7 in 30 tens, 4 tens times, and 2 tens, equal to 20 units, remaining. We write the 4 tens, and add the 20 units to the 2 units, making 22 units. 7 in 22 units, 3 units times, and 1 unit remaining. We write the 3 units, and indicating the division of the 1 unit, we annex the fractional expression, unit, to the integral part of the quotient. Therefore, 1702 divided by 7 is equal to 2434, read two hundred forty-three and one seventh. The solution by the second kind of Division is: One seventh of 17 hundreds is 2 hundreds, and a remainder of 3 hundreds, equal to 30 tens. We write the 2 hundreds, and add the 30 tens to the 0 tens, making 30 tens. One seventh of 30 tens is 4 tens, and a remainder of 2 tens, equal to 20 units. We write the 4 tens, and add the 20 units to the 2 units, making 22 units. One seventh of 22 units is 3 units, and a remainder of 1 unit. We write the 3 units, and indicating a seventh of the 1 unit, we annex the expression,, to the integral part of the quotient. Therefore, 1702 divided by 7 is 2434. 2. Let it be required to obtain the quotient of 763 divided by 7. OPERATION. Divisor, 7) 763 Dividend. 109 Quotient. For convenience, we write the divisor, and begin to divide, as in the preceding operation. 7 in 7 hundreds, 1 hundreds time, which we write. 7 in 6 tens 0 tens times, and 6 tens, equal to 60 units, remaining. We write the 0 tens, and add the 60 units to the 3 units, making 63 units. 7 in 63 units, 9 units times, which we write. In practice we may say, 7 in 7, 1, which we write; 7 in 6, 0, which we write; prefix the 6 to the figure 3; 7 in 63, 9, which we write; answer, 109. When, as in the above operations, the dividing is performed men、 tally, except in writing the quotient figures, the process is called SHORT DIVISION. Give the solution by the first ind of Division. The second. When is the process called Short Division? For, the divisor must then denote the number of equal parts into which the dividend is to be divided. Hence, 3. If the divisor and dividend are not like numbers, the quotient and dividend will be like numbers. For, the quotient will denote one of the equal parts into which the dividend is divided. 4. The remainder and dividend must always be like numbers. For, the remainder is evidently a part of the dividend. 5. Division may be regarded as the reverse of multiplica tion. For, the dividend corresponds to the product, and the divisor and quotient to the two factors. 61. From the foregoing principles, it follows that there may be two kinds of division: FIRST KIND - when the size or value of the equal parts of a quantity is given, to find their number; and SECOND KIND when the number of equal parts of a quantity is given, to find their size or value. CASE I. 62. To divide by Short Division. 1. Let it be required to divide 1702 by 7. OPERATION. Divisor, 7) 1702 Dividend. 2434 Quotient. For convenience, we write the divisor at the left of the dividend, and begin at the left to divide. 7 is contained in 1 thousand no thousands times; therefore, there will be no thousands in the quotient. Try 17 hundreds; 7 is contained in 17 hundreds, 2 hundreds times, and 3 hundreds, equal to 30 tens, remaining. We write the 2 hundreds, and add the 30 tens to the O tens, making 30 tens. What is the third Principle? The fourth? The fifth? Name the two kinds of Division. 7 in 30 tens, 4 tens times, and 2 tens, equal to 20 units, remaining. We write the 4 tens, and add the 20 units to the 2 units, making 22 units. 7 in 22 units, 3 units times, and 1 unit remaining. We write the 3 units, and indicating the division of the 1 unit, we annex the fractional expression, unit, to the integral part of the quotient. Therefore, 1702 divided by 7 is equal to 2434, read two hundred forty-three and one seventh. The solution by the second kind of Division is: One seventh of 17 hundreds is 2 hundreds, and a remainder of 3 hundreds, equal to 30 tens. We write the 2 hundreds, and add the 30 tens to the 0 tens, making 30 tens. One seventh of 30 tens is 4 tens, and a remainder of 2 tens, equal to 20 units. We write the 4 tens, and add the 20 units to the 2 units, making 22 units. One seventh of 22 units is 3 units, and a remainder of 1 unit. We write the 3 units, and indicating a seventh of the 1 unit, we annex the expression,, to the integral part of the quotient. Therefore, 1702 divided by 7 is 2434. 2. Let it be required to obtain the quotient of 763 divided by 7. OPERATION. Divisor, 7) 763 Dividend. 109 Quotient. For convenience, we write the divisor, and begin to divide, as in the preceding operation. 7 in 7 hundreds, 1 hundreds time, which we write. 7 in 6 tens 0 tens times, and 6 tens, equal to 60 units, remaining. We write the 0 tens, and add the 60 units to the 3 units, making 63 units. 7 in 63 units, 9 units times, which we write. In practice we may say, 7 in 7, 1, which we write; 7 in 6, 0, which we write; prefix the 6 to the figure 3; 7 in 63, 9, which we write; answer, 109. When, as in the above operations, the dividing is performed men、 tally, except in writing the quotient figures, the process is called SHORT DIVISION. Give the solution by the first kind of Division. The second. When is the process called Short Division? RULE. Write the divisor at the left of the dividend. Begin at the left, divide the numbers expressed by each figure of the dividend by the divisor, and write the result beneath. If there be a remainder after any division, regard it as prefixed to the figure of the next lower order, and divide as before. If any partial dividend be less than the divisor, write a cipher in the quotient, and prefix such dividend to the next figure, if any, for a new dividend. If there be a final remainder, write it, with the divisor beneath, after the integral part of the quotient. PROOF. Multiply the integral number of the quotient by the divisor, and to the product add the remainder, if any; and the result will equal the dividend, if the work is right. 12. 321001 by 7. Ans. 458574. 17. 161413 by 6. Ans. 26902. 13. 447078 by 8. 18. 9080706 by 5. Repeat the Rule. What is the Proof? Required 19. One third of 189. Ans. 63. 21. One eighth of 9872. 1. How many cords of wood can be bought for 1965 dollars, at 5 dollars a cord? SOLUTION. Since the cost of 1 cord is 5 dollars, as many cords can be bought for 1965 dollars as 5 dollars are contained times in 1965 dollars, or 393. Therefore, there can be bought 393 cords of wood for 1965 dollars, at 5 dollars a cord. 2. If 4 bushels of wheat make 1 barrel of flour, how many barrels will 9650 bushels make? 3. At 7 cents a pound, how many pounds of rice can be bought for 363 cents? Ans. 51 pounds. 4. If 9 horses cost 2025 dollars, how much must 1 horse cost? SOLUTION. If 9 horses cost 2025 dollars, 1 horse must cost one ninth of 2025 dollars, or 225 dollars. Therefore, if 9 horses cost 2025 dollars, one horse must cost 225 dollars. 5. When 3168 dollars are paid for 6 bales of cloth, how much is paid for 1 bale? Ans. 528 dollars. 6. How many cords of wood, at 5 dollars a cord, can be bought for 1965 dollars? Ans. 393 cords. 7. When a carpenter is paid 581 dollars for 7 months' labor, how much is that a month? 8. 7 times a certain number is equal to 22134