EXAMPLES FOR PRACTICE. Ex. 7. How many times is 48 contained in 28618? Ans. 596. PROOF BY MULTIPLICATION Divisor 4 8 2 8 618 (5 9 6 Quotient. 5 9 6 Quotient. 2 40 48 Divisor. * This sign of addition denotes the several products to be added. 145 55 Quotients. Rem. 12. Divide 3051 by 21. 6 13. Divide 190850 by 25. 7634 0 14. Divide 218579 by 42. 5204 11 15. Divide 9012345 by 31. 290720 25 16. Divide 6717890 by 98. 68549 88 17. Divide 4567890 by 19. 240415 18. Divide 1357901 by 87. 15608 5 19. Divide 9988891 by 77. 129725 66 20. Divide 9999999 by 69. 144927 36 21. Divide 867532 by 59. 14703 55 22. Divide 167008 by 87. 1919 23. Divide 345678 by 379. 912 30 24. Divide 3456789567 by 987. 3502319 714 25. Divide 8997744444 by 345. 26080418 234 26. Divide 4500700701 by 407. 11058232 277 27. Divide 6789563 by 1234. 95 28. Divide 78112345 by 8007. 4060 29. Divide 34533669 by 9999. 7122 30. Divide 99999999 by 3333. 0 31. Divide 47856712 by 1789. 962 32. Divide 345678901765 by 4007. 86268755 480 33. Divide 478656785178 by 56789. 8428688 22346 34. Divide 678957000107 by 10789561. 62927 2295060 35. Divide 990070171009 by 900700601. 1099 200210510 36. Divide three hundred twenty-one thousand three hundred dollars equally among six hundred seventy-five men. Ans. 476. 37. Four hundred seventy-one men purchase a township containing one hundred eighty-six thousand forty-five acres; what is the share of each ? Ans. 395 acres. 38. A railroad, which cost five hundred eighteen thousand seventy-seven dollars, is divided into six hundred seventy-nine shares ; what is the value of each share ? Ans. 763 dollars. 39. Divide forty-two thousand four hundred thirty-five bushels of wheat equally among one hundred twenty-three men. Ans. 345 bushels each. 40. A prize, valued at one hundred eighty-four thousand seven hundred seventy-five dollars, is to be divided equally among four hundred seventy-five men; what is the share of each? Ans. 389. 41. A certain company purchased a valuable township for nine millions six hundred ninety-one thousand eight hundred thirty-six dollars ; each share was valued at seven thousand eight hundred fifty-four dollars ; of how many men did the company consist ? Ans. 1234 men. 42. A tax of thirty millions fifty six thousand four hundred sixty-five dollars is assessed equally on four thousand five hundred ninety-seven towns; what sum must each town pay? Ans. 65381217 dollars. 55. When the divisor is a composite number. OPERATION. Ex. 1. A merchant bought 15 pieces of broadcloth for 1440 dollars ; what was the cost of each piece ? Ans. 96 dollars. The factors of 15 are 3 3) 1 4 4 0 dolls., cost of 15 pieces. and 5. Now, if we divide the 1440 dollars, the cost of 5 ) 48 0 dolls., cost of 5 pieces. 15 pieces, by 3, we obtain 480 dollars, the cost of 5 pieces, because there are 5 6 dolls., cost of 1 piece. times 3 in 15. Then, dividing 480 dollars, the cost of 5 pieces, by 5, we get the cost of 1 piece. RULE. — Divide the dividend by one of the factors, and the quotient thus found by another, and thus proceed till every factor has been made a divisor. The last quotient will be the true quotient required. EXAMPLES FOR PRACTICE. 