From these illustrations we derive the following RULE. 1. How do you write the numbers down, and divide ? A. As in whole numbers. II. How many figures do you point off in the quotient for decimals? A. 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. III. Suppose that there are not figures enough in the quotiert for this purpose, what is to be done ? A. Supply this defect by prefixing ciphers to said quotient. IV. What is to be done when the dirisor has more decimal places than the diridend? A. Annex as many ciphers to the dividend as will make the decimals in both equal. V. What will be the valuc of the quotient in such cases ? A. A whole number. VI. When the decimal places in the divisor and dividend are equal, and the divisor is not contained in the dividend, or whin there is a remainder, how do you proceed ? A. Annox ciphe : to the remainder, or dividend, and divide as before. VII. What places in the dividend do these ciphers taise ! A. Decimal places. More Escercises for the Slate. 4. At $,25 a bushel, how many bushels of oats may be bought for $300,50A. 1202 bushels. 5. Ai *,12, or $,125 a yard, how many yards of cotton cloth may be bougisi for $16 ? A. 128 yards. 6. Bought 128 yards of tape for $,64 ; how much was it a yard ? A. $,005, or 5 mills. 7. If you divide 116,5 barrels of flour equally among 5 men, how many barrels will each have ? A. 23,3 barrels. Note. The pupil must continue to bear in mind, that before he proceeds to add together the figures in the parentheses, he must prefix ciphers, 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) For $56,375? (25) For $112,75? (50) For $338,25? (150) A. 237,5 gallons. I. If $2,85 will boará one man a weck, how many weeks ran he be boarded for $1001,25 ? (445) For $500,85? (2226) for $200,7? (892) For $100,35 ? (446) For $60,75? (27} 828,4 weeks. 10. If 3,355 bushels of corn will fill one barrel, how many 1 pels will 3,52275 bushels fill? (105) Will ,4026 of a pushel? Will 120,780 bushels ? (36) Will 63,745 bushels ? (19) WH 40,260 bushels ? (12) A. 68,17 barrels. 11. What is the quotient of 1561,275 divided by 24,3? (6425) by 48,6? (32125) By 12,15? (1285) By6,075 ? (257) Ails. 481,875. 12. What is the quotient of ,264 divided by ,2? (132) By ,4? 06) By ,02? (132)" By ,04 ? (66) By ,002? (132) By ,0017 (66) Ans. 219,78. REDUCTION OF DECIMALS. | LVII. To change a Vulgar or Common Fraction to its equal Decimal. 1. A man divided 2 dollars equally among five men; what part of a dollar did he give cach? and how much in 10ths, or decimals ? In common fractions, each man cridently has g of a dollar, the answer; but, to express it decimally, we proceed thus :OPERATION. In this operation, we cannot divide Numer. dollars, the numorator, by 5, the denom. Denom. 5)2,0(,4 inator ; but, by annexing a cipher to 2, 20 (that is, multiplying by 10,) we have 26 tenths, or dimes; then 5 in 20, 4 times; Ans. 4 tenths,=,4 that is, 4 tenths, ,4: Hence the common fraction , reduced to a decimal, is ,4, Ans. 2. Reduce 3's to its equal decimal. In this example, by annexing one cipher 32)3,000,09375 to 3, making 30 tenths, we find that 32 is 288 not contained in the 10ths; consequently, a cipher must be written in the 10ths' placo 120 in the quotient. These 30 tenths may be brought into 100ths by annexing another 96 cipher, making 300 hundredths, which con tain 32, 9 times; that is, 9 hundredths. By 240 continuing to annex ciphers for 1000ths, 224 &c., dividing as before, we obtain ,09375 Ans. By conting the ciphers annered to 16) the numerator, 3, we shall find them equei 162 to the decimal places in the quotient OPERATION. Note In the last answer, we have five places for decimals; but, as the 5 in the fifth place is only todooo of a unit, it will be found sufficiently exact for most practical purposes, to extend the decimals to only threo or four places. To know whether you have obtained an equal decuinal, change the decimal into a common fraction by placing its proper denominator under it, and reduce the fraction to its lowest terms. If it produces the same common fraction again it is right; thus, taking the two foregoing examples, ,4=1= Again, ,09375—188760= From these illustrations we derive the following RULE. 1. How do you proceed to reduce a common fraction to its equal decimal ? A. Annex ciphers to the numerator, and divide by the denominator. II. How long do you continue to annex ciphers and divide? 4 Till there is no remainder, or until a decimal is obtained sufficiently exact for the purpose required. III. How many figures of the quotient will be decimals : 4. As many as there are ciphers annexed. IV. Suppose that there are not figures enough in the quotient for this purpose, what is to be done ? A. Prefix ciphers to supply the deficiency. More Exercises for the Slate. 