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. 65381279 dollars. ART. 55. Method of operation, when the divisor is a composite number. Ex. 1. A merchant bought 15 pieces of broadcloth for 1440 dollars, what was the value of each piece? OPERATION. 3) 1 4 4 0 dolls., cost of 15 pieces. 5) 480 dolls., cost of 5 pieces. 96 dolls., cost of 1 piece. Ans. 96 dollars. The factors of 15 are 3 and 5. Now, if we divide the 1440 dollars, the cost of 15 pieces, by 3, we obtain 480 dollars, which is evidently the cost of 5 Then, dividing 480 dol pieces, because there are 5 times 3 in 15. lars, the cost of 5 pieces, by 5, we get the cost of 1 piece. Hence we deduce the following 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, and 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 ART. 56. Method of finding the true remainder, when here are several 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. QUESTIONS. Art. 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? What is the rule for dividing by a composite number? OPERATION. 7) 298 4) 42-4 days 18 days. 10 2 weeks Since there are 7 days in 1 week, we first divide the 298 days by 7, and have 42 weeks and a remainder of 4 days. Then, since 4 weeks make 1 month, we divide the 42 weeks by 4, and have 10 months and a remainder of 2 weeks. Now, to find the true remainder in days, it is evident that we must 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 remainder. Hence the propriety of the following RULE. Multiply each remainder 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. OPERATION. 2)789 Ans. 33. 2, 2d Product. 6) 131-1, 2d Rem. 21-5, 3d Rem. 1, 1st Remainder. 33, true Rem. 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. 5. Divide 5873 by 84, using the factors 3, 4, and the true remainder. 6. Divide 249237 by 1728, using the factors and find the true remainder. Ans. 20. 7, and find Ans. 77. 12, 6, 6, and 4, Ans. 405. ART. 57. To divide by 10, 100, &c., or 1 with ciphers at the right. Ex. 1. Divide 356 dollars equally among 10 men, what will each man have? Ans. 35 dollars. QUESTIONS. Art. 56. When there are several remainders, what is the rule for finding the true remainder? Will you give the reason for this rule ? OPERATION. 110) 3 5/6 Quotient 35 .6 Rem. It will be remembered, that, to multiply by 10, we annex one cipher, which removes the figures one place to the left, and thus increases their value ten times. 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 of each ten times, and thus divide the whole number 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 dollars. RULE. Cut off as many figures from the right hand of the dividend as there are ciphers in the divisor, and the figures on the left hand of the separatrix will be the quotient, and those on the right hand the remainder. Ex. 1. If I divide 5832 pounds of bread equally among 600 soldiers, what is each one's share? OPERATION. 1,00) 583 2 6) 58-32, 1st Rem. 9. 4, 2d Rem. Or thus, 600) 5 83 2 Ans. 9488 pounds. The divisor, 600, may be resolved into the factors 6 and 100. We first divide by the factor 100, by cutting off two figures at the right, and get 58 for the quotient and 32 for a remainder. We then divide the quotient, 58, by 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 9433 pounds. 9-432 QUESTIONS. Art. 57. How do you divide by 10? How does it appear that this divides the number by 10? What is the rule for dividing by 10, 100, &c.? Art. 58. How do you divide by 600 in the example? How does it appear that this divides the number? RULE. Cut off the ciphers from the divisor, and the same number of figures from the right hand of the dividend. Then divide the re maining figures of the dividend by the remaining figures of the divisor, and the result will be the quotient. To complete the work, annex to the last remainder found by the operation the figures cut off from the div idend, and the whole will form the true remainder. 497654325 306787 6. Divide 71897654325 by 700000000. 44916000000 RULE. - Annex two ciphers to the multiplicand, and divide it by 4, and the quotient is the product required. *If the principles on which these contractions depend are considered too difficult for the young pupil to understand at this stage of his progress, they may be omitted for the present, and attended to when he is further advanced. QUESTIONS. What is the rule for dividing when there are ciphers on the right of the divisor? — Art. 59. What is the rule for multiplying by 251 What is the reason for the rule ? EXAMPLES FOR PRACTICE. 2. Multiply 76589658 by 25. 3. Multiply 567898717 by 25. 4. Multiply 123456789 by 25. ART. 60. To multiply by 33. OPERATION. 3) 876789 6300 2922632100 Product. Ans. 1914741450. Ans. 14197467925. Ans. 3086419725. Ans. 2922632100. We first multiply by 100, as before, and since 33, the multiplier, is only one third of 100, we divide by 3 to obtain the true product. RULE. Annex two ciphers to the multiplicand, and divide it by 3, and the quotient is the product required. ART. 61. To multiply by 125. OPERATION. 8) 7896538000 987067250 Product. Ans. 19454930400. Ans. 987067250. We multiply by 1000, by annexing three ciphers to the multiplicand, and since 125, the multiplier, is only one eighth of 1000, we divide by 8 to obtain the true product. RULE. — Annex three ciphers to the multiplicand, and divide by 8, and the quotient is the product. QUESTIONS. Art. 60. What is the rule for multiplying by 33? What is the reason for this rule? Art. 61. What is the rule for multiplying by 125 ? Give the reason for the rule ? |