be, of once or twice the multiplicand. The same principle may be applied to a variety of other numbers, such as 68 to 72; 78 to 82; 98 to 102; 90; 180; 270; 360, &c., the last four numbers being the same as 100, 200, 300, 400, less one tenth. 21. How many are 8X24 ? 16X37 ? 27X24 ? 36 X 25 ? 44X15? 72X30 ? 64X24 ? 85X25 ? 92x27? 76X28? 116 X 25 ? 47 X32 ? 94X38 ? 77X28 ? 56 x 19 ? 49 X 49 ? 52X47 ? 80X27? 43 X 26 ? 18X144 ? 55 (50+io of 50) X86 ? 32X24 ? 78x7 ? 45 X 45 (50— To of 50) ? 23X72 ? 99 X 28 ? SECTION XIX. - Practical Questions. 1. A FARMER sold a flock of 300 sheep at 2 dollars a head, and bought 25 cows at 18 dollars each. How much money had he left? 2. What is the cost of 28 bushels of oats at 32 cents a bushel ? 3. What cost 27 bushels of corn at 49 cents per bushel ? 4. How much must be paid for 15 thousand feet of boards at 18 dollars a thousand; and 6 thousand shingles at 3 dollars a thousand ? [16X18. Why ?] 5. How much is due to a laborer for working 24 days at 75 cents per day? 6. What cost 18 bushels of corn at 58 cents a bushel ? 7. A man had 40 barrels of flour. He sold 16 of them at 6 dollars a barrel, and the rest at 7 dollars a barrel. How much did he get for the whole ? [40X6+24.) Why? 8. What is the cost of 14 bureaus at 15 dollars each ? 9. What cost 24 bedsteads at 23 dollars each ? 10. A man bought 7 barrels of sugar at 13 dollars a barrel, and paid 48 dollars. How much remained due ? 11. A farmer had an apple orchard, consisting of 16 rows of trees, and 14 in each row; and an orchard of peaches, of 13 rows and 17 in each. Which orchard had the greater number of trees, and what was the difference? 12. Two trains of cars leave a depot in different directions, one going eastward for 14 hours at the rate of 18 miles an hour, and the other westward for 18 hours at 16 miles an hour. How far were the trains then apart? [(32X16)+(14x2.)] Why? 13. How much must be paid for 16 tierces of rice at 15 dollars a tierce, and 18 barrels of sugar at 17 dollars a barrel ? [(34X15) + (2x18.)] Why? 14. What would be the cost of 23 pounds of black tea at 34 cents a pound, and 27 pounds of green tea at 44 cents a pound? [50 X 44-230.] Why? 15. What would be the cost of 28 yards of cloth at 84 cents a yard, and 27 yards at 44 cents per yard ? [(55 X 84) (27X 40.)] Why? 16. A man bought 23 bushels of corn at 60 cents a bushel, and 17 bushels of rye at 64 cents a bushel. What was the cost of the whole ? [..X..-92.) Why? 17. What must I give for 19 bushels of buckwheat at 45 cents a bushel, and 25 bushels of oats at 43 cents a bushel ? [..X..-50.] Why? 18. A farmer bought from a merchant 23 yards of satinet at 38 cents a yard, and paid him 27 bushels of oats at 42 cents a bushel. What is the balance, and to whom must it be paid ? [65 X 4.] Why? 19. What is the difference in value of 24 pounds of tea at 35 cents per pound, and 30 bushels of oats at 40 cents a bushel ? [(24 x 5) + (6 X 40.)] Why? SECTION XX. – Division of Large Numbers by a Single Digit. Division of large numbers mentally by a single digit is generally considered more difficult than any of the other elementary operations. But it may be very much facilitated by some previous practice in the rapid resolution of a dividend, mentally, or by inspection, into separate MULTIPLES of the divisor; that is, into numbers containing the divisor an exact number of times, when its respective digits are not so divisible. Thus, neither of the digits in 92 are divisible by 4; but the number can be readily resolved by the eye, or in the mind, into 8ty and 12; and 192, in like manner, is readily perceived to be resolv. able into 16ty and 32. Again, 84 requires no resolution for division by 4; but, if 3 be the divisor, it should be considered as 6ty and 24 ; while, if the divisor be 7, 84 should be considered as 7ty and 14. This process of rapid resolution is useful, even when the dividend is not exactly a multiple of the divisor, and consequently produces a remainder. Thus, for division by 3, 44 may be resolved into 30, 12, and 2; and for 6, 93 may be resolved into 60, 30 and 3. 1. Prepare 72 for division by 3; by 4; by 6. 2. Prepare 54 for division by 2; by 6. 3. Prepare 57 for division by 5; by 4; by 6 [30, 24, 3]. 4. Prepare 48 for division by 3; by 5. 5. Prepare 87 for division by 7; by 6; by 5. 6. Prepare 174 for division by 8; by 7; by 6; by 5; by 4; by 3. 7. Prepare each of the following numbers for division by 9: 100; 114; 125 [90, 27, 8]; 136; 153; 142; 171 ; 281. 8. Prepare each of the following numbers for division by 8: 176 ; 196; 232 ; 352; 268; 576; 455. 