3. Change 4, 4, 4, and z's to equal decimals. A. ,5, ,75, ,25, ,04. 4. What deciinal is equal to zb? (5) What=? (5) What =10? (75) What=? (4) Ans. 1,34. 5. What decimal is equal to 18o? (5) What=?? (25) What=1? (5) What=%(175) What=18? (625) A. 1,6 6. What decimal is equal to $? (1111). What=f? (1444) What=g? (10101! What=$? (3333)* A. ,898901. to * When decimal fraccions continue to repeat the same figure, like 333, &c in this example, they aro calied Repetends, or Circulating Decimals. When only one figure repeats, it is called a single repetend; but, if two or more figures repcat, it is called a compound repetend: tbus, ,333 &c. is a single copetend, ,010101, &c. a compound repetend. When other decimals come beforo circulating decimals, as , in ,8333, the Cecimal is called a mixed repetend. It is the common practice, instead of writing the repeating figures several times, to plar a dotorer the repeating ligare in a single repotcrd ; thus, 11., &e 13 ILVIII. To reduce Compound Numbers to Deci mals of the highest Denomination. Reduce 15 s. (d. to the decimal of a pound OPERATION. 12)6, 0d. In this example, 6 d.=; ne of a shilling and Theme reduced to a decimal by si LVII. 20)15, 5s. is equal to ,5 of a shilling, which, joined with 15 s., makes=15, 5g. In the same ,775£. nianner, 15,5 s. :-20 s.=,775 £, Ans. is written i; also over the first and last repeating figure of a compound rope tend ; thus, for ,230303, Sic. we write, ,03. The vo lue of any repetend, not withistanding it repeate one figure or more av infinite number of times, coming nearer and neurer to a unit each time, though never r' saching, it, may be easily determined by coinınon fractions; as wir appear from what follows. By educing $ to a decimal, we have a quotient consisting of ,1111, &c., thal is, the repotend, ,i ; $, then, is the value of the repetend i, the value of ,333, &c ; that is, the repetend 3 must be three times as much ; that is, and =*;,5=%; and ,j== whole. Hence, we have the following RULE for changing a single repetend to its equal common fraction,-Make the given repetend a numerator, writing 9 underneath for a denominator, and it is durie. What is the valus of ,i? Of ,ż? Ct ,À? Of ,1? 01,8? orð? A. $, g, By changing gt va decimal, we shall havc, ,010101, that is, the repetend „ói. Then, the .eretend ,0i, being 4 times as much, must be of, and „30 must be , alu, ,46=5 If yg be reduced to a docimul, it produces ,ooi. Then the decimal ,004, Deing 4 cimes as much, is ging, and ,036 = . This principle will be true for any pumber of placca. Hence we derive tbe following RULE for reducing a circulating decimal to a common fraction,--Make she given repetend a numerator, and iko denominator will be as in any 9s as there are figures in the repetend Change ,iš to a common fraction. A. }=ir: In the following cxampla, viz. chango ,83 to a common fraction, the reBeating figure is 3 that is, g. and ,£ is %; then g instead of being if Hence we derive the following RULE. ! How must the several denominations be placed ? A. Ono above another, the highest at the bottom. II. How do you divide ? A. Begin at the top, and divide as in Reduction; that is, shillings by shillings, ounces by ounces, &c., annexing ciphers. III. How long do you continue to do so? A. Till the denom. inations are reduced to the decimal required. More Exercises for the Slate. 2. Reduce 7 s. 6 d. 3 qrs. to the decimal of a pound. A. ,378125 £ 3. Reduce 5 s. to the decimal of a pound. A. ,25 £ 4. Reduce 3 farthings to the decimal of a pound. .4. ,003125 £. 5. Reduce 2 qrs. 3 na. to the decimal of a yard. A. ,6875 yd. 6. Reduce 2 s. 3. d. to the decimal of a dollar. A. $,375. 7. Reduce 3 qrs. 3 na. to the decimal of a yard. l. ,9375 yd. 8. Reduce 8 oz. 17 pwts. to the decimal of a pound Troy. A. ,7375 lb. Reduce 3 £, 17 s. 6 d. 3 qrs. to the decimal of a pound. A. 8,878125 £ • unit, is, by being in the second placo, g of th = y; then ana osadiled together, thus, stof=78=, Ans. Hence, to find the value of a mixed repetend—First find the value of the repeating decimals, then of the other decimals, and add these results together. 2. Change ,916 to a common fraction. A. % +980=438=. Proof, 11•12=,910. 3. Change 203 to a common fraction. A. 3. 1'o know if the result be right, change tho common fraction to a decima) grin. If it produces the same, the work is right. Repenting dicci nals may be easily multiplied, subtracted, &o. by first rad de ng them to their equal common fractious. |