decimals place the figures of the same denomination under each other; then subtract as in whole numbers, and place the decimal point under the other points.

Ex. From 61.3 take 42.012.

These, reduced to a common denominator, are 61.300 and 42.012; therefore their difference is 19-288 (Art. 34). In the operation the cyphers may be omitted, thus,

61.3

42.012

19.288

MULTIPLICATION.

46. To multiply one decimal by another multiply the figures as in whole numbers, and point off as many decimal places in the product as there are in the multiplier and multiplicand together.

tion) 235.98.

cases.

And a similar proof may be given in all other

47. When there are fewer figures in the product than there are decimals in the multiplier and multiplicand together, cyphers must be annexed to the left of the product, that the decimal places may be properly represented.

48.

Division in decimals is performed as in whole numbers, observing to point off as many decimals in the quotient as the number of decimal places in the dividend exceeds the number in the divisor.

here there are three decimals in the dividend and one in the divisor; therefore, there are two in the quotient.

The truth of this rule is apparent from the nature of multiplication; for the product of the divisor and quotient is the dividend; there are, therefore, as many places of decimals in the dividend, as there are in the divisor and quotient together (Art. 46); consequently there are as many in the quotient as the number in the dividend exceeds the number in the divisor.

49. If figures be wanting in the quotient to make up the proper number of decimal places, cyphers must be added to the left.

Ex. Divide 336 by 42.

336

8;

42

and as the quotient of 336 divided by 42 must contain three decimal places, that quotient is '008.

50.

When the dividend does not contain as many deci

mals as the divisor, cyphers must be added to the right of the

decimals in the dividend, till that is the case.