From these illustrations we derive the followiog RULE. 4. Enough to make the number of decimal places in the divisor and quotient, counted together, equal to the number of decimal places in the dividend. Q. Suppose that there are not figures enough in the quotient for purpose, what is to be done ? A. Supply this defect by prefixing ciphers to said quotient. Q. What is to be done when the divisor has more decimal places than the dividend ? A. Annex as many ciphers to the dividend as will make the decimals in both equal. Q. What will be the value of the quotient in such cases ? Q. When the decimal places in the divisor and dividend are eginal and the divisor is not contained in the dividend, or when there is a po mainder, how do you proceed ? A. Annex ciphers to the remainder, or divi dend, and divide as before. Q. What places in the dividend do these ciphers take ? More Exercises for the Slate. 4. At $,25 a bushel, how many bushels of oats may be bougbe for $300,50 ? A. 1202 bushels. 5. At $,12%, or $,125 a yard, how many yards of cotton cloth may be bought for $16? A. 12 yards. 6. Bought 128 yards of tape for $,64; how much was it e yard? A. $,005, or 5 mills. 7. If you divide 116,5 barrels of four equally among 5 men, how many barrels will each have ? A. 23,3 barrels. Note.-Tho papil must continue to bear in mind, that, before he proceede to add together the figures annexed to each question, he must prefis cipbom when required by the rule for pointing off. 8. At $2,255 a gallon, how many gallons of rum may be bought for $28,1875-125. L'or $56,3757-85. For $112,75 la 60. For $338,25 ?-150. A. 237,5 gallons. 9. If $2,25" will board one man a week, how many weeko can be be boarded for $1001,25 :-445. For $500,862-22 Pere $200,7 ?-892. For $100,35 ?-446. For $60,75?-27. 4. 829,4 weeks. 10. If 3,3355 bushels of corn will fill one barrel, how many barrels will 3,52275 bushels fill ?-105. Will ,4026 of a bushel ?12. Will 120,780 bushels ?-36. Will 63,745 bushels ?-19. Will 40,260 bushels ?-12. H. 68,17 barrels. 11. What is the quotient of 1561,275 divided by 24,37-6425. By 48,6?-32125. By , 12,15 ?-1285. By 6,075 -257. 8. 481,875. 12. What is the quotient of ,264 divided by 22-132. By ,4?56. By ,022-132. By ,042-66. By ,0027-132. By ,004 ?-16 9. 219,78. ILVII. TO CHANGE A Vulgar OR COMMON FRAQ TION TO ITS EQUAL DECIMAL. . 1. A man divided 2 dollars equally among 5 men ; what part of a dollar did he give each and how much in 10ths, or decimals' In common fractions, each man evidently has of a dollas the answer; but, to express it decimally, we proceed thus :OPERATION. In this operation, we cannot diNumer. vide 2 dollars, the numerator, by 5, 3 Denom. 5)2,0(,4 the denominator; but, by anne . 20 ing, a cipher to 2, (that is, multi. plying by 10,) we have 20 tenths, Ans. 4 tenths,=,4. or dimes; then 5 in 20, 4 times ; that is, 4 tenths,=,4: Hence why common fractions, reduced to a decimal, is ,4, Ans. 2. Reduce 3 to its equal decimal. OPERATION. In this example, by annexing ono 32)3,00 (,09375 cipher to 3, making 30 tenths, we 288 find that 32 is not contained in the 10ths; consequently, a cipher must 120 be written in the 10ths' place in 96 the quotient. These 30 tenths may be brought into 100ths by annex. 240 ing another cipher, making 300 224 hundredths, which contain 32, 9 times; that is, 9 hundredths. By 160 continuing to annex ciphers for 160 1000ths, &c. , dividing as before, we obtain 09375, Ans. By counting the ciphers annexed to the numerator, 3, we shall find then nged to the decimal places in the quotiepl Wole. - In the last mswor, we have five places for decimalo; but, u thot - the fifth place is only too oo of a unit, it will be found sufficiently exact. mout practical purposes, to extend the decimals to ouly three or four placu To know whether you have obtained an equal decimal, change the decimal into a common fraction, by placing its proper de nominator under it, and reduce the fraction to its lowest terms. If it produces the same common fraction again, it is right; thus, making the two foregoing examples, ¡4==. Again, 09375=183338=. From these illustrations we derive the following RULE. Q. How do you proceed to reduce a common fraction to its eqnal hecimal ? A. Annex ciphers to the numerator, and divide by the denominator. Q. How long do you continue to annex ciphers and divide ? A. Till there is no remainder, or until a decimal is obtained sufficiently exact for the purpose required. Q. How many figures of the quotient will be decimals ! Q. Suppose thai there are not figures enough in the quotient too this purpose, what is to be done ? 