7. How many times is the number 43 contained in the number 12986 ? 8. How many times is the number 627 contained in the number 657723 ? 9. What is one of the equal parts of 256 barrels of flour, divided equally among 16 families? 10. How many times is the number 804 contained in the number 320796? X 11. Divide 147735 by 45. 16. Divide 14420946 by 74. 17. Divide 295470 by 90. 18. Divide 1874774 by 162. Divide 435780 by 216. 20. Divi. 119836687 by 3041. 21. Divide 203812983 by 5049. 30. Divide 558001172606176724 by 2708630425. 32. Divide 6754371495671594 by 678957. . 38. Divide 121932631112635269 by 987654321. 39. In a hogshead there are 63 gallons: how many hogs heads are there in a reservoir, containing 2645750 gallons? 40. A drover wishes to divide 15600 cattle into 75 droves how many cattle must he put in each drove ? 73. Principles resulting from Division. 1. When the divisor is 1, the quotient will be equal to the dividend. 2. When the divisor is equal to the dividend, the quotient will be 1. 3. When the divisor is less than the dividend, the quotien will be greater than 1. 4. When the divisor is greater than the dividend, the quotient will be less than 1. Proof of Multiplication. 74. In Division, the divisor and quotient are factors of the dividend. In Multiplication, the multiplicand and multiplier are factors of the product: Hence, If the product of two numbers be divided by the multiplicand, the quotient will be the multiplier; or, if the product be divided by the multiplier, the quotient will be the mul tiplicand. 73. When the divisor is 1, what is the quotient? When the divisor is equal to the dividend, what is the quotient? When the divisor is less than the dividend, how does the quotient compare with 1? When the divisor is greater than the dividend, how does the quotient compare with 1? 74. In Multiplication, what are the factors of the product? If the product be divided by the multiplicand, what is the quotient? If it be divided by the multiplier, what is the quotient? 96 2. The product of two factors is 68959488; one factor, ; what is the other? 3. The multiplier is 270000; now, if the product be 1315170000000, what will be the multiplicand? Contractions in Division. 75. CONTRACTIONS IN DIVISION, are short methods of finding the quotient, when the divisor is a composite number. CASE I. 76. When the divisor is any composite number. 1. Let it be required to divide 1407 dollars equally among Here the factors of the divisor are 7 and 3. 21 men. ANALYSIS.-Let the 1407 dollars be first divided into 7 equal piles. Each pile will contain 201 dollars. Let each pile be now divided into 3 equal parts. Each part will contain 67 dollars, and the number of parts will be 21: hence the following Rule. OPERATION. 7) 1407 3) 201 1st quotient. 67 quotient sought. Divide the dividend by one of the factors of the divisor; then divide the quotient, thus arising, by a second factor, and so on, till every factor has been used as a divisor; the last quotient will be the answer. Examples. Divide the following numbers by the factors: 1. 1260 by 12 = 3 × 4. 5. 55728 by 4×9×4=144. 6. 92880 by 2×2×3×2×2. 75. What are contractions, in Division? What is a composite number? 76. What is the rule, when the divisor is any composite number? True Remainder, when the divisor is a composite number. Let it be required to divide 755 grapes into 24 equal parts. 242 × 3 × 4. 125 piles, each containing 3 bunches, and 2 bunches over = 2 × 2 = 4 grapes. If we divide 125 piles into 4 equal parts, we shall have 31 new piles, and 1 pile over 3 x 26 grapes. Hence, to find the true remainder, we have the following Rule. To the first remainder add the products which arise by multiplying each of the following remainders by all the preceding divisors, except its own; their sum will be the true remainder. Examples. 1. Let it be required to divide 43720 by 45 = 3 × 5 × 3. What is the rule for finding the true remainder? 77. When the divisor is 10, 100, 1000, &c. 1. In 476 yards of cloth, how many pieces are there of 0 yards each? ANALYSIS.-There will be one-tenth as nany pieces as there are yards; 47 tens is one-tenth of 47 hundreds: if, then, we strike off the right-hand figure, we obtain one-tenth of 476, which is 47, and 6 over. OPERATION. 10) 476 47-6 remainder. 47% quotient. If the divisor is 100, the quotient is one-hundredth of the dividend; 4 is one-hundredth of 4 hundreds: if, then, we strike off the two right-hand figures, thus, 476, we obtain one-hundredth of 476, and 76 over. Hence, the following Rule. I. From the right hand, cut off, by a line, as many figures as there are ciphers in the divisor: II. The figures at the left will be the quotient, and those at the right, the remainder. 78. When the divisor contains significant figures, with ciphers on the right of them. 77. What is the rule when the divisor is 1, with any number o ciphers annexed? |