OF THE COMPLEMENT AND THE NUMBER CARRIED TO THE QUOTIENT, INSTEAD OF THE NUMBER CARRIED, ITSELF. ART. V. When we employ long division, we multiply the divisor by each quotient figure, and set down the product before subtracting it from the partial dividend. We may somewhat shorten the process, by subtracting each figure of this product, as we multiply, and noting down only the remainder, thus : Here our first quotient figure is 1. We multiply the divisor 764)98532(128 by it, and sabtract each figure of the product, as we multiply, 2213 setting down only the remainder, 221. To this we annex the next figure of the dividend, 3. We then multiply and subtract, 6852 as before, and so on. Our final remainder is 740; our quotient 740 128. In this operation, there is so much to be retained in the mind, that it re. quires considerable acquaintance with numerical calculations, to render it very useful. ART. VI. When the divisor and dividend are large, it will often be found useful to make a table of products of the divisor, by the nine digits. For each quotient figure must be one of these digits or a cypher; and, by means of the table, we can tell , at a glance, how many times the divisor is contained in each partial dividend. We have, then, the quotient figure and its corresponding product, and, of course, have nothing to do but subtract. These are the most important contractions in Division. We have now been attending to Notation or Numeration, Addition, Multiplication, Subtraction, and Division. These are called the fundamental or ground rules of Arithmetic, because, by their various combinations, all arithmetical operations are performed. EVOLUTION might, perhaps, seem an exception to this remark; but a little reflection will convince any one, that this process is only a case of division, in which the divisor and quotient are both unknown, but are required to be equal. For division consists in resolving a number into factors, and evolution, in resolving a number into equal factors. COMPOUND NUMBERS. ADDITION. MENTAL EXERCISES. XXXV. 1. Four boys gathered chesnuts. The first gathered 2 qts. ; the second, 3 qts. ; the third, 7 qts., and the fourth, 5 qts. How many pecks did they all gather? Then, 2 qts. and 3 qts. and 7 qts. and 5 qts. are how many pecks ? 2. A silversmith melted together several pieces of silver, as follows; one weighing 10 oz. ; another 11 oz. ; another, 5 oz. ; and another 6 oz. How many lbs. did he melt. Then, 10 oz. and 11 oz. and 5 oz. and 6 oz. are how many lbs. ? 3. In 3s. Ad. and 5s. 6d. and 2s. ld. how many shillings and pence? 4. In 4s. 1d. and Ss. 3d. and 2s. 11d. how many shillings and pence? 5. In 16s. 9d. and 7s 10d. how many pounds, shillings and pence? 6. In 3 qts. 1 pt. and 5 qts 1 pt. how many gal. and qts. ? 7. In 3 qts. 1 pt. and 2 qts. 1 pt. and 1 qt. 1 pt. how many gal. qts. and pts.? 8. In 2 yds. 2 ft. and 7 yds. 2 st. how many yds. and ft. ? 9. In 6 fur. 35 rds. and 7 fur. 15 rds. how many mls. fur. and rds. ? In the above examples, the numbers have consisted of sereral denominations. Such numbers are called COMPOUND NUMBERS. Numbers consisting of one denomination only, are called SIMPLE NUMBERS. Write the following examples. 10. A boy paid for a book 58. 4d. 2 qrs.; for a bunch of quills, 28. 3d. 3 qrs.; and for a penknife 38. 5d. I qr. What cost the whole ? Place like denominations under each other ; d. qrs. then add the qrs. 1+3+2=6qrs. But 6 qrs.=-1d. 4 2 2 qrs., because 4 qrs.