2. Divide 765325 by 25 = 5 x 5 Quotients 30613 1469 7546 7901 182 264 56. To find the true remainder when there are several remainders in the operation. Ex. 1. How many months of 4 weeks each are there in 298 days, and how many days remaining ? Ans. 10 months and 18 days. 55. What are the factors of 15? What do you get the cost of, in this example, when you divide by the factor 3? What, when you divide by 5? Why? The rule for dividing by a composite number? OPERATION. we re OPERATION. Since there are 7 days in 1 week, 7) 2 98 first divide the 298 by 7, and have 42 weeks and a 4) 42, 4 days 18 days. mainder of 4 days. Then, since 10, 2 weeks 4 weeks make 1 month, we di vide the 42 by 4, and have 10 months and a remainder of 2 weeks. Now, to find the true remainder in days, we multiply the 2 weeks by 7, because 7 days make a week, and to the product add the 4 days; thus 2 X 7=14, and 14+4=18 days, for the true remainder. RULE. — 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. Ex. 2. Divide 789 by 36, using the factors 2, 3, and 6, and find the true remainder. Ans. 33. Dividing by 2 2) 7 8 9 5 X 3 X2=30, 1st Prod. gives 394 twos, 3) 3 9 4, 1, 1st Rem. 1 X 2 = 2, 2d Prod. and 1 unit re 1, 1st Rem. maining; divid6) 1 3 1, 1, 2d Rem. ing the twos by 3 gives 131 sixes, 2 1, 5, 3d Rem. 33, true Rem. and 1 two 2 remaining; and dividing the sixes by 6 gives 21 thirty-sixes, and 5 sixes 30 remaining; therefore 1 + 2 + 30, or 33, is the true remainder. EXAMPLES FOR PRACTICE. 3. Divide 934 by 55, using the factors 5 and 11, and find the true remainder. Ans. 54. 4. Divide 5348 by 48, using the factors 6 and 8, and find the true remainder. Ans. 20. 5. Divide 5873 by 84, using the factors 3, 4, and 7, and find the true remainder. Ans. 77. 6. Divide 249237 by 1728, using the factors 12, 6, 6, and 4, and find the true remainder. Ans. 405. 57. When the divisor is 1, with one or more ciphers at the right. Ex. 1. Divide 356 dollars equally among 10 men ; what will each man have ? Ans. 35 dollars. 56. When there are several remainders, what is the rule for finding the true remainder ? The reason for this rule } OPERATION. To multiply by 10 we annex one 110) 3516 cipher, which removes each figure one place to the left, and thus makes the Quotient 35, 6 Rem. value denoted tenfold. Now, if we Or thus, 3 5 | 6. reverse the process, and cut off the right-hand figure of the dividend by a line, we remove each remaining figure one place to the right, and consequently diminish the value denoted 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 the share of each man is 35,8 dollars. EXAMPLES FOR PRACTICE. 2. Divide 6892 by 10. Quotient. Rem. 689 2 75 815 54321123 58. When the divisor has one or more ciphers on the right, and is not 10, 100, &c. Ex. 1. If I divide 5832 pounds of bread equally among 600 soldiers, what is each one's share? Ans. 9433 pounds. OPERATION. The divisor, 600, may be re1100) 583 2 solved into the factors 6 and 100. 6 ) 5 8, 32, 1st Rem. We first divide by the factor 100, by cutting off two figures at the 9, 4, 2d Rem. right, and get 58 for the quotient and 32 for a remainder. We Or thus, 6 100 ) 5 8 1 3 2 then divide the quotient, 58, by 9, 4 3 2 the other factor, 6, and obtain 9 for the quotient and 4 for a remainder. The last remainder, 4, being multiplied by the divisor, 100, and 32, the first remainder, added, we obtain 432 for the true remainder (Art. 56). Hence each soldier receives 948 pounds. 59. Rule. — Cut off the ciphers from the divisor, and the same number of figures from the right of the dividend. Then divide the remaining figures of the dividend by the remaining figures of the divisor. 57. How do you divide by 10? How does it appear that this divides the number by 10 ? 58. How do you divide by 600 in the example ? How does it appear that this divides the number? - 59. The general rule ? |