barrels, and 188 of 10ths, by annexing a cipher (that is, multiplying by 10), placing a decimal point at the right of 4, a whole number, to keep it separate from the 10ths, which are to follow. The separatrix may now be retained in the divisor and dividend, thus: OPERATION. 2,50)11,00 (4,4 Ans. 1000 1000 1000 We have now for an answer, 4 barrels and 4 tenths of another barrel. Now, if we count the decimals in the divisor and quotient (being 3), also the decimals in the dividend, reckoning the cipher annexed as one decimal (making 3), we shall find again the decimal places in the divisor and quotient equal to the decimal places in the dividend. We learn, also, from this operation, that, when there are more decimals in the divisor than dividend, there must be ciphers annexed to the dividend to make the decimal places equal, and then the quotient will be a whole number. Let us next take the 3d example in Multiplication, (¶ LV.) and see if multiplication of decimals, as well as whole numbers, can be proved by Division. 3. In the 3d example we were required to multiply,15 by ,05; now we will divide the product 0075 by ,15. OPERATION ,15),0075,05 Ans. 7.5 We have, in this example, (before the cipher was placed at the left of 5), four decimal places in the dividend, and two in the diviser; hence, in order to make the decimal places in the divisor and quotient equal to the dividend, we must point off two places for decimals in the quotient. But, as we have only one decimal place in the quotient, the deficiency must be supplied by prefixing a cipher. 75 The above operation will appear more evident by commen fractions, thus: 0075-1000, and 15; now Tosad 100 X 75 is divided by fős by inverting 1% (TXLVII.), thus, 5X100X =138885=155,05, Ans., as before. T From these illustrations we derive the following RULE. I. 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 quotient 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 divisor has more decimal places than the dividend? A. Annex as many ciphers to the dividend as will make the decimals in both equal. V. What will be the value 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 when there is a remainder, how do you proceed? A. Annex ciphers to the remainder, or dividend, and divide as before. VII. What places in the dividend do these ciphers take? A. Decimal places. More Exercises for the State. 4. At $25 a bushel, how many bushels of oats may be bought for $300,50? A. 1202 bushels. 5. At $,124, or $,125 a yard, how many yards of cotton cloth may be bought for $16? A. 128 yards. 6. Bought 128 yards of tape for $,64; how much was it a yard? . $,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. 9. If $2,25 will board one man a week, how many weeks can he be boarded for $1001,25? (445) For $500,85? (2226) For $200,7? (892) For $100,35? (446) For $60.75? (27) A. 828,4 weeks. 10. If 3,355 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) 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) By 6,075? (257) Ans. 481,875. 12. What is the quotient of,264 divided by 2? (132) By ,4? (66) By ,02? (132) By,04? (66) By,002? (132) By,004? (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 each? and how much in 10ths, or decimals? In common fractions, each man eridently has & of a dollar, the answer; but, to express it decimally, we proceed thus: OPERATION. Denom. 5)2,0(,4 20 In this operation, we cannot divide 2 dollars, the numerator, by 5, the denominator; but, by annexing a cipher to 2, (that is, multiplying by 10,) we have 20 tenths, or dimes; then 5 in 20, 4 times; that is, 4 tenths, =,4: Hence the common fraction, reduced to a decimal, is,4, Ans. 2. Reduce to its equal decimal. Ans. 4 tenths,=,4 OPERATION. 288 120 96 In this example, by annexing one cipher 32)3,00 (,09375 to 3, making 30 tenths, we find that 32 is not contained in the 10ths; consequently, a cipher must be written in the 10ths' place in the quotient. These 30 tenths may be brought into 100ths by annexing another cipher, making 300 hundredths, which contain 32, 9 times; that is, 9 hundredths. By continuing to annex ciphers for 1000ths, &c., dividing as before, we obtain ,09375, Ans. By counting the ciphers annexed te the numerator, 3, we shall find them equa to the decimal places in the quotient. 240 224 160 160 Note. In the last answer, we have five places for decimals; but, as the 5 in the fifth place is only 1000 of a unit, it will be found sufficiently exact for most practical purposes, to extend the decimals to only three or four places. To know whether you have obtained an equal decimal, 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==}. Again, ,09375 88680=34. 93 From these illustrations we derive the following RULE. I. 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? A. 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,,, and to equal decimals. A.,5,,75, ,25,,04. 4. What decimal is equal to ? (5) What? (5) What ? (75) What? (4) Ans. 1,34, 5. What decimal is equal to. Too? (5) What? (25) What? (5) What? (175) What? (625) A. 1,6. 6. What decimal is equal to ? (1111) What? (4444) What? (10101) What? (3333)* A.,898901. + *When decimal fractions continue to repeat the same figure, like 333, &c., in this example, they are called Repetends, or Circulating Decimals. When only one figure repeats, it is called a single repetend; but, if two or more figures repeat, it is called a compound repetend: thus, ,333, &c. is a single repetend, 010101, &c. a compound repetend. When other decimals come before circulating decimals, as ,8 in ,8333, thọ Cecimal is called a mixed repetend. It is the common practice, instead of writing the repeating figures, sovora times, to place a dot over the repeating figure in a single repetend; thus, .1, &c. 1 LVIII. To reduce Compound Numbers to Decimals of the highest Denomination. Reduce 15 s. 6d. to the decimal of a pound. OPERATION. 12)6, 0 d. 20)15, 5 s. In this example, 6 d. of a shilling, and, reduced to a decimal by ¶ LVII., is equal to 5 of a shilling, which, joined with 15 s., makes 15, 5s. In the same manner, 15,5 s.÷20 s.-,775 £, Ans. 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 to a unit each time, though never 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 is, the repetend,,;, then, is the value of the repetend 1, the value of ,333, &c.; that is, the repetend 3 must be three times as much; that is, and ‚4—§;‚5—3 ; and,9—3—1 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 done. What is the value of,i? Of‚1⁄2? Of‚Ã? Of,7? Of‚8? Of‚¿? A. †,§‚†, 7, 8, 8. By changing to a decimal, we shall have, ,010101, that is, the repetend ,ói. Then, the repetend,0i, being 4 times as much, must be, and,36 must be38, also,45-45. If be reduced to a decimal, it produces,001. Then the decimal,004, being 4 times as much, is 5, and,036. This principle will be true for any number of places. 36 999. Hence we derive the following RULE for reducing a circulating decimal to a common fraction,-Make the given repetend a numerator, and the denominator will be as many 9s as there are figures in the repetend. Change,18 to a common fraction. A. = In the following example, viz. change ,83 to a common fraction, the repeating figure is 3 that is, 3, and ,8 is 10 ; then, instead of being of |