=ld. Set down the 2 qrs. 2 3 3 and carry the ld. to the pence. There are then 3 5 1 13d.=1s. id. Set down the ld. and carry the ls. to the shillings. The shillings, added, are 11 1 2 11s., which, as they are not enough to maxe a £. set down under shillings. In like manner, perform the following. 11. Bought a watch for 5£. 68. and a chain for 1£. 19s. How much did I give for both? Ans. 7£. 58. 12. A man has four farms. The first contains 100 acres, 1 rood, 20 rods; the second 75 acres, 2 roods, 10 rods; the third 150 acres, 15 rods; the fourth 125 acres. 1 rood, 5 reds. How many acres in all ? · Ans. 451 acres, 1 rood, 10 r ds. 13. A merchant bought 4 pieces of cloth, the first containing 10 yds. 2 qrs. 1 nl.; the second 25 yds. 2 qrs. 1 pl.; the third 20 yds. 2 qrs. 2 nls. and the fourth 10 yds. 1 qr. 1 nl. How much in all ? Ans. 67 yds. 1 nl. 14. Add together 38 gals. 2 qts. 1 pt. 2 gi. ; 16 gals. 1 qt. 20 gals. 2 qts. 1 pt. 1 gi. ; 18 gals. 1 qt. 1 pt.; 7 gals. 1 qt. 2 gi. ; 30 gals. 2 qts. 1 pt. Ans. 132 gals. From these examples we derive the rule to perform Addition of Compound numbers. I. PLACE THE SAME DENOMINATIONS UNDER EACH OTHER. II. ADD, FIRST, THE LOWEST DENOMINATION. FIND HOW MANY OF THE NEXT HIGHER ARE CONTAINED IN THE SUM, whicu NUMBER CAR. RY TO THE NEXT HIGBER, AND PLACE THE REMAINDER UNDER THE DEYOMINATION ADDED. III. PROCEED THUS WITH ALL THE DENOMINATIONS, gi. ; EXAMPLES FOR PRACTICE. 15. A man bought land to the amount of 69£ 13s. 5d.; farming implements to the amount of 11£ 10s.; a yoke of oxen for 15£ 6s.; a horse for 13£. Os. 4d.; a cart for 4€ 178. 8d. and a saddle for 19s. 4d. 2 qrs. What did the whole cost him; Ans. £115; 6; 95. s. 16. A silversmith purchased seven ingots of silver. The first reighed 11 lb. 6 oz. 18 dwt. ; the second 9 lb. 3 oz. 7 dwt., the third 4 lb. 7 oz. 9 dwt. 5 gr.; the fourth, 8 lb. 4 oz. 6 dwt. ; the fifth, 10 lb. 3 oz. 5 dwt. 19 gr.; the sixth, 7 lb. 9 oz. 0.dwt. 18 gr.; and the seventh 8 lb. 11 oz. 10 dwt. What did the whole weigh; Ans, 60 lb. 9 oz. 16 dwt. 18 gr. 17. George lived in Hartford until he was 14 yrs. 3 mo. 4 d. old; then he went to New Haven, where he remained 8 yrs. 5 mo. ; then he went to New York, where he remained 3 yrs.; then to Phila. delphia, where he staid 3 yrs. 2 mo. 1 d. His journeys occupied 4 days ? How old was he then ? Ans. 28 yrs. 10 mo. 1 w. 2 d. 18. A man has, in real estate £304 ; 5, in one place, and £247 ; 0; 11, in another; and in personal property, the several sums £34; 19; 7, £7; 18; 5, £45; 0; 6, and 19s. Od. 3 qrs. How much in all ? Ans. £640; 3; 5; 3. 19. A man brings to market 4 loads of wood, containing, the first 1 cord 60 ft. 860 in.; the second l cord 67 ft. 68 in.; the third I cord 30 ft. 300 in.; the fourth 1 cord 30 ft. 631 in. How much in all ? Ans. 5 cords, 60 ft. 131 in 20. Bought a quantity of goods for £125; 10: paid for freight £3; 19; 6: for transportation, £2; 5: for duties £1; 15; 10: my ex. penses were £2; 13; 9. What was the whole expense of the goods to me ? MULTIPLICATION. MENTAL EXERCISES. ♡ XXXVI. 1. A man gave 6d. apiece to four of his children. How many shillings did he give them? 2. 5 boys gathered 3 quarts of walnuts apiece. How many pecks did they all have?. 3. Three baskets hold 1 pk. 4 qts. each. How many pks. and qts. will all hold ? 4. If one bushel of grain cost 2s. 6d. how much will 2 cost? How much will 3? will 4?..5?. 6? 7? 8? 