13. 15. 17) 6 7 8 9 16 ( 35) 910 6013 ( 91 ) 6210011( Quotients. Rem. 16. Divide 671678953 by 6. 111946492 1. 17. Divide 166336711 by 7. 23762387 2. 18. Divide 161331793 by 8. 20166474 1. 19. Divide 161677678 by 9. 17964186 4. 20. Divide 363895678 by 11. 33081425 3. 21. Divide 164378956 by 12. 13698246 4. 22. Divide 78950077 by 3. 1. 23. Divide 678956671 by 4. 3. 24. Divide 667788976 by 5. 1. 25. Divide 777777777 by 6. 3. 26. Divide 888888888 by 7. 6. 27. Divide 789636 by 46. 28. Divide 7967848 by 52. 153227 44. 29. Divide 16785675 by 61. 275175 30. Divide 675753 by 39. 31. Divide 5678911 by 82. 1. 32. Divide 6716394 by 94. 71451 33. Divide 1167861 by 135. 8650 111. 34. Divide 7861783 by 87. 90365 28. 35. Divide 1678567 by 365. 4598 297. 36. Divide 87635163 by 387. 226447 174. 37. Divide 34567890 by 6789. 5091 5091. 38. What is the value of 213255467083 ; 30204 ? 4267. 39. What is the value of 395020613 ; 4444 ? 2341. 40. What is the value of 7207276639 ; 9009 ? 4567. 41. What is the value of 454115186870257 ; 500123 ? 87 65. 42. How many barrels of flour, at 9 dollars a barrel, can be bought for 18621 dollars ? 43. How much sugar at 15 dollars a hundred may be bought for 405 dollars ? 44. A tailor has 938 yards of broadcloth; how many cloaks can be made of the cloth, if it require 7 yards to make one cloak? 45. What number multiplied by 1728 will produce 1705536? Ans. 987. 46. A. Hartmann has sold his wagon to J. Herr for 85 dol., lars. He is to receive his pay in wood at 5 dollars a cord. How many cords will it require to pay for the wagon ? Ans. 17 cords. 47. The Bible contains 31,173 verses; how many must be read each day, that the book may be read through in a year of 365 days? Ans. 85143 verses. 48. A train on the Liverpool Railroad runs at the rate of 65 miles an hour; how long would it take at that velocity to pass round the earth, the distance being about 25,000 miles ? Ans. 384,8 hours. 49. A gentleman possessing an estate of 66,144 dollars, bequeathed one fourth to his wife, and the remainder was divided among his 4 children ; what was the share of each? Ans. 12,402 dollars. 50. If the dividend is 6756785 and the quotient 193051, what is the divisor ? Ans. 35. 51. A's age multiplied by 17, or B's age multiplied by 19, is equal to 1292 years, and she sum of their ages is equal to C's age multiplied by 3. What is the age of each? Ans. A's 76 years; B's 68 years; C's 48 years. 78. When the divisor is a composite number. Ex. 1. A farmer bought 21 horses for 2625 dollars; how many dollars did each cost ? Ans. 125 dollars. OPERATION. The factors of 21 are 3) 2 6 2 5 dolls., cost of 21 horses. 3 and 7. Now, if we divide the 2625 dollars, 7) 8 7 5 dolls., cost of 7 horses. the cost of 21 horses, by 1 2 5 dolls., cost of 1 horse. 3, we obtain 875 dollars, the cost of 7 horses, since there are 7 times 3 in 21. Then, dividing the 875 dollars, the cost of 7 horses, by 7, we obtain 125 dollars, the cost of 1 horse. 2. Divide 3515 by 42. Ans. 8371 OPERATION, FINDING THE TRUE REMAINDER. 2) 3515 4 X 3 X 2 = 24, 1st Product. 3) 1 7 5 7, 1, 1st Rem. 2 x 2 4, 2d Product. 7) 5 8 5, 2, 2d Rem. 1, 1st Remainder. 83, 4, 3d Rem. 29, true Remainder. Or, (4 X 3 X 2) + (2 X 2)+1=29, true Rem. Using as divisors 2, 3, and 7, the factors of 42, we obtain for remainders, 1, 2, and 4. The first remainder, 1, is a unit of the given dividend, since it is a part of it (Art. 71). The second remainder, 2, is units of the quotient 1757, whose units are 2 times as great as those of the given dividend. The third remainder, 4, is units of the quotient 585, whose units are 3 times as great as those of the quotient 1757, of which the units are 2 times as great as those of the given dividend. Now, these remainders must be all of the same units as the given dividend, to constitute the whole or true remainder. We therefore multiply the third remainder by 3 and 2, the divisors used in producing the quotient of which it is a part; and the second remainder by 2, the divisor used in producing the quotient of which it is a part ; and the products with the first remainder added together give 29, the whole or true remainder sought. Hence, as shown by these illustrations, when the divisor is a composite number, we may Divide the dividend by one of the factors, and the quotient thus found by another, and thus proceed till each of the factors has been made a divisor. The last quotient will be the quotient required. If there be remainders, multiply each remainder, except the first, by all the divisors preceding the one which produced it; and the first remainder being added to the sum of the products, the amount will be the true remainder. NOTE. — There will be but one product to add to the first remainder when there are only two divisors and two remainders. EXAMPLES. 