18. .08.008 ? Ans. 10. 20. .163÷121? Ans..01331. 19. .005÷.0015? Ans. 3. 21. ÷.00? Ans. 112.5. Ans. .00008. Ans. .000005. Ans. .000325. Ans. .038833. Ans. .41883. Ans. .0375. Ans. .0000000375. Ans. .0000000146484375. Ans. 4286.931. 31. Divide four thousand three hundred and sixty-two and five hundredths by six hundred and ninety-five millionths. Ans. 6276330.935-35 CONTRACTIONS IN DIVISION OF DECIMALS. 139 278. Certain Abbreviations may be made in the division of decimals, which will facilitate the operation. 1. Divide 35.765342 by 8.76347, extending the quotient to four decimal places. CONTRACTED OPERATION. 8.76347)35.765342(4.0811 7114 7010 104 88 16 9 7 SOLUTION.-In the first method of contraction, we compare 8 units, the first term of the divisor, with 35 tens, the first two terms of the dividend, and find that the first quotient place will be units, and since four decimal places are required, it will contain five terms. Taking, therefore, the five left-hand terms of the divisor, we find that 87634 is contained in 357653, 4 times; multiplying the contracted divisor by 4 and carrying 3 from the rejected part, and subtracting from the dividend, we have 7114 for a new dividend. Dropping the right-hand term of the divisor, and dividing by 8763, we find it is not contained in the dividend; we therefore place a zero in the quotient, and dropping another term from the divisor, we find it is contained in the dividend 8 times. Multiplying the divisor by 8, carrying 2 from the rejected part as in Contracted Multiplication, and subtracting, we have 104 left for a new dividend. Continuing this process till all the terms in the divisor are rejected, we have a quotient 4.0811, with a remainder of 7, and as this is more than SECOND METHOD. 104 16 7 5, we may make the last quotient figure 2. By comparing the contracted with the common method, we shall see how much the work is abbreviated and how closely the intermediate results agree. The second contracted method differs from the first in writing the quotient under the divisor in a reverse order, each term of the quotient being placed under that term of the divisor by which it is first multiplied, and the remainder only being set down, according to Case III. in Contracted Division, Art. 131. Rule.-I. Compare the divisor with the dividend to ascertain the number of terms in the quotient. II For the first contracted divisor, take as many terms of the divisor, beginning with the first significant term on the left, as there are terms in the quotient; and for each successive divisor, reject the right-hand term of the previous divisor, until all the terms of the divisor have been rejected. III. In multiplying by the several terms of the quotient, carry from the rejected terms of the divisor as in contracted multiplication. NOTES.-1. Annex ciphers to either divisor or dividend, if necessary, before beginning the work. We take a divisor containing as many terms as the quotient, in order that all the terms of the divisor may be exhausted when we have obtained the required number of terms in the quotient. 2. It will be found convenient to write each term of the quotient as soon as found below the first term of the divisor into which it is first multiplied, as in the second contracted method, since greater accuracy is likely to be thus attained. 3. If a divisor is a little less than 1, the rule given in Art. 132 may be used, placing the dividing line to the right of that term of the dividend, which, multiplied by the difference between 1 and the divisor, gives a decimal of the required place. If the number of places in the quotient is not mentioned, the decimal point may be used as the dividing line. The last three examples may be most readily solved in this manner. Find the quotient of Ans. 7.66+. Ans. 484.607+. 2. 36.7345÷4.7932, to 2 decimal places. 4. .847963.92579, to 3 places. 5. 57.643987.63975, to 4 places. 8. 473 641.999, to 4 places. Ans. .916-. Ans. 90.1039+ Ans. 2.99797-. Ans. .413568+. Ans. 474.1151+. 9. 97.68397.9994, to 5 decimal places. Ans. 97.74261+. 10. 8574.3965÷.99997, to 6 decimal places. Ans. 8574.653740-. 7. Of (.2x.02x.002)-(.01 x .001 x .0001 x 103)? 14. of 5.0356 is contained how many times in & of 23.79321? 231-4.6, 3.1515+3.08 3.5 Ans. .06.1627 22508 1.125 .18 Ans. 4.8. 15. Find the value of + + 1212-.005 1.1-3 MISCELLANEOUS PROBLEMS. 1. If digging 26.54 rods of ditch cost $176.25, what will 39.81 rods cost? Ans. $264.37. 2. How many solid feet in a pile of wood 7.3 feet long, 5.7 feet wide, and 6.5 feet high? Ans. 270.465. 3. From a cistern containing 2765 gallons, 56.25 barrels, of 31.5 gal. each, are drawn off; how many gallons remain? Ans. 993.125 gallons. 4. A and B divide 897.25 bushels of corn between them, A taking .371 and B .621; how many bushels belong to each? Ans. A, 336.467; B, 560.781. 5. If I buy 4 loads of wood, the first containing 1.34 cords, the second 1.4 cords, the third .995 cords, and the fourth 1.16 cords; what would it cost at $3.75 a cord? Ans. $18.35g. 6. Which will contain the most, a box 5.5 inches long, 4 inches wide, and 4.25 inches deep, or one 6.5 inches long, 4.5 inches wide, and 3.5 inches deep, the contents being equal to the product of the three dimensions? Ans. 2d, 8.875 cu. in. 7. Mr. Jones gives .13 of his income in charity, spends .15 for books, .16 in traveling, .52 for his household expenses, and saves $276.84; what is his income? Ans. $6921. 8. How many barrels of flour, at $9.66 a barrel, must a man give for 75.25 bushels of wheat at $1.75 a bushel, 57.5 bushels of corn, at $0.85 a bushel, and 65.75 bushels of oats at $0.56 a bushel? Ans. 22.5+ barrels. 9. A ship whose cargo was worth $15,000, being disabled by a storm, .564 of the whole cargo was thrown overboard; how much would a merchant lose who owned .25 of the cargo? Ans. $2109.375. 10. The circumference of the fore wheel of a carriage is 13.25 ft., and of the hind wheel 15.75 ft.; how many revolutions will each make in going 25 miles, there being 5280 ft. in a mile? Ans. Fore, 996214; hind, 838020. 11. A grocer wished to buy an equal number of pounds of rice, hominy, and dried apples; the rice being 9 cents a pound, the hominy 13 cents, and the apples 15 cents; how many pounds of each can he buy for $7.03? Ans. 191b. 12. Mr. Bowman laid out $779 in groceries, of the whole quantity being sugar at $0.16 a pound, being tea at $0.95 a pound, being coffee at $0.35 a pound, and the remainder being starch at $0.13 a pound to the amount of $19.50; how many pounds of each did he buy? Ans. 700 lb. sugar, 350 lb. tea, 900 lb. coffee, 150 lb. starch. 13. Mr. Thompson's will gave .5 of his property to his wife, of the remainder to each of his two sons, and the remainder to his daughter, who received $1666.663; what was the amount of his property and the share of each? Ans. Amt., $10,000; wife, $5,000; each son, $1666.663. 14. James Williams left .2 of his property to his son John, .25 of the remainder to his son James, and of the remainder to his daughter, making his wife residuary legatee. The difference between the wife's and the daugh ter's share was $1245.36; what was the whole amount of the property, and what did each receive? Ans. Amount, $3113.40; wife, $1556.70; John, $622.68; James, $622.68; daughter, $311.34. 13 15. A, B, C, and D, having built a stone wall, received a certain sum which was to be divided as follows: A received $90.09 and of the remainder, B $100.10 and of the remainder, C $110.11 and of the remainder, and D what was left, when it was found that each received the same sum; what was the amount of their wages? Ans. $480.48. 3 |