4. Prefix ciphers to supply the deficiency. More Exercises for the Slate. 3. Change 4, 1, , and y's to equal decimals. A. 75 25, ,04. 4. What decimal is equal to a'5 ?-5. What = $?-5. What ig?-75. What=562-4. A. 1,34. 5. What decimet is equal to 1o ?-5. What = What=17?-5. What=72-175. What=12-62). A. 1,6 6. What decimal is equal to ?-1111. What = $-4444 What=g5?-10101. What=*?-3333.* A. ,898901.+ 32-35 • When decimal fractious continue to repeat the same tiguro, like 333, &c. In this example, they are called Repelends, or Circulating Decimals. When only one figure nepoats, it is called a single repetend; but if two or more Aguren repeal, it is called a compound repetend: thus, ,333, &c. is a single repetend, 2010101, &c. & compound repetend. When other decimale come before circulating decimals, as ,8 in 8333, the dorimial is called a mixed repotond ILVIII. TO REDUCE COMPOUND NUMBERS TO Deck MALS OF THE HIGHEST DENOMINATION. 1. Reduce 15 s. 6 d. to the decimal of a pound. OPERATION. In this example, 6 d. = of a shit12 ) 6,0d. ling, and , reduced to a decimal by 1 LVII., is equal to 15 of a shilling, 20 ) 15,5 s. which, joined with 15 s., makes=15,58. In the same manner, 15,5 8. + 20 s. ,775 £. ,775£, Ans. It is tho common practice, instead of writing the repeating figures arveral umoa, to place a dot over the repeating figure in a single repetend ; thus, 111, ke., is written i; also over the first and last repeating figure of a compound repetend; thus, for ,030303, &c. we write, 03. The value of any repetend, notwithstanding it repeats one figure or more an Infinite number of times, coming nearer and nearer io a unit each time, though dover reaching it, may be easily determined by common fractions ; as will appear from what follows. By reducing $ to a decimal, we have a quotient consisting of ,1111. , &c., that le, the repetend ,i ; $, then, is the value of the repetend i, the value of 333, ke.; that is, the repetend 3 must be three timos as much; that is, it and i= $; s= }; and ,1-3 = 1 whole. Honce we have tho following RULE for changing a single repetend to its •qual common fraction :-Make the given repetond a dumerator, writing 9 oderneath for a denominator, and it is dono. What is the value of ,i? or i? Of ,4Or ,1? Or ,8? Of ,6? 4. po By changing og to a decimal, we shall have ,010101, that is, the repetend ói. Then, the repetend , di, being 4 times a much, must be om, and 36 must be , also ,45=$. if gto be reduced to a decimal, it produces ,öoi. Then the decimal ,004, being 4 times as much, is ggg, and ,036= . This principle will be bruo for any number of places. Henco we derive the following RULE for reducing a circulating decimal to & common fraction :- Make the given repetend a numerator, and the denow.d aator will be as many 98 as there are figures in the repotend. Change ,18 to' a common fraction. : for Change , 12 to a common fraction. A. J = 4 Chango „003 ro a common fraction. A. fig=: le the following example, viz. Change ,83 to a common fraction, the repenting figure is 3, that is, t, and 8 io Yo then $, instead of being it as A. Hence we derive the following RULE. A. One above another, the highest at the bottom. Q. How do you divide ? A. Begin at the top, and divide as in ReducLion; that is, shillings by shillings, ounces by ounces, &c., annexing ciphers. Q. How long do you continue to do so? A. Till the denominations are reduced to the decimal required. More Exercises for the Slate. 2. Reduce 7 8. 6 d. 3 qrs. to the decimal of a pound. A. 3781252 3. Reduce 5 s. to the decimal of a pound. A: 25£ 4. Reduce 3 farthings to the decimal of a pound. A. ,003125£. 5. Reduce 2 qrs. 3 na. to the decimal of a yard. A. ,6875 yd 6. Reduce 2 si 3 d. to the decimal of a dollar. A. $,375. 7. Reduce 3 qrs. 3 na. to the decimal of a yard. A.,9375 yd. 8. Reduce 8 öz. 17 pwts. to the decimal of a pound Troy. A. ,7375 Ib. 9. Reduce 8 £. 17 s. 6 d. 3 qrs. to the decimal of a pound. A. 8,878125£ a unit, is, by being in the second place, f of It=g; then and added together, thus, & top==ff, Ans. Hence, to find the valuo of a mixed repetend-First find the value of the repeating decimals, then of the other decimals, and arld these results together 2. Change ,916 to a common fraction. A. 1906 +960=8&t=1t: Proof, 11 + 12=,916. 3. Change ,203 to a commou fraction. A. To know if the renule be right, change the common fraction to a decima again. If it produces the same, the work is right. Repeating decimals may be easily multiplied, subtracted, &c. by fint reden thom to theu equal common fractions. 30 |