Ans. 32 years. in the lower line be greater than the figure standing over it, suppose ten to be added to the upper figure, and the next significant figure in the upper line to be diminished by 1, (96) regarding ciphers, if any come between, as 9s,(97); or, which gives the same result, suppose 10 to be added to the upper figure, and the next figure in the lower line to be increased by 1, with which proceed as before, and so on till the whole is finished. PROOF. 100. Add together the remainder and the subtrahend, and if the work be right, their sum will equal the minuend. QUESTIONS FOR PRACTICE. 6. In 1810, Montpelier con- 12. What number is that tained 1877 inhabitants, and which taken from 365, leaves in 1820, 2308 inhabitants; 159? Ans. 206. what was the increase, and in what time? 13. Supposing a man to 2308 have been born in 1796, how 1810 14-1877 old war he in 1828? 14. If a man have 125 head 7. Dr. Franklin died in of cattle, how many will he 1790, and was 84 years old, have after selling 8 oxen, 11 COWS, 9 steers and 13 heifers? in what year was he born? Ans. 84. Ans. 1706. 8. A man deposited 9000 which if you add 643, it will 15. What number is that to the flight of Mahomet in 622, 1828? Ans. 1206. 17. America was discoverwill there be left? Ans. 47. ed by Columbus in 1492; how 10. Sir Isaac Newton was many years since? born in the year 1642, and 18. If you lend 3646 dollars died in 1727; how old was he and receive in payment 2998 when he died? Ans. 85 years. dollars, how much is still due? 11. If you lend a neighbor Ans, 648 dolls. 765 dollars, and he pay you 19. A owed B. $4850, of at one time, 86 dollars, and at which he paid at one time another 125 dollars,how much $200, at another, $475, at anis still due? Ans. 554 dolls, other $40, at another $1200, and at another $156; what 21. Gunpowder was inremains due? Ans. 2779. vented in the year 1330; then 20. The sum of two num- how long was this before the bers is 64892, and the greater | invention of printing, which number is 46234: what is the was in 1441 ? smallest number? Ans. 18658. 22. 23. 24. From 3287625 5327467 7820004 12345678 Take 2343756 2100438 2780009 4196289 Ans. I11 years. 25. Rem. 943869 Proof., 3287625 26. 6485_4293–2192. 27. 900000—1=899999. 28. 4876479313906. DIVISION. ANALYSIS. 101. 1. Divide 24 apples equally among 6 boys, how many will each eceive? The most simple way of doing this would be, first to give each boy 1 *pple, then each boy l'apple more, and so on, till the whole were distribated, and the number of i's, which each received, would denote his share of the apples, which would in this case bę 4. Or as it would take 6 apples to give each boy one, each boy's share will evidently contain as inany apples as there are sixes in 24.' Now this may be ascertained by suba tracting 6 from 24, as many times as it can be done, and the number of subtractions will be the number of times 6 is contained in 24; thus, 24 b=18, 18-6=12, 612=6, and 6–6=0. Here we find that by performing 4 subtractions of 6, the 24 is completely exhausted, which shows that 24 contains 6 just 4 times. Now as Subtraction is the reverse of Addition, '94) it is evident that the addition of 4 sixes, (6767676=24) must recompose the number, which we have separated by the subtraction of 4 sixes. But when the nunibers to be added are all equal, Addition becomes Multiplication,(83) and 24 is therefors the product of 4 and 6, (4X6=24). A number to be divided, and which is called a dividend, is then to be regarded as the product of two factors, one of which, called the divisor, is given to find the other, called the quotient; and the inquiry how many times one number is contained in another, as 6 in 24, is the same as how many times the one will make the other, as how many times 6 will make 24, and both must receive the same answer, viz. 4. Hence to prove Division, we multiply the divisor and quotient together, and if the work be right, the product will equal the dividend. 4 How many yards of cloth will 63 dollars buy, at 9 dollars a yard? 102 102. When the dividend does not exceed 100, nor the divisor exeeed 10, the whole operation may be performed at once in the mind: but when either of them is greater than this,' it will be found most convenient to write down the numbers before performing the operation, 3. Divide 552 dollars equally between 2 men, how many dollars will each have? Here we cannot say at once how many times 2 is con2)552 2 tained in 552, we therefore write down the dividend, 552, and place the divisor, 2, at the left hand. We then pro400-200 ceed to separate the dividend into such parts as may read140-70 ily be divided by 2. These parts we find to be 400, 149, 12 6 and 12. Now 2 is contained in 4, 2 times, and therefore in 400, 200 times; 2 in 14, 7 times, and in 140, 70 times, 552-276 and 2 in 12, 6 times; and since these partial dividends, 400+140+12=552, the whole dividend, the partial quotients, 200+70+6=276, the whole quotient, or whole number of times 2 is contained in 552. But in practice we separate the dividend into parts no faster than we proceed in the division. Having written down the divi dend and divisor as before, we first seek how many