3. Divide 7704 by 24 4 X 6. Ans. 321. 4. Divide 8317 by 27 3 x 9. 5. Divide 3116 by 81 9 x 9. 6. Divide 61387 by 121 11 x 11. Ans. 507421 7. Divide 19917 by 144 12 x 12. Ans. 1381444 8. Divide 91746 by 336 6 X 7 X 8. Ans. 273336 9. At 45 dollars an acre, a farm of how many acres can be bought for 5464 dollars ? Ans. 1211% acres. OPERATION. OPERATION, 79. When the divisor contains one or more ciphers at the right hand. Ex. 1. If 10 men receive 792 dollars for a job of work, what will be each man's share of it ? Ans. 79 dollars. To multiply by 10, we annex one 110 ) 7 912 cipher, which removes the figures one Quotient 7 9, 2 Rem. place to the left, and thus makes the value denoted tenfold (Art. 65). Or thus: 7 9 | 2. Now, it is obvious, that, if we reverse the process, and cut off the righthand figure by a line, we remove the remaining figures one place to the right, and consequently diminish the value denoted by each the same as dividing by 10. The figures on the left of the line are the quotient, and the one on the right is the remainder, which may be written over the divisor and annexed to the quotient. Hence each man's share is 79%. 2. How many years will it take a man at a yearly salary of 700 dollars to earn 3664 dollars ? Ans. 5766 years. The divisor, 700, may 1100) 36164 be resolved into the factors 7 and 100. We first divide 7) 3 6, 6 4, 1st Rem. by the factor 100, by cut5, 1, 2d Rem. ting off two figures at the right, and get 36 for the Or thus : 7100 ) 3 6164 quotient, and 64 for a re mainder. We then divide 5, 1 6 4. the quotient, 36, by the other factor, 7, and obtain 5 for a quotient and 1 for a remainder. The last remainder, 1, being multiplied by the divisor, 100, and 64, the first remainder, added, we obtain 164 for the true remainder (Art. 77); and for the answer required, 5263 9787 years. Hence, when the divisor contains one or more ciphers at the right, we may, to perform the division, Cut off the ciphers from the right of the divisor, and the same number of figures from the right of the dividend; and then divide the remaining figures of the dividend by the remaining figures of the divisor. NOTE. — When, by the operation, there is a remainder, to it must be annexed the figures cut off from the dividend to form the true remainder. Should there be no last remainder, then the significant figures, if any, cut off from the dividend, will form the true remainder. EXAMPLES Quotient. 12345678 Rem. 9. 3. Divide 123456789 by 10. 4. Divide 987654300 by 100. Quotients Rem. 5. Divide 32100 by 6000. 5 2100. 6. Divide 3678953 by 326100. 11 91853. 7. Divide 1637851 by 500000. 3 137851. 8. Divide 41111111 by 1100000. 37 411111. 9. Divide 89765432156 by 1000000. 10. The entire annual loss to the United States in consequence of intemperance has been estimated to be about 98,400,000 dollars. How many schools at a yearly expense of 600 dollars would that sum support? Ans. 164,000 schools. 11. The late war with Russia was carried on at a cost of 600,000,000 dollars to Great Britain. Allowing that country to have a population of 28,000,000, what was the cost to each individual ? 12. If light moves at the rate of 192,000 miles in a second, how long is it in passing from the sun to the earth, a distance of 95,000,000 of miles. Ans. 49415 seconds. GENERAL PRINCIPLES AND APPLICATIONS. 80. In division, the value of the quotient depends upon the relative values of the divisor and dividend. 81. If the dividend be multiplied, or the divisor divided, by any number, the quotient is multiplied by the same number. Thus, if the dividend be 20 and the divisor 4, the quotient will be 5; but if the dividend be multiplied by any number, as 2, and the divisor remain unchanged, the quotient will be 2 times as large as before, or 10; as (20 X 2) = 4 10; and if the divisor be divided by the 2, and the dividend remain unchanged, the quotient will be, likewise, 2 times as large, or 10; as 20 ; (4 2) 10. 82. If the dividend be divided, or the divisor multiplied, by any number, the quotient is divided by the same number. Thus, if the dividend be 32 and the divisor 8, the quotient will be 4. But if the dividend be divided by any number, as 2, and the |