barrels? (3548475) In 1,75 barrels? (55125) In 125,626789 barrels? (39572438535). Ans. 39574,9238535 gallons. 6. What will 8,6 pounds of flour come to, at $,04 a pound (344) At $,03 a pound? (258) At $,035 a pound? (301) At $,0455 a pound? (3913) At $,0275 a pound? (23650) Ans. $1,5308. 7. At $9 a bushel, what will 6,5 bushels of rye cost? (585) What will 7,25 bushels? (6525) Will 262,555 bushels? (2362995) Will,62 of a bushel? (558) Will 76,75 bushels? (69075) Will 1000,0005 bushels (90000045) Will 10,00005 bushels? (9000045) A. $1227,307995. DIVISION OF DECIMALS. LVI. In Multiplication, we point off as many decimals in the product as there are decimal places in the multiplicand and multiplier counted together; and, as division proves multiplication by making the multiplier and multiplicand the divisor and quotient, hence, there must be as many decimal places in the divisor and quotient, counted together, as there are decimal places in the dividend. 1. A man bought 5 yards of cloth for $8,75; how much was t a yard? $,8,75-875 cents, or 100ths; now, 875÷5-175 cents, or 100ths, $1,75 Ans. OR By retaining the separatrix, and dividing as in whole numDers, thus : OPERATION. 5)8,75 Ans. $1,75 As the number of decimal places in the divisor and quotient, when counted together, must always be equal to the decimal places in the dividend, therefore, in this example, as there are no decimals in the divisor, and two in the dividend, by pointing off two decimals in the quotient, the number of decimals in the divisor and quotient will be equal to the dividend, which produces the same result as before. 2. At $2,50 a barrel, how many barrels of cider can I have for $11? $11-1100 cents, or 100ths, and $2,50-250 cents, or 100ths; then, dividing 100ths by 100ths, the quotient will evident ly be a whole number, thus 13 10ths, by annexing a cipher (that is, multiplying by 10), placing 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. 75 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 divi. sor; 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 common fractions, thus 0075=100, and,15-10%; now To000 100 X 75 is divided by by inverting To (↑ XLVII.), thus, 15X10000 =178880=18,05, Ans., as before. 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 ciphe 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 Slate. 4. At $25 a bushel, how many bushels of oats may be bought for $300,50? A. 1202 bushels. 5. At 8,124, 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? 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. 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 oushel? (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? (06) By ,02? (132) By,04? (66) By ,002? (132) By,004? (66) Ans. 219,78. REDUCTION OF DECIMALS. I 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 evidently has of a dollar, the answer; but, to express it decimally, we proceed thus: OPERATION. Denom. 5)2,0(,4 20 Ans. 4 tenths,=,4 In this operation, we cannot divide 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, Hence the =,4: common fraction, reduced to a decimal, is,4, Ans. 2. Reduce to its equal decimal. 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 ,03375, Ans. By counting the ciphers annexed to the numerator, 3, we shall find them equal 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 Too 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=1=f. Again, ,09375188360-32. 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, 1, and 25 to equal decimals. A. ‚5,‚75, ,25,,04. 4. What decimal is equal to ? (5) What? (5) What 12? (75) What? (4) Ans. 1,34. 5. What decimal is equal to 80' (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. + &c, *When decimal fraccions cominne to repeat the same figure, like 333, 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, in ,8333, the Zecimal is called a mixed repetend. It is the common practice, instead of writing the repeating figures several times, to place a dot over the repeating figure in a single repetend ; thus, 11!, &e |