From these illustrations we derive the following RULE. 1 How are the numbers to be written down? A. Terths under tenth, hundredths under hundredths, and so on. II. How do you proceed to add? A. As in Simple Addition. III. Where do you place the separatrix? A. Directly under the separating points above. More Exercises for the Slate. 2. James bought 2,5 cwt. of sugar, 23,265 cwt. of hay, and 4,2657 cwt. of rice; how much did he buy in all? A. 30,0307 cwt. 3. James is 14 years old, Rufus 155, and Thomas 1675; what is the sum of all their ages? A. 46,5 years. 4. William expended for a chaise $255, for a wagon $37, for a bridie $10, and for a saddle $11; what did these amount to? A. $304,455. 000 5. A merchant bought 4 hhds. of molasses; the first contained 621 gallons, the second 7265 gallons, the third 50 gallons, and the fourth 55 gallons; how many gallons did he buy in the whole ? A. 240,6157 gallons. 6. James travelled to a certain place in 5 days; the first day he went 40 miles, the second 28 miles, the third 42 miles, the fourth 22 miles, and the fifth 2910 how far did he travel in all? .A. 162,0792 miles. miles, 7. A grocer, in one year, at different times, purchased the fol lowing quantity of articles, viz. 427,2623 cwt., 2789,00065 cwt., 42,000009 wt., 1,3 cwt., 7567,126783 cwt., and 897,62 cwt.; how much did he purchase in the whole year? 9. 11724,309742 cwt 8. What is the amount of, 215fvo, 610‰0, 24570000, 1108800, 1000, 42710000, 40, 10000, 62 and 1925 ? 89 A. 2854,492472. 9. What is the amount of one, and five tenths; forty-five, and three hundred and forty-nine thousandths; and sixteen hundredths? A. 47,009. SUBTRACTION OF DECIMALS. ¶ LIV. 1. A merchant, owing $270,12, paid $192,625 how much did he then owe? OPERATION $270,42 $192,625 Ans. $77,795 For the reasons shown in Addition, we proceed to subtract, and point off, as in Sub traction of Federal Money. Hence we derive the following 1 How do you write the numbers down? A. As in Addition af Decimals. II. How do you subtract? A. As in Simple Subtraction. III. How do you place the separatrix? A. As in Addition of Decimals. 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. Ilow 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 in the parentheses 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,20 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 pipes, (41928) take 419,89 pipes; (101) take 419,8999 pipes; (10001). Ans. 1536,7951 pipes. MULTIPLICATION OF DECIMALS. ↑ LV. 1. How many yards of cloth 'n 3 pieces, each piece containing 20 yards? OPERATION 20,75 3 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 Ans. 62,25 yds. made the multiplicand, und geen two decimals in the multiplier also, we must have pointed off tw. 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 evi. dent by considering both the factors common fractions, and multiplying by ¶ XLI., thus ;,25=10%; and,5=1%; now 25 X 5 125 which, written decimally, is,125, Ans., as before. 100X10 1000' 3. Multiply,15 by,05. 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 This will appear evident by thə 15 X 5 75 following;,15% and ,05=1Ỗo; then 0075, Ans., the same as before. Ans. ,0075 many. 100X100 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. 21 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 125,620799 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 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, Lence, 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 ta 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 numcers, 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 decinals in the divisor, and two in the dividend, by pointing off two decinials 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 OPERATION. 250)1100(4188 barrels, Ans. 1000 100 250 In this example, we have for an answer 4 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 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, reck oning 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 aivisor 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 decima. place in the quotient, the deficiency must be sapplied by prefixing a cipher. 75 The above operation will appear more evident by common fractions, thus ,0075=1%%, and,15=1; now TooOO is divi·led by by inverting (5 XLVII.), thus, 15 X 10000 3888=180,05, Aus., as vefore. 100 X 75 |