104. 5. A certain cornfield contains 2688 hills of corn planted in rows, which are 56 hills long, how many rows are there? Here, as 56 is not contained in 26, it is necessary 56 ) 2688 ( 48 to take three figures, or 268, for the first partial divi224 dcnd; but there may be some difficulty in finding how many times the divisor may be had in it. It will, 448 however, soon be seen by inspection, that it cannot be less than 4 times, and by making trial of 4, we find that we cannot have a larger number than that in the ten's place of the quotient, because the remainder, 44, is less than 56, the divisor. In multiplying the divisor by the quotient figure, if the product be greater than the part of the dividend used, the quotient rure is too great ; and in subtracting this product, if the remainderex ceed the divisor, the quotient figure is too small; and in each case the operation must be repeated until the right figure be found. SIMPLE DIVISION. DEFINITIONS. 105. Simple Division is the method of finding how many times one simple number is contained in another; or, of separating a simple number into a proposed number of equal parts. The number which is to be divided, is called the dividend ; the number by which the dividend is to be divided, is called the divisor; and the number of times the divisor is contained in the dividend, is called the quotient. If there be any thiog left after performing the operation, that excess is called the remainder, and is always less than the divisor, and of the same kind as the dividend. RULE. 106. Write the divisor at the left hand of the dividend ; find how many times it is contained in as many of the left hand figures of the dividend, as will contaip it once, and not more than nine times, and write the result for the highest figure of the quotient. Multiply the divisor by the quotient figure, and set the product under the part of the dividend used, and subtract it therefrom. Bring down the next figure of the dividend to the right of the remainder, and divide this number as before ; and so on till the whole is finished. NOTE.-If after bringing down a figure to the remainder, it be still less ihan the divisor, place a cipher in the quorient, and bring down another figure. (103.) Should it still be too small, write another cipher in the What is Simple Division ? | How may the division of the reWhat is meant by the dividend? | mainder he denoted ?(103) by the divisor? by the quotient ? | How do you place the numbers for by the remainder ? division? where the quotient? Of what kind is the remainder? How is the operation performed ? quotient, and bring down another figure, and so on till the number shall contain the divisor. PROOF. 107. Multiply the divisor by the quotient, (adding the re. mainder, if any) and, if it be right, the product will be equal to the dividend. QUESTIONS FOR PRACTICE. 6. If 30114 dollars be divid. I 12. If a certain number of ed equally among 63 men, how men, by paying 33 dollars each, many dollars will each one re-paid 726 dollars, what was the ceive ? number of men ? Ans. 22. 63 ) 30114 ( 478 dolls. Ans. | 13. The polls in a certain 252 town pay 750 dollars, and the number of polls is 375, what 491 does each poll pay ? 441 Aps. 2 dolls. 14. If 45 horses were sold 504 in the West Indies for 9900 dollars, what was the average | price of each ? Ans. $220. - 7. If a man's income be 1460 15. An army of 97440 men dollars a year, how much is was divided into 14 equal divi. that a day? Ans. 4 dolls. | sions, how many men were 8. A man dies leaving an there in each ? Ans. 6960. estate of 7875 dollars to his 7 16. A gentleman, who own. sons, what is each son's share ? | ed 520 acres of land, purchas: Ans. 1125 dolls. ed 376 acres more, and then 9. A field of 34 acres pro- | divided the whole into 8 equal duced 1020 bushels of corn, farms, what was the size of how much was that per acre ? ) each? Ans. 112 acres. Aps. 30 bush. I 17. A certain township con. 10. A privateer of 175 men tains 30000 acres, how many took a prize worth 20650 dol. lots of 125 acres each does it lars, of which the owner of the contain ? Ans. 240. privateer had one half, and the ! 18. Vermont contains 247 rest was divided equally among townships, and is divided into the men; what was each man's 13 counties, what would be the share ? Ans. 59 dolls. I average number of townships 11. What number must I in each county ? Ans. 19. multiply by 25, that the pro. 519. Vermont contains 5640duct may be 625? Ans. 25. 1000 acres of land, and in 1820 What is the method of proof? is expressed by writing the divi: 1 dend, what is the expression call. ed? 108, 109. CONTRACTIONS OF DIVISION. contained 235000 inhabitants, to the moon's distance from the what was the average quantity earth? Ans. 30. of land to each person ? 21. Divide 17354 by 86. Ans. 24 acres. Quot. 201. Rem. 68. 20. The distance of the 22. Divide 1044 by 9. moon from the earth is 240000 Quot. 116. miles, and the diameter, or dis- | iameter, or dis. 23. Divide 34748748 by 24. tance through the earth, is 1 Quot. 1447864. Rem. 12. 8000 miles ; how many diame. 24. 29702-6=49504 Ans. ters of the earth will be equal | 25. 279060=39865 Ans. CONTRACTIONS OF DIVISION. 108. 1. Divide 867 doliars equally among 3 men, what will each receive Here we seek how many times 3 in 8, and finding, Divis. 3 ) 867 Divid. it 2 times and 2 over, we write 2 under 8 for the first figure of the quotient, and suppose the 2, which 289 Quot. remains, to be joined to the 6, making 26. Tben 3 in 26, 8 times, and 2 over. We write 8 for the next figure of the quotient, and place 2 before the 7, making 27, in which we find 3, 9 tinies. We therefore place 9 in the unit's place of the quna tient, and the work is done. Division performed in inis mauner, without writing down the whole operation, is called Short Division. 1. When the divisor is a single figure; Rule.- Perform the operation in the mind, according to the general rule, writing down only the quotient figures. 