5. Multiply 3 qrs. 4 nls. by 2; by 3; by 4; by 5; by 6; by 7; by. 8; 6. Multiply 15 min. by 15 sec. by 4; by 8; by 12; 7. Multiply 1 pt. 3 gi. by 2; by 3; 4; 5; 6; 7; 8; 9; 8. Multiply 9 oz. 12 dwt. 13 gr. by 2; by 3; 4; 5 6; 7; 8; 9; 9. Multiply 2 in. 2 b. c. by 2; by 3; by 4; 5; 6; 7; S; 9; 10. Multiply 3 yds. 2 st. 8 in. by 2; 3; 4; 5; 6; 7; 8; 9; Perform the following on your slate, 11. A man bought 3 sheep at 1£. 10s. apiece. What cost the whole ? Set the multiplier under the multiplicand, £. and multiply as in simple numbers, observing only 1 10 to carry from one denomination to another, as in Ad. 3 dition of Compound Numbers. 4 10 12. How much wool in 3 packs, each pack weighing 2 cwt. 2 qrs. 13 lb. A. 7 cwt. 3 qrs. 11 lb. 13. What is the value of 5 cwt. of raisins, at 2£. ls. 8d. pr. cwt. A. 10£. 8s. 4d. 14. What is the weight of 3 doz. silver spoons, each doz. weighing 2 lb. 6 oz. 12 dwt. 3 gr. A. 7 Ib. 7 oz. 16 dwt. 9 grs. 15. In 8 bales of cloth, each bale containing 12 pieces, and each piece 27 yds. 1 qr. 2 nls., how many yds. ? A. 2,628. From these examples, we dorive the following rule. PLACE THE MULTIPLIER UNDER THE MULTIPLICAND, MULTIPLY EACII DENOMINATION SEPARATELY, BEGINNING WITH THE LOWEST, AND CARRK AS IN ADDITION. EXAMPLES FOR PRACTICE. 16. What cost 12 bu. of apples, at ls. 9d. per bu. ? A. £1. ls. 17. What cost 9 lbs. of cinnamon, at Ils. 4!d. per lb. ? A. £5; 2; 4; 2. 18. FORMS OF BILLS. Sheffield, Jan. 1, 1828. BOUGHT OF WM. B. SAXTON. Rec'd. payment, £20; 2; 2, WM. B. Saxton. Hartford, March 7, 1830. BOUGHT OF PACKARD & BUTLER. £5; 8; 10 3 Atlantic Souvenir, for 1830, at 13s. Gd. 2; 0; 6 1 Byron's works, 1; 0; 0) 1 Johnson's do. 1; 15 Rec'd. payment, £10; 4; For P. & B. WILLIAM TRUSTY. 19. If I load of hay weigh 1 T. 10 cwt. 2 qrs. 20 lb. 5 oz. 15 dr. what will 33 loads weigh ? 20. Multiply 27 gals. 1 qt. 1 pt. 3 gi. by 28. 21. Multiply 67 yds. 3 qrs. 3 nls. by 72. SUBTRACTION. MENTAL EXERCISES. s XXXVII. 1. A goldsmith having 1 lb. of silver, melted up 7 oz. of it. How much had he left ? 2. A boy having 3 shillings, gave away 1s. 6d. How much had he left ? 3. A pitcher contains 3 quarts of cider, and a man fills 3 pint tumblers from it. How much cider is left in the pitcher ? 4. Take 2 gi. from 1 pt. From 1 qt. From 1 gal. 5. Take 3 nls. from 1 qr. From 1 yd. From 1 E. E. From 1 E. F. 6. Take 3 oz. from 1 lb. From 1 lb. 1 oz. From 2 lb. 2 oz. 12 gr. 7. Take 19 gr. from 1 9. From 1 9. 6 gr. From 33. 8. Take 3d. from 1 shilling. From Is. 1d. From 3s. 2d. 9. Take 15 minutes from 1 hour. From lh. 5m. From 7h. 7m. 10. Take 5 drams from 116. 13., From 63.33. From 91. 13. 43. Write the following examples 11. A man's property amounted to 6,872£. 158., and he lost a ship worth 1,539£. 175. What was he then worth? As we cannot take 178, from 159. we must borrow f. a £.=20s. This 20s. added to 159. makes 358., from 6872 ; 15 which, 17 being taken, 18 are left. As we borrowed 1539; 17 a £, we must make the pounds 1 less ; that is, 6,871 instead of 6,872. 6871-1539=5332. 5332 ; 18 12. George had 16s. 8d. and he gave 4s. 9d. for a sled. How much had he left ? Ans. 116. lld. It will be better, instead of diminishing the next higher denomina. tion of the minuend, to increase that of the subtrahend by 1; and the result will be the same, for 5 taken from 16, evidently leaves the same remainder, as 4 from 15, viz. 11. This is the best methoä in compound numbers, because, sometimes, the figure of the minuend may be a cypher. We should not be able to diminish this. 6. |