To do this on the slate, write the numbers as in the margin, and say, 3 in 3, once; and as the 3 is hundreds, the 1 is also hundreds, and is written in the hundreds' place. 3 in 9, 3 times, to be written in the tens' place. 3 in 6, 2 times, to be written in the units' place. 3)396 132 2. How many yards of cloth, at 2 dollars a yard, can be bought for $4628? 3. How many bushels are there in 4808 pecks? Since there are 4 pecks in 1 bushel, there will be as many bushels in 4808 pecks as there are times 4 in 4808. 4. How many yards are there in 3693 feet? 5. How much is 24682? 4884484? 12660606? 164804804? 50055? 12639? 248844? NOTE. If the divisor is not contained in the first figure of the dividend, find how many times it is contained in the first two figures. = At 2 cents a pound, how many pounds of squashes can be bought for 3153 cents? Ans. As many pounds as there are times 2 in 3153. 3153 2000+1000+140+13. 2 is contained in 2000, 1000 times; in 1000, 500 times; in 140, 70 times; in 13, 63 times; which, being added, make 1576 pounds. 6. How many pounds at 3 cents? 3153-3000+150+ 3. How many at 4 cents? 3153-2800+320+33. How many at 5 cents? At 6 cents? At 7 cents? To perform such divisions, we may write the 2)3153 numbers as in the margin, and, 15761 If after dividing any figure there is a remainder, prefix it mentally to the next figure of the dividend, and divide as before. In this example, after dividing 3 by 2, 1 remains. Prefixing it to the next figure, makes 11. 2 in 11, 5 times, and 1 remains. Prefixing the 1 to 5, makes 15. 2 in 15, 7 times, and 1 remains. Prefixing this to the 3, makes 13. 2 in 13, 6 times. 7. How many times is 5 contained in 16278? Ans. 3255g times. 8. Divide 5649 by 3. 85732 by 4. 514087 by 5. 761834 by 6. 5519875 by 8. 9. Divide 30807510 by 3; by 4; by 5; by 6; by 7; by 8; by 9; by 10; 11; 12. 10. How much is 384599? 10010009 ÷ 9? 11. In 184765 inches how many feet, and how many inches over? In these feet how many yards? Since there is 1 foot in 12 inches, there will be as many feet as there are times 12 in 184765; and since there is 1 yard in 3 feet, there will be as many yards as there are times 3 in the number of feet. 12)184765 3)15397 ft. and 1 inch over. 5132 yd. and 1 ft. over. 12. In 48967 furlongs how many miles, and how many furlongs over? In these miles how many leagues? Learn the table, Art. 55. 13. In 860419 nails how many quarters? How many yards? 14. In 33167 pints how many quarts? pecks? bushels? 15. In 115418 gills how many pints? quarts? gallons? 16. In 35145 farthings how many pence? shillings? (49.) 17. At 7 dollars a barrel, how many barrels of flour can be bought for 8640 dollars? 38. If there are decimals in the dividend only, divide as in whole numbers, and point off as many decimal places in the quotient as there are in the dividend. EXAMPLES FOR PRACTICE. 1. How many times is 4 contained in 3416.8? 2. How many times is 4 contained in 5028.16? In 30449.28? 4)3416.8 854.2 3. Divide 2084.25 by 5. Divide 4080.012 by 6. Divide 619.0048 by 8. 304186.4084÷7?, 4. Divide $362.168 equally among 8 men. 5. How much is 80416.008÷ 9? 4180.7616÷6? 6. Divide 4061.709 by 7. 7) 4061.7090000 580.2441428+ NOTE. If there is a remainder after dividing, and more decimals are desired in the quotient, naughts may be annexed as decimals to the dividend, as in the margin. The sign is to be annexed to the quotient if there is a remainder after performing the division as far as it is desired. In the remaining examples of this Art. carry the quotient to at least 5 places of decimals, if there are remainders. 7. Divide 1843.07 by 2; by 3; by 4; by 5; by 6. 8. Divide 5106.847 by 7; by 8; by 9; by 10. 9. Divide 800700.5001 by 2; by 6; by 7; by 9. 10. Divide 501.080701 by 2; by 4; by 6. It was shown (27) that a product must have as many decimal places as there are in its factors; therefore, the divisor and quotient, being factors of the dividend, (33,) must together have as many decimal places as the dividend. GENERAL RULE FOR DIVISION OF DECIMALS. Divide as in whole numbers, and point off in the quotient as many places for decimals as the decimal places in the dividend exceed those in the divisor. The dividend must contain at least as many decimal places as the divisor. If it has not so many, anner as many decimal naughts as are needed. 