2. Divide 78904 by 4. 13. Divide 234567 by 9. Quot. 19726. Quot. 26063. 109. 4. Divide 238 dollars into 42 equal shares; how many dollars wul there be in each? If there were to be but 7 shares, we should 42—6x7 divide by 7, and find the shares to be $33 each, 7) 233-6 rem. 1st. with a remainder of 6 Jollars; but as there are to be 6 times 7 shares, each share will be only one 6) 33–3 rem. 20. sixth of the above, or a little more than 5 dollars. In the example there are two remainders; the first, 6, is evidently 6 units of the given dividend, 7x3+6=27 rem. or 6 dollars; but the second, 3, is evidently units Ans. 5 27 dolls. of the second dividend, which are 7 times as great as those of the first, or equal to 21 units of ibę first, and 21+6=27 dolls. the true remainder. II. When the divisor is a composite number. (90.) What is Short Division ? | How do you multiply by a compo. site number? quotient by another, and so on, if there be more than two, the last quotient will be the answer. 5. Divide 31046835 by 56=76. Divide 84874 by 48=6*8. ><8. Quo. 554407, Rem. 43. 1 Quo. 176818 110. 7. Divide 45 apples equally among 10 children, how many will. each child receive? As it will take 10 apples to give each child 1, each child will evidently, receive as many apples as there are 10's in the whole number; but all the figures of any number, taken together, may be regarded as tens, excepting that which is in the vuit's place. The 4 theu is the quotieni, and the 5 is in the remainder; that is, 45 apples will give 10 children 4 apples and 5 tenths, or 4, each. And as all the figures of a number, higher than in the ten's place, inay be considered hundreds, we may in like man. ner divide by 100, by cutting off two figures from the right of the divi. Jend; and generally, III. To divide by 10, 100, 1000, or 1 with any number of ci-. phers annexed; RULE.- Cåt off as many figures from the right hand of the dividend as there are ciphers in the divisor; those on the left will be the quotient, and those on the right, the remainder. 8. Divide 46832101 by mong 100 men, how much 10000. Quot. 46832106will each receive ? 9. Divide 1500 dollars a Ans. 15. dolls. 111. 10. Divide 36556 into 3200 equal parts. Here 3200 is a composite number, whose 32000 ) 365156 ( 11 Quot. component parts are 100 and 32; we therefore 32 divide by 100, by cutting off the two right hand figures. We then divide the quotient, 365, by 32, and find the quotient to be 11, and remain. der 13; but this remainder is 13 hundred, (109) and is restored tu ils proper place by bringing 1356 Rem. down the two figures which remained after dividing by 100, making the whole remainder, 1356. Hence IV. To divide by any number whose right hand figures are . ciphers ; RULE. ---Cut off the ciphers from the divisor, and as many figures from the right of the dividend; divide the remaining figures of the dividend by the remaining figures of the divisor, and bring down the figures cut off from the dividend to the right of ihe remainder. What is the rule for multiplying by | How do you proceed when the divi. 1, with ciphers annexed ? I sor has ciphers in the right hand ? Give the reason for the operation. Give the reason. 11. Divide 738064 by 2300. | 12. Divide 6095146 by 5600, Quot. 320, Rem. 2064. 1 Quot. 108836 REVIEW, 112, 1. What are the fundamen- | 20. What would you call the tal operations in this Section ? other number? Ans. Addition and Subtraction. 21. By what name would you 2. What relation have Multipli- call the result of the operation ? cation and Division to these? (83, 22. Where there is a part of the 101.) dividend left after performing the 3. When two or more numbers | operation, what is it called ? are given, how do you find their sum? 23. How can you denote the di 4. What is the method of per vision of this remainder? (103.) forming the operation? (81.) 24. If the divisor and dividend 5. When the given numbers are | were given, how would you find the all equal, what shorter method is | quotient ? there of finding their sum ? (83.) : 25. If the dividend and quotient 6. How is Multiplication per- | were given, how would you find the formed? (88) divisor? 7. What are the given numbers em 26. If the divisor and quotient ployed in Multiplication called? (87.) were given, how would you find the 8. What is the result of the ope- | dividend ? ration called ? (87.) 27. If the multiplicand and mul. 9. How would you find the diffe- tiplier were given, how would you rence between two numbers? (94.) I find the product? 10. By what names would you 28. Jf the multiplica’nd and procall the iwo numbers ? (98.) duct were given, how would you 11. What is the difference called? find the multiplier ? 12. If the minuend and subtra 29. If the multiplier and produce heud were given, how would you were given, how would you find the find the remainder? multiplicand ? 13. If the minuend and remain. 30. When the price of an article cler were given, how would you find is given, how do you find the price the subtrahend? of a number of articles of the same 14. If the subtrahend and remain- I kind ? (83.) der were given, how would you find 31. Does the proof of an ariththe minuend? metical operation demonstrate its lá. If the sum of two numbers, correctness ? (82.) What then is its and one of them were given, how use? would you find the other? 16. If the greater of two numbers NOTE.--The definitions of such and their difference be given, how of the following terms as have not would you find the less? been already explained, may be found 17. If the less of two numbers in a dictionary. and their difference be given, how would you find the greater ? What is Arithmetic? What is a 18. How would you find how ma: Science ? Number? Notation ? Nuny times one number is contained meration? Quantity ? Question? in another? Rule ? Answer? Proof? Principle > 19. By what name would you | Illustration? Explanation. call the number divided? (105.) |