is greater than the partial dividend, the quotient figure is too large, and must be diminished. NOTE 4. If the remainder equals or exceeds the divisor, the quotient is too small, and must be increased. 77. Division is the reverse of multiplication. In multiplication the two factors are given, and the product is required; in division the product and one factor are given, and the other factor is required. The dividend is the product, and the divisor and quotient are the factors; thus, 78. PROOF. Multiply the divisor by the quotient, and to the product add the remainder; the SUM should be the dividend. 51. A flock of 1728 sheep were divided equally in 9 different pastures, how many sheep were there in each pasture? 77. What is said of Division and Multiplication? In Multiplication what is given? What required? In Division what is given? Required? 78. How is Division proved? 68. Divide four hundred eighteen thousand, six hundred and forty-eight, by twenty-four. Ans., Quo. 17443, Rem. 16. 69. Divide two hundred one thousand, five hundred and ninety-five acres of land, into twenty-three equal parts. 70. A railroad that cost $3576500 was divided into 7153 equal shares; what was the cost of each share? 71. A farmer raised 2001 bushels of wheat on 87 acres of land; how many bushels did he raise per acre? 72. In how many days will a ship sail 3456 miles, if it sails 144 miles per day? 73. A farmer raised 4088 bushels of corn, his crop 56 bushels per acre; how many acres did he plant? averaging 74. A drover paid $2175 for 29 oxen; how many dollars did he pay for each ox? 75. The product of two numbers is 10707, and one of the numbers is 129; what is the other number? 76. The earth, in its revolution round the sun, moves about 1641600 miles in one day; how far does it move in one second. there being 86400 seconds in a day? 77. Divide $1064 equally among 8 men. Ans. $133. 79. To divide by a composite number. Ex. 78. Divide $1855 equally among 35 men. The 35 men may be separated into 71 groups of 5 men each. Then divid ing by 7 gives $265 for each group, and each man. NOTE. When a composite number is made up of different sets of factors, as in Ex. 79, it is immaterial which set is taken. It is also immaterial in what order the factors are taken. 79. Divide 10656 by 288. 2884 X 6 X 12 6 X 6 X 88X3 X 12, etc. From these examples we have the following RULE. Divide the dividend by one factor of the divisor, and the quotient so obtained by another factor, and so on till all the factors of the set have been used. true quotient. 80. Divide 1551 by 33. 81. Divide 31794 by 42. 82. Divide 47936 by 56. 83. Divide 24840 by 72. 81. Divide 7665 by 105. 85. Divide 1064 by 56. 83. Divide 1984 by 64. 87. Divide 3321 by 81. The last quotient will be the 88. Divide 187236 by 252. 79. Rule for dividing by a Composite Number? Is it material which factor of the divisor is used first? 80. In dividing by the factors of the divisor, there may be a remainder after either or each of the divisions. Should the learner find a difficulty in determining the true remainder, he has but to remember that it is always of the same kind as the dividend (Art. 69, Note). divisor, is the other part; i. c. 5 + 3 × 6 = 23, is the true remainder. The same species of reasoning applies when there are more than two divisors. Hence, To obtain the true remainder when division is performed by using the factors of the divisor: RULE. Multiply each remainder, except that left by the first division, by the continued product of the divisors preceding that which gave the remainders severally, and the sum of the products, together with the remainder left by the first division, will be the true remainder. NOTE 1. When there are but two divisors and two remainders, the rule 80. Rule for finding the true remainder when the factors of the divisor are used separately? The reason? What is meant by a continued product? only requires the addition of the first remainder, to the product of the first divisor and second remainder. NOTE 2. When three or more factors are multiplied together, the product is called a continued product. 102. Divide 46987 by 504, using the factors of the divisor. 81. To divide by 10, 100, 1000, etc.: RULE. Cut off, by a point, as many figures from the right hand of the dividend as there are ciphers in the divisor. The figures at the left of the point are the quotient, and those at the right are the remainder. 113. Divide 756 by 10. Ans. 75.6, i. e. 75 Quo., 6 Rem. 81. Rule for dividing by 10? By 100? |