11. How much is 3104.64÷4? .30415÷.7? 12. How much is .000805.005? 13.041÷.006? 14÷ .0005? 13. Divide 3104.57 by 6; by .6; by .06; by .006; by .0006. 14. Divide 510.867 by 5; by .5; by .05; by .005. 15. Divide 87165.0008 by 7; by .08; by .009. 39. It was shown (28) that removing the decimal point one place to the right multiplies the number by 10; removing it two places, multiplies it by 100, &c. For a like reason, removing the decimal point one place to the left, divides the number by 10, since it makes units tenths, tens units, &c. Hence the following RULE. TO DIVIDE BY 10, 100, 1000, &c. Remove the decimal point in the dividend as many places to the left as there naughts at the right of the divisor. EXAMPLES. 1. Divide 304617 by 10. Ans. 30461.7. By 100; by 1000. (2.) By 10000; by 100000. 3. Divide 30.4671 by 10; by 100; by 1000. Prefix naughts to the figures of the dividend if necessary; thus, 30.4671÷ 1000.0304671. 4. Divide 85.1865 by 10000; by 100000. 5. Divide 81564 by 100; by 1000; by 10000. 6. Divide 304.06 by 20. First remove the decimal point in the dividend one place to the left, which will divide it by 10; then cancel the 0, and divide by 2. See margin. 20) 30.406 15.203 RULE. When the divisor has naughts on the right, remove the decimal point in the dividend as many places to the left as there are naughts on the right of the divisor; cancel the naughts, and divide by the remaining figure or figures. Carry the quotient in the 7th, 9th, and 10th examples to at least 6 decimal places if there are remainders. 7. Divide 51761.7 by 300; by 4000; by 60000. 8. Divide 104.57604 by 600000. Removing the decimal point 5 places to the left; thus, .0010457604 divides the number by 100000; then dividing by 6, gives the quotient required. 9. Divide 510756 by 50; by 700; by 80000; by 90000. 10. Divide 30.1457 by 30; by 900; by 50000. NOTE. If decimals are not desired in the quotient, the figures pointed off by the rule may be annexed to the remainder, after dividing the other figures. 11. Divide 351285 by 500. See margin. 500) 3512.85 702285 12. Divide 4160841 by 8000; by 60000; by 800. 13. How many minutes are there in 6148578 seconds, and how many seconds over? In these minutes how many hours? Since there is 1 minute in 60 seconds, there will be as many minutes as there are times 60 in 6148578; and since there is 1 hour in 60 minutes, there will be as many hours as there are times 60 in the minutes. Ans. 1707 h. 56 min. 18 sec. 60)6148578 60) 102476 min. and 18 sec. 1707 h. and 56 min. over. NOTE. Lower denominations are reduced to higher by division. 14. How many minutes in 784206 seconds? How many hours? 15. How many barrels of pork will 75740 lb. make, allowing 200 lb. to a barrel? Ans. 378 barrels and 140 lb. over. 16. Divide 18457 by 20. Ans. 92217. 200) 757.40 378148 20) 1845.7 92217 17. Divide 351743 by 300; by 4000; by 60. 20. Divide 341608 by 7000; by 80000; by 600. NOTE. The answers to the 20th, 21st, and 22d are to be carried to six decimals, if there are remainders. 21. Divide 310.45 by 1000; by 300; by 90000. 22. Divide 54716 by 8000; by 12000. 40. All the preceding examples are performed by what is called Short Division; but where the divisor consists of significant figures higher than 12, it is more convenient to perform the work by Long Division. How many times is 24 contained in 3373689 ? 24 X 2 24 X 1: = 24 24)3373689(1405702 = 48 24 72 = 96 97 96 136 = 192 120 168 24 X 7 = 168 To perform this, first write in a column the products of the divisor by each of the 9 digits. Then, having written the divisor and dividend as in the margin, take as many figures at the left of the dividend as will contain the divisor once or more, and divide them by it; 24 in 33, once. Place the 1, as the first figure of the quotient, to the right of the dividend, and subtract once 24 from 33. Write the remainder, 9, and to it annex the next figure of the dividend for another partial dividend. By examining the column of products, see how many times 24 is contained in 97, which is found to be 4. Write the 4 in the quotient, and subtract the product of 4 times 24 from 97. Write the remainder, 1, and annex to it the next figure in the dividend. As 24 is not contained in 13, place a naught in the quotient, and bring down and annex the next figure of the dividend. Proceed 168 9 |