9. Prepare each of the following numbers for division by 7: 244; 638; 272; 289; 356; 376; 485. 10. Prepare each of the following numbers for division by 6: 532; 236; 748; 255 ; 192. 11. Prepare each of the following numbers for division by 4: 236 ; 347; 252 ; 375; 458; 137. 12. Prepare each of the following numbers for division by 3: 226; 452; 478; 246; 329. 13. Divide the following numbers severally by 9: 192; 353; 468; 736; 228. 14. Divide the following numbers severally by 8: 176; 325 ; 424; 636; 478; 528. 15. Divide the following numbers severally by 7: 325 • 236; 124; 242; 188; 378; 434; 272. 16. Divide the following numbers severally by 6: 532, 312; 636; 234; 152; 444; 228. 17. Divide the following numbers severally by 5: 325; 432; 876; 152; 645; 138. 18. Divide the following numbers severally by 4: 316; 237; 251; 138; 246. 19. Divide the following numbers severally by 3: 213; 231; 316; 415; 632; 174. 6* SECTION XXI. - Practical Questions. 1. A COMPANY of 9 men undertook to build a bridge, for which they were paid 836 dollars. How much would each receive if the money was equally divided among them? 2. How much more would each have received if the company had only consisted of 6 persons instead of 9 ? 3. How many weeks are there in a year of 365 days? 4. Five brothers sold a piece of property, in which each had an equal interest, for 3215 dollars. How much would each receive? 5. Four equal partners found, at the close of a year's business, that the profits on their sales amounted to 8426, while the expenses amounted to 2000 dollars. How much would be the dividend of each partner? 6. A merchant owned an eighth of a vessel which traded to the West Indies. In one of her voyages the net profits amounted to 3424 dollars. What would be the amount of that merchant's share ? 7. Three men went to fish for mackerel, and at the close of the day found they had caught 456 fish. How many would each man have, if they were equally shared ? 8. The property of a man dying intestate was found to amount to $3492, of which, by law, one-third goes to his widow, and the rest is divided equally among his 6 children. What will be the share of the widow, and of each child ? 9. But, supposing the expenses of settling the estate amount to $450, what would then be the respective shares of the widow, and of each child ? 10. A captain and 3 men went out on a fishing excursion, in which 498 fish were taken. The owner of the boat and fishing apparatus was to have one share, the captain two shares, and each of the seamen one share. What would be the respective shares of each person interested ? 11. Four men contracted to build 2 bridges. For one of them, the price was to be $1500; and for the other, $1450. The materials, tools, &c., cost them $400. What would be each man's share of the remainder ? 12. A farm, on which was a mortgage of $300, was sold for $1400. It belonged to three brothers. How much would be each one's share of the net proceeds ? CHAPTER II. INCREASE AND DECREASE OF FRACTIONS, OR BROKEN NUM BERS. SECTION I. - Fractions of Numbers exceeding Unity. number, is divided into two equal parts, each of these parts is called the half of the thing, or the number. When any thing, or any number, is divided into three equal parts, each of these is called the third of the thing or number. When divided into four equal parts, one of them is called the fourth part. In like manner, when a thing or number is divided into five, ten, twenty, or any other number of equal parts, each of these parts is called the fifth, tenth, twentieth, &c., part. Such numbers as these are called fractions, a word which means broken into parts. They are represented by figures as follows: signifies one-half; } signifies a third ; 4 a fourth ; To a tenth, and so on, When we wish to express more than one of the parts, they are represented in a similar manner. Thus, stands for two-thirds; for three-fifths; 34, fifteen twenty-fourths, &c. [Here the teacher will write a variety of fractions on the blackboard, to be named by the class till the subject is familiar.] Fractions always arise from division, and are, therefore, very properly expressed by the same character, namely, by a hori. zontal line between the dividend and the divisor. [See Chap. I., Sect. xvi. 8.] Thus, in ș, the figure above the line stands for the thing, or number ; the line itself expresses division ; and the figure under the line shows into how many parts the thing or number is to be divided. Thus, also, the upper figure, which is the dividend, is called the numerator (or num |