Divis. Divid. Quot. times 2 in 5, and find it to be completely contained 2,552 ( 276 in it only 2 times. We therefore writě 2 for the 500, is evidently 200; but we leave the place of ing part of the dividend. We now multiply the 12 divisor 2, by the 2 in the quotient, and write the 12 product, 4, (400) under the 5 hundred in the divi dend. We have thus found that 400 contains 2, 200 times, and by subtracting 4 from 5, we find that there are 1 hundred, 5 tens, and 2 units, remaining to be divided. We next bring down the 5 tens of the dividend, by the side of the 1 hundred, making 15 tens, and find 2 in 15, 7 times. But as 15 are so many tens, the 7 musi be tens also, and must occupy the place next below hundreds in the quotient. We now mnltiply the divisor by 7, and write the product, 14, under the 15. Thus we find that 2 is contained in 15 tens 70 times, and subtracting 14 from 15, find that I ten remains, to which we bring down the 2 units of the dividend, making 12, which contains 2, 6 times; which 6 we write in the unit's place of the quotient, and multiplying the divisor by it, find the product to be 12. Thus have we completely exhausted the dividend, and obtained 276 for the quotient as before. 103. 4. A prize of 3349 dollars was shared equally among 16 men, how many dollars did each man receive? We write down the numbers as before, and find 16 16)3349 209,5Ans. in 32, 2 times,—we write 2 in the quotient, multiply 16 32 the divisor by it, and place the product, 32, under 33, the part of the dividend used, and subtracting, 149 find the remainder to be 1, which is 1 hundred. To the 1 we bring down the 4 tens, making 14 teng; but in the quotient. We therefore put a cipher in the ren's place in the quotient, and bring down the 9 units of the dividend to the 1,4 teny, making 149 units, which contain 16 somewhat more than 9 times. Placing 9 in the unit's place of the quotient, and multiplying the divisor by it, the product is 444, which, subtracted from 149, leaves a remainder of 5. The division of these 5 dollars may be denoted by writing the 5 over 16, with a line between, as in the example. Each man's share then will be 209 dollars and 5 sixteenths of a dollar.(21) The division of any number by another may be denoted by writing the divi. dond over the divisor, with a line between, and an expression of that kind is called a Vulgar Fraction. 104. 5. A certain cornfield contains 2638 hills of corn planted in rows, which are 56 bills long, how many rows are there? Here, as 56 is not contained in 26, it is necessary to take -56)2688(48 three figures, or 268, for the first partial dividend: but 224 there may be some difficulty in finding how many times the divisor may be had in it. It will, however, soon be seen by 418 inspection, that it cannot be less than 4 times, and by making 448 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 figure is too great; and in subtracting this product, if the remainder exceed the divisor, the quotient figure is too small; and in each case the operation must be repeated until the right figue.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 thing 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 contain 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 ased, 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.-!f after bringing down a figure to the remainder, it be still lege than the divisor, place a cipher in the quotient, and bring down another figure.[103] Should it still be too small, write another cipher in the quotient, and bring down another figure, and so on till the number shall cuntain the divisor. PROOF. 107. Multiply the divisor by the quotient, (adding the remainder, if any) and, if it be right, the product will be equal to the dividend. QUESTIONS FOR PRACTICE. 6. If 30114 dollars be die 11. What number must I vided equally among 63 men, multiply by 25, that the prohow many dollars will each duct may be 625? Ans. 25. one receive? 12. If a certain number of 63)30114(478 dolls. Ans. 252 men, by paying 33 dollars each, paid 1726 dollars, wbat was the number of men? 491 Ans. 22. 441 13. The polls in a certain 504 town pay 750 dollars, and the number of polls is 375, what 504 does each põll pay? Ans. 2 dolls. 7. If a man's income be 14. If 45 horses were sold 1460 dollars a year, how much in the West Indies for 9900 is that a day? Ans. 4 dolls. dollars, what was the average 8. A man dies leaving an price of each? Ans. $220. estate of 7875 dollars to his 7 15. An army of 97440 men sons,what is each son's share? Ans. 1125 dolls. was divided into 14 equal di visions, how many men were 9. A field of 34 acres pro- there in each? Ans. 6960. duced 1020 bushels of corn, 16. A gentleman,who ownhow inuch was that per acre? ed 520 acres of land, purAns. 30 bush. chased 376 acres more, and 10. A privateer of 175 men then divided the whole into took a prize worth 20650 dol- eight equal farms; what was Jars, of which the owner of the size of each? the privateer had one half, Ans. 112 acres. and the rest was divided c- 17. A certain township qnally among the men; what contains 30000 acres, how was each man's share? many lots of 125 acres each Ans. 59 dolls. does it contain? Ans. 240, |