observing to carry for the increase of the figures cut off, as in contraction of multiplication. In the example above, the order of the first quotient figure was obviously tens; hence, as there were three decimal places required in the quotient, five figures of the divisor must be used. 2. Divide 59 by .74571345, and let the quotient contain four places of decimals. 3. Divide 17493.407704962 by 495.783269, and let the quotient contain four places of decimals. 4. Divide 98.187437 by 8.4765618, and let the quotient contain ten places of decimals. 5. Divide 47194.379457 by 14.73495, and let the quotient contain as many decimal places as there will be integers in it. REDUCTION OF VULGAR FRACTIONS TO DECIMALS. 158. The value of every vulgar fraction is equal to the quotient arising from dividing the numerator by the denominator (Art. 94). EXAMPLES. 1. What is the value in decimals of 2? We first divide 9 by 2, which gives a quotient 4, and 1 for a remainder. Now, 1 is equal to 10 tenths. If, then, we add a cipher, 2 will divide 10, giving the quotient 5 tenths. Hence, the true quotient is 4.5. 2. What is the value of 13? OPERATION. 1334; but 31 3100 3.25. We first divide by 4, which gives a quotient 3 and a remainder 1. But 1 is equal to 100 hundredths. If, then, we add two ciphers, 4 will divide the 100, giving a quotient of 25 hundredths. QUEST.-What is the order of the first quotient figure in Ex. 2? In 3? 158. What is the value of a fraction equal to? What is the In 4 Hence, to reduce a vulgar fraction to a decimal, I. Annex one or more ciphers to the numerator and then divide by the denominator. II. If there is a remainder, annex a cipher or ciphers, and divide again, and continue to annex ciphers and to divide until there is no remainder, or until the quotient is sufficiently exact: the number of decimal places to be pointed off in the quotient is the same as the number of ciphers used; and when there are not so many, ciphers must be prefixed to supply the deficiency. We here use two ciphers, and therefore point off two decimal places in the quotient. 2. Reduce and 12 to decimals. 18 1129 27 3 OPERATION. 125)635(5.08 625 1000 1000 Ans. 3. Reduce 80, 33, Tobo, and to decimals. 6 4809 60006 1375 3265 8436' 4121' 123 Ans. Ans. 6. Reduce, 18, 2295, 52 to decimals. Ans. 7. Reduce 30 to decimals. 1280 8. Reduce 347 to decimals. 2560 9. Reduce 10000 10. Reduce 3476 to decimals. Ans. Ans. Ans. Ans. 15625 11. Reduce 204000 to decimals. Ans. Ans. Ans. Ans. Ans. Ans. Ans. QUEST. What is the decimal value of one-half? Of three-fourths? Of six-fourths? Of nine-halves? Of seven-halves? Of five-fourths? Of onefourth? Give the rule for reducing a vulgar fraction to a decimal 5 18. What is the decimal value of 2 of 3 multiplied by 2 19. What is the value in decimals of by 3 of 2? of of 7 divided 4 22 20. A man owns of a ship; he sells of his share what part is that of the whole, expressed in decimals? 3 8 9 20 21. Bought of 87 bushels of wheat for of 7 dollars a bushel: how much did it come to, expressed in decimals? 22. If a man receives another, and 83 at a third: decimals? 5 of a dollar at one time, 71⁄2 at how many in all, expressed in TI 23. What decimal is equal to 1⁄2 of 18, and of 11 of 711 added together? 24. What decimal is equal to 3 of 6 taken from 3 of 83? 25. What decimal is equal to 21, 13, 3, added together? REDUCTION OF DENOMINATE DECIMALS. 159. We have seen that a denominate number is one in which the kind of unit is denominated or expressed (Art. 14). A denominate decimal is a decimal fraction in which the kind of unit that has been divided is expressed. Thus, .5 of a £, and .6 of a shilling are denominate decimals: the unit that was divided in the first fraction being £1, and that in the second 1 shilling. CASE I. 160. To find the value of a denominate number in decimals of a higher denomination. 1. Reduce 9d. to the decimal of a £. We first find that there are 240 pence in £1. We then divide 9d. by 240, which gives the quotient .0375 of a £. This is the true value of 9d. in the decimal of a £. OPERATION. 240d. £1 240)9(.0375 Ans. £.0375. What is a denominate QUEST.-159. What is a denominate number? decimal? In the decimal five-tenths of a £, what is the unit? In the decimal six-tenths of a shilling, what is the unit? Hence, to make the reduction, I. Consider how many units of the given denomination make one unit of the denomination to which you would reduce. II. Divide the given denominate number by the number so found, and the quotient will be the value in the required de nomination. EXAMPLES. 1. Reduce 14 drams to the decimal of a lb. avoirdupois. 2. Reduce 78d. to the decimal of a £. 3. Reduce .056 poles to the decimal of an acre. 4. Reduce 42 minutes to the decimal of a day. 5. Reduce 63 pints to the decimal of a peck. 10. Reduce 4ft. 6in. to the decimal of a yard. 11. Reduce 7oz. 19pwt. of silver to the decimal of a pound. 12. Reduce 9 months to the decimal of a year. 13. Reduce 62 days to the decimal of a year of 365 days. 14. Reduce £25 19s. 6d. to the decimal of a pound. 15. Reduce 3qr. 21lb. to the decimal of a cut. 16. Reduce 5fur. 36rd. 2yd. 2ft. 9in. to the decimal of a mile. 17. Reduce 4cwt. 23qr. to the decimal of a ton. 18. Reduce 3cwt. 7lb. 8oz. to the decimal of a ton. 19. Reduce 17hr. 6m. 43sec. to the decimal of a day. CASE II. 161. To reduce denominate numbers of different denominations to an equivalent decimal of a given denomination. QUEST.-160. How do you find the value of a denominate number in a decimal of a higher denomination? OPERATION. d..75d.; hence, 93d. - 9.75d. 12)9.75d. 1. Reduce £1 4s. 93d. to the denomination of pounds. We first reduce 3 farthings to the decimal of a penny, by dividing by 4. We then annex the quotient .75 to the 9 pence. We next divide by 12, giving .8125, which is the decimal of a shilling. This we annex to the shillings, and then divide by 20. Hence, to make the reduction, .8125s., and 20)4.8125s. £.240625; therefore, £1 4s. 93d. = £1.240625. Divide the lowest denomination named, by that number which makes one of the denomination next higher, annexing ciphers if necessary; then annex this quotient to the next higher denomination, and divide as before: proceed in the same manner through all the denominations to the last: the last result will be the answer sought. EXAMPLES. 1. Reduce £19 17s. 31d. to the decimal of a £. 4. Reduce 2lb. 5oz. 12pwt. 16gr. troy to the decimal of a lb. 5. Reduce 7 feet 6 inches to the denomination of yards. 6. Reduce 16. 12dr. avoirdupois to the denomination of pounds. 7. Reduce 10 leagues 4 furlongs to the denomination of leagues. 8. Reduce 7s. 5d. to the decimal of a pound. 9. What decimal part of a pound is three halfpence? 10. Reduce 4s. 7d. to the decimal of a pound. 11. Reduce loz. 11pwt. 3gr. to the decimal of a pound troy. QUEST.-161. How do you reduce denominate numbers of different denominations to equivalent decimals of a given denomination? |