decimals in the multiplier also, we must have pointed off two more places for decimals, which, counting both, would make 4 Hence we must always point off in the product as many places for decimals, as there are decimal places in both the factors. 2. Multiply,25 by,5. ,5 Ans. ,125 In this example, there being 3 decimal places in both the factors, we point off 3 places in the product, as before directed. The reason of this will appear more evident by considering both the factors common fractions, and multiplying by XLI., thus; 25, and 5; now 25 X 5 125 which, written decimally, is,125, Ans, as before. 100X10 1000' OPERATION. ,05 In this case, there not being so many figures in the product as there are decimal places in both the factors (viz. 4), we place two ciphers on the left of 75, to make as many. This will appear evident by the following;,15% and ,05=8; then ,0075, Ans., the same as before. Ans. ,0075 15 X 5 100X100 75 10000 From these illustrations we derive the following RULE. 1. How do you multiply in Decimals? A. As in Simple Multiplication. II. How many figures do you point off for decimals in the product? A. As many as are in both the multiplicand and multiplier. III. If there be not figures enough in the product for this purpose, how would you proceed? A. Prefix ciphers enough to make as many. Q. What is the meaning of annex? A. To place after More Exercises for the Slate. 4. What will 5,66 bushels of rye cost, at $1,08 a bushel? A. $6,1128, or $6. 11 c. 2% m. 5. How many gallons of rum in ,65 of a barrel, each barrel containing 31 gallons? (20475) In,8 of a barrel? (252) In ,42 of a barrel? (1323) In,6 of a barrel? (189) In 1126,5 barrels? (3548475) In 1,75 barrels? (55125) In 125,626729 barrels? (39572433535). 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 bushe!, 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 10,0000 bushels? (9000045) A. $1227,307995. DIVISION OF DECIMALS. ILVI. 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, ence, there must be as many decimal places in the divisor and quotient, counted together, as there are decimul places in the dividend. 1. A man bought 5 yards of cloth for $8,75; how much was ta yard? $,8,75-875 cents, or 100ths; now, 875÷5=175 Tents, or 100ths, $1,75 Ans. = OR By retaining the separatrix, and dividing as in whole numpers, thus: OPERATION. 5)8,75 As the number of decimal places in the divisor and quotient, when counted together, must always be equal to the Ans: $1,75 decimal places in the dividend, therefore, in this example, as there are no decunals in the divisor, and two in the dividend, by pointing off two decr mals 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 Jy be a whole number, thus 13 barrels, and 188 of another barrel. But, instead of stopping here in the process, we may bring the remainder, 100, into 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 sepa rate 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 quo tient 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. 75 100 X 75 The above operation will appear more evident by common fractions, thus; .0075–10%, and,15—165; now todJO is divided by ros by inverting Too (¶XLVII.), thus, 15X1x08 =138930=3,05, Ans., us 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 $,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 how many barrels will each have? A. 23,3 barrels. men, 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 pushel? (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 Décimal. 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 denoninator; but, by annexing a cipher to 2 (that is, multiplying by 10,) we have 20, tenths, or dimnes; then 5 in 20, 4 times; Hence the that is, 4 tenths, ,4: common fraction, reduced to a decimal, is,4, Ans. 2. Reduce to its equal decimal. Ans. 4 tenths,: OPERATION. 288 120 96 ,4 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 to the numerator, 3, we shall find them equal to the decimal places in the quotient. 240 224 160 160 |