3. Find how many times 36 is contained in 11798. 4. Find how many times 82 is contained in 89634. 5. Find how many times 154 is contained in 32740. 25385201. 974. 23434402. 645. 826451. 70404. 901. 5009. 346. 7. 22. Divide 28101418481 by 1107. 27. A railroad cost one million eight hundred fifty thousand four hundred dollars, and was divided into eighteen thousand five hundred and four shares; what was the value of each share? Ans. 100 dollars. 28. If a tax of seventy-two million three hundred twenty thousand sixty dollars be equally assessed on ten thousand seven hundred thirty-five towns, what amount of tax must each town pay? Ans. 6736 dollars. 29. In 1850 there were in the United States 213 college libraries, containing 942321 volumes; what would be the average number of volumes to each library? Ans. 4424 vols. 30. The number of post offices in the United States in 1853 was 22320, and the entire revenue of the post office department was 5937120 dollars; what was the average revenue of each office? Ans. 266 dollars. CONTRACTIONS. CASE L 81. When the divisor is a composite number. 1. If 3270 dollars be divided equally among 30 men, how many dollars will each receive? OPERATION. 5)3270 6) 654 109 Ans. ANALYSIS. If 3270 dollars be divided equally among 30 men, each man will receive as many dollars as 30 is contained times in 3270 dollars. 30 may be resolved into the factors 5 and 6; and we may suppose the 30 men divided into 5 groups of 6 men each; dividing the 3270 dollars by 5, the number of groups, we have 654, the number of dollars to be given to each group; and dividing the 654 dollars by 6, the number of men in each group, we have 109, the number of dollars that each man will receive. RULE. Divide the dividend by one of the factors, and the quotient thus obtained by another, and so on if there be more than two factors, until every factor has been made a divisor. The last quotient will be the quotient required. Ans. 1246. Ans. 2735. Ans. 125. 5. Divide 6. Divide 33642 by 27=3 × 9. 7. Divide 153160 by 56=7×8. 8. Divide 15625 by 125-5 × 5 × 5. 82. To find the true remainder. 1. Divide 1143 by 64, using the factors 2, 8, and 4, and find the true remainder. What are contractions? Case I is what? Give explanation? Rule. 571 being a quotient arising from dividing by 2, its units are 2 times as great in value as the units of the given dividend, 1143. Dividing the 571 by 8, we have a quotient of 71, and a remainder of 3 undivided. As this 3 is a part of the 571, it must be multiplied by 2 to change it to the same kind of units as the 1. This makes a true remainder of 6 arising from dividing by 8. Dividing the 71 by 4, we have a quotient of 17, and a remainder of 3 undivided. This 3 is a part of the 71, the units of which are 8 times as great in value as those of the 571, and the units of the 571 are 2 times as great in value as those of the given dividend, 1143; therefore, to change this last remainder, 3, to units of the same value as the dividend, we multiply it by 8 and 2, and obtain a true remainder of 48 arising from dividing by 4. Adding the three partial remainders, we obtain 55, the true remainder. RULE. I. Multiply each partial remainder, except the first, by all the preceding divisors. II. Add the several products with the first remainder, and the sum will be the true remainder. 5. Divide 9078126 by 90=3×5×6. 6. Divide 18730627 by 120-4 × 5 × 6. 7. Divide 7360479 by 96=2×6×8. 8. Divide 24726300 by 70=2×5×7. 9. Divide 5610207 by 84-7×2×6. Explain the process of finding the true remainder when dividing by the factors of a composite number. 6. 67. 63. 60. 15. CASE II. 83. When the divisor is 10, 100, 1000, etc. 1. Divide 374 acres of land equally among 10 men ; how many acres will each have? OPERATION. 10)374 Quotient. 37...4 Rem. or, 37 acres. ANALYSIS. Since we have shown, that to remove a figure one place toward the left by annexing a ciphe increases its value tenfold, or multiplies it by 10 (68), so, on the contrary, by cutting off or taking away the right hand figure of a number, each of the other figures is removed one place toward the right, and, consequently, the value of each is diminished tenfold, or divided by 10 (32). For similar reasons, if we cut off two figures, we divide by 100, if three, we divide by 1000, and so on. RULE. From the right hand of the dividend cut off as many figures as there are ciphers in the divisor. Under the figures so cut off, place the divisor, and the whole will form the quotient. 2. Divide 4760 EXAMPLES FOR PRACTICE. 3. Divide 362078 4. Divide 1306321 by 10. by 100. by 1000. 5. Divide 9760347 by 10000. 6. Divide 2037160310 by 100000. CASE III. 84. When there are ciphers on the right hand of the divisor. |