102. When the dividend does not exceed 100, nor the divisor exceed 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? 2)552 400-200 140- -70 126 552-276 Divis. Divid.Quot. 2,552 (276 2 15 552 Here we cannot say at once how many times 2 is contained in 552, we therefore write down the dividend, 552, and place the divisor, 2, at the left hand. We then proceed to separate the dividend into such parts as may readily be divided by 2. These parts we find to be 400, 140, 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, and 2 in 12, 6 times; and since these partial dividends, 400+140+12-552, the whole dividend, the partial quotients, 20070-4-6-276, 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 dividend and divisor as before, we first seek how many times 2 in 5, find it to be completely contained in it only 2 times. We therefore write 2 for the highest figure of the quotient, which, since the 5 is 500, is evidently 200; but we leave the place of tens and units blank to receive those parts of the 14 proof. quotient which shall be found by dividing the remaining part of the dividend. We now multiply the divisor 2, by the 2 in the quotient, and write the product, 4, (400) under the 5 hundred in the dividend. 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 must be tens also, and must occupy the place next below hundreds in the quotient. We now multiply 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. 12 103. 4. A prize of 3349 dollars was shared equally among 16 men, how many dollars did each man receive? 32 149 16 We write down the numbers as before, and find 16 16)3349 209 5 Ans. in 32, 2 times,-we write 2 in the quotient, multiply the divisor by it, and place the product, 82, under 33, the part of the dividend used, and subtracting, find the remainder to be 1, which is 1 hundred. To the 1 we bring down the 4 tens, making 14 tens; but as this is less than the divisor, there can be no tena in the quotient. We therefore put a cipher in the ten's place in the quotient, and bring down the 9 units of the dividend to the 14 tens, making 149 units, which contain 16 somewhat more than 9 5 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 dividend 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 hills long, how many rows are there? 224 448 448 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 there may be some difficulty in finding how many times the divisor may be had in it. It will, 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 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 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 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 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 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 contain 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. 7. If a man's income be 1460 dollars a year, how much is that a day? Ans. 4 dolls. 8. A man dies leaving an estate of 7875 dollars to his 7 sons, what is each son's share? Ans. 1125 dolls. 9. A field of 34 acres produced 1020 bushels of corn, how much was that per acre? Ans. 30 bush. 10. A privateer of 175 men took a prize worth 20650 dollars, of which the owner of the privateer had one half, and the rest was divided equally among the men; what was each man's share? Ans. 59 dolls. 11. What number must I multiply by 25, that the product may be 625? Ans. 25. 12. If a certain number of men, by paying 33 dollars each, paid 726 dollars, what was the number of men? Ans. 22. 13. The polls in a certain town pay 750 dollars, and the number of polls is 375, what does each poll pay? Ans. 2 dolls. 14. If 45 horses were sold in the West Indies for 9900 dollars, what was the average price of each? Ans. $220. 15. An army of 97440 men was divided into 14 equal divisions, how many men were there in each? Ans. 6960. ed 520 acres of land, pur16. A gentleman, who ownchased 376 acres more, and then divided the whole into eight equal farms; what was the size of each? Ans. 112 acres. 17. A certain township contains 30000 acres, how many lots of 125 acres each does it contain? Ans. 240. 108. 1. Divide 867 dollars equally among 3 men, what will each receive? Divis. 3) 867 Divid. Here we seek how many times 3 in 8, and finding it 2 times and 2 over, we write 2 under 8 for the 289 Quot. first figure of the quotient, and suppose the 2, which remains, to be joined to the 6, making 26. Then 8 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 times. We therefore place 9 in the unit's place of the quotient, and the work is done. Division performed in this manner, without writing down the whole operation, is called Short Division. I. 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. Quot. 19726. 109. 4. Divide 237 dollars into 42 equal shares; how many dollars will there be in each? 42=6X7 7)237-6 rem. 1st. 6 33-3 rem. 2d. In If there were to be but 7 shares, we should divide by 7, and find the shares to be $33 each, with a remainder of 6 dollars; but as there are to be 6 times 7 shares, each share will be only one sixth of the above, or a little more than 5 dollars. the example there are two remainders; the first, 6, is evidently 6 units of the given dividend, or 6 dollars; but the seeond, 3, is evidently units. of the second dividend, which are 7 times as great as those of the first, or equal to 21 units of the first, and 21+6=dollars, the true remainder. 5 7X3+6=27 rem. Ans. 5 dolls. II. When the divisor is a composite number.(90) RULE.-Divide first by one of the component parts, and that quotient by another, and so on, if there be more than two; the last quotient will be the answer. 5. Divide 31046835 by 56=7 | 6. Divide 84874 by 48=6X8. X8. Quot. 554407, Rem. 43. Quot: 176848. 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 unit's place. The 4 then is the quotient, and the 5 is the remainder; that is, 45 apples will give 10 children 4 apples and 5 tenths, or each. And as all the figures of a number, higher than in the ten's place, may be considered hundreds, we may in like manner divide by 100, by cutting off two figures from the right of the dividend; and, gene. rally, III. To divide by 10, 100, 1000, or 1 with any number of eiphers annexed: RULE.-Cut off as many figures from the right hand of the dividend as there arc 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. 4683. will each receive? 9. Divide 1500 dollars a 111. 10. Divide 36556 into 3200 equal parts. 82 45 82 Ans. 15 dolls. Here 3200 is a composite number, whose com32:00)365 56(11 Quot. ponent parts are 100 and 32; we therefore 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 remainder 13; but this remainder is 18 hundred,[109], and is restored to its proper place by bringing 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: 1356 Rem. 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 divi sor, and bring down the figures cut off from the dividend to the right of the remainder. 11. Divide 738064 by 2300. | 12. Divide 6095146 by 5600. Quot. 320, Rem. 2064. Quot. 10888 |