Hence we derive the following RULE. Q. How do you write the numbers down? A. As in Simple Subtraction. More Exercises for the Slate. 1. Bought a hogshead of molasses, containing 60,72 gallons, how much can I sell from it, and save 19,999 gallons for my own use? A. 40,721 gallons. 2. James rode from Boston to Charlestown in 4,75 minutes, Rufus rode the same distance in 6,25 minutes; what was the difference in the time? A. 1,5 min. 3. A merchant, having resided in Boston 6,2678 years, stated his age to be 72,625 yrs. How old was he when he emigrated to that place? A. 66,3572 yrs. Note. The pupil must bear in mind, that, in order to obtain the answer, the figures annexed to each question, are first to be pointed off, supplying ciphers, if necessary, then added together as in Addition of Decimals. 4. From ,65 of a barrel take ,125 of a barrel-525; take 2 of a barrel-45; take ,45 of a barrel-2; take ,6 of a barrel-5; take ,12567 of a barrel-52433; take 26 of a barrel-39. A. 2,13933 barrels. 5. From 420,9 pipes take 126,45 pipes-29445; take ,625 of a pipe-420275; take 20,12 pipes-40078; take 1,62 pipes41928; take 419,89 pipes-101; take 419,8999 pipes-10001. A. 1536,7951 pipes. MULTIPLICATION OF DECIMALS. LV. 1. How many yards of cloth in 3 pieces, each piece containing 20% yards? OPERATION. 20,75 Ans. 62,25 yds. the multiplier also, we In this example, since multiplication is a short way of performing addition, it is plain that we must point off as in addition, viz. directly under the separating points in the multiplicand; and, as either factor may be made the multiplicand, had there been two decimals in 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 deci mals, as there are decimal places in both the factors. 2. Multiply,25 by,5. OPERATION. 25 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 evi. dent by considering both the factors common fractions, and multiplying by ¶ XLI., thus, 25%, and ,5=fo; now, % X 10= 10, which, written decimally, is ,125, Ans., as before. 125 3. Multiply,15 by ,05. OPERATION. ,15 ,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 T80; then 10 X 150=10000=,0075, Ans., the same as before. Ans.,0075 ,05 From these illustrations we derive the following RULE. Q. How do you multiply in Decimals? A. As in Simple Multiplication. Q. How many figures do you point off for decimals in the product? A. As many as are in both the multiplicand and multiplier. Q. If there be not figures enough in the product for this purpose, how would you proceed? A. Prefix ciphers enough to make as many. A. To place after. Q. What is the meaning of prefix? A. To place before. More Exercises for the Slate. 4. What will 5,66 bushels of rye cost, at $1,08 a bishel? A. $6,1128, or $6, 11 c., 21 m. 5. How many gallons of rum in,65 of a barrel, each barrel containing 31 gallons 2-20475. 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 125,626789 barrels ?-39572438535. A. 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. A. $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. 1 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 it 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 numbers, thus: OPERATION. 5)8,75 Ans. $1,75 As the number of decimal places in the divisor and quotient, when counted together, mur 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 e al to the dividend, which produces the same result hefore. 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 evidently be a whole number, thus: OPERATION. 250) 1100 (4188 barrels, Ans. 1000 100 250 In this example, we have for an answer 4 barrels, and 288 of another barrel. But, instead of stopping here in the process, we may bring the remain der, 100, into tenths, 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, (TLV.) 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 divisor; 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. The above operation will appear more evident by common fractions, thus: ,0075-100%0, and,15; now TOOOO divided by by inverting (¶ XLVII.), thus, =138880=180=,05, Ans., as before. = 75 is From these illustrations we derive the following Q. How do you write the numbers down, and divide ? Q. 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. Q. 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. 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? A. A whole number. Q. 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 re 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 bought for $300,50? A. 1202 bushels. 5. At $,123, 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 annexed to each question, 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 |