1000, &c., may be shortened by removing the decimal point as many places to the right, as there are ciphers in the multiplier: and, If there be not so many figures on the right of the point, annex ciphers to supply the deficiency. Thus, 2.07X10=20.7 For additional problems, see Ray's Test Examples. DIVISION OF DECIMALS. ART. 183. Decimals may be divided when the divisor, or dividend, or both, are decimals. Since the dividend is equal to the product of the divisor and quotient, it must contain as many decimal places as there are decimals in both divisor and quotient. Art. 181: hence, There must be as many decimals in the quotient as the decimal places in the dividend exceed those in the divisor. 1. Divide 2.125 by 5 tenths. SOLUTION.-Divide as in Simple Numbers; then, since there are three decimal places in the dividend, and one in the divisor, point off two decimals in the quotient. 2. Divide 21 units by .5 OPERATION. .5)2.125 Ans. 4.25 OPERATION. .5)21.0 Ans. 42. SOLUTION. In dividing a whole number by a fraction, the whole number is reduced to the same parts of a unit as the fraction, that the divisor and dividend may be of the same denomination. So, in dividing a whole number by a decimal; reduce the dividend to the same denomination as the divisor, by annexing to it as many ciphers as there are decimal places in the divisor, and the quotient is a whole number. 3. Divide 83.1 by 4. SOLUTION.-Divide the figures of the dividend by the divisor, as in whole numbers, and a remainder occurs. Then, to continue REVIEW.-181. To what is the number of decimal places in the product equal? What is the General Rule for multiplication? 182. How multiply a decimal by 10, 100, &c.? 183. When may decimals be divided? How many decimal places must the quotient contain? Why? the division, annex ciphers to the dividend, which does not alter its value, (Art. 172), and divide as before. Continue the division until there is no remainder, or until the quotient is sufficiently exact. OPERATION. 4)83.100 Ans. 20.775 As there are three places of decimals in the dividend, and nons in the divisor, there must be three in the quotient. 4. Divide 2.11 by .3 In this example, the division will not terminate. In such cases, it is to be carried to sufficient exactness: the sign+is OPERATION. .3)2.11000 Ans. 7.0333+ annexed to denote that the division is not complete. General Rule for Division.-Divide as in Simple Numbers, and point off from the right hand of the quotient as many places for decimals as the decimal places in the dividend exceed those in the divisor; if there be not so many places, supply the deficiency by prefixiny ciphers. PROOF.-The same as in Division of Simple Numbers. NOTES.-1. When the divisor has more decimals than the dividend, annex ciphers to the dividend until its decimal places equal those of the divisor; the quotient will be a whole number. 2. After dividing all the figures of the dividend, if there be a remainder, annex ciphers to it, and continue the division till there is no remainder, or until the quotient is sufficiently exact. In pointing the quotient, regard the ciphers annexed as decimal places. REVIEW.-183. What is the General Rule for Division? NOTE 1. When the number of places in the divisor exceeds those in the dividend, what is required? Why? What will be the quotient? Ans. 1. 24. Divide 1 thousandth by 1 thousandth. 25. 1 ten-thousandth÷1 ten-millionth. Ans. 1000. 26. 1 hundredth÷4 millionths. Ans. 2500. Ans. 2.142857+ ART. 184. To divide a Decimal by 10, 100, 1000, &c., remove the decimal point as many places to the left as there are ciphers in the divisor: And, if there are not so many figures on the left of the point, supply the deficiency by prefixing ciphers. ART. 185. REDUCTION OF DECIMALS. CASE I. To reduce a common fraction to a decimal. 1. Reduce the fraction to a decimal. SOL.-The numerator, 3, will not be changed by writing ciphers in the place of tenths, hundredths, &c. Since a fraction is expressed in the form of an unexecuted division (Art. 125), regard the operation as division of decimals, and perform it according to the rule, (Art. 183). Since the divisor has no decimal places, the quotient must have as many places as there are ciphers annexed. *2. Reduce to a decimal. OPERATION. 4)3.00 Ans. .75 Ans. .125 Rule for Case I.-Annex ciphers to the numerator, divide by the denominator, and point off in the quotient as many pluces for decimals as there are ciphers annexed to the numerator. NOTE. When common fractions can not be exactly expressed in decimals, continue to divide till the quotient is sufficiently exact. REDUCE THESE COMMON FRACTIONS TO DECIMALS: ART. 186. To reduce a decimal to a common fraction, and to its lowest terms. 1. Reduce .75 to a common fraction in its lowest terms. *2. Reduce .6 to a common fraction, in its lowest terms. Ans. . Rule for Case II.-Write the denominator under the decimal; it will then be a common fraction, which reduce to its lowest terms, (Art. 138). REDUCE TO COMMON FRACTIONS, IN THEIR LOWEST TERMS, 191 NOTE. When a decimal contains several places of figures, their value can be found nearly by inspection. To do this, take the first or the first two figures, as the numerator, of which 10 or 100 is the denominator, then reduce it to its lowest terms: Thus .332125 is nearly 1 third; .258321 is nearly 1 fourth. CASE III. ART. 187. To reduce a decimal of one denomination to an equivalent decimal of another denomination. 1. Reduce .25 qt. to the fraction of a pt. SOLUTION. To reduce quarts to pints, we multiply by 2, the number of pints in a quart. Therefore, multiply the fraction of a quart by 2, to reduce it to the fraction of a pint. Art. 81. 2. Reduce .5pt to the fraction of a qt. SOLUTION. To reduce pints to quarts, we divide by 2, the number of pints in a quart. Therefore, divide the fraction of a pint by 2, to reduce it to the fraction of a quart. Art. 81. OPERATION. qt. .25 2 .50 Ans. .5 pt. OPERATION. pt. 2).5 Ans. .25 qt. *3. Reduce .125 bu. to the fraction of a pk. Ans. .5 *4. Reduce .7 pk. to the fraction of a bu. Ans. .175 Rule for Case III.—Multiply or divide as in Reduction of Whole Numbers, Art. 81, according to the rules for the Multiplication and Division of Decimals. Arts. 181 and 183. 5. Reduce .0625 lb. Troy to the frac. of an oz. Ans. .75 Ans. .8 REVIEW.-185. What is Case 1? Rule for Case 1? NOTE. When com. fractions can not be exactly expressed in decimals, what is to be done? 186. What is Case 2? Rule for Case 2? NOTE. How find the value nearly of a decimal? 187. How reduce a decimal from a higher to a lower denomination? From a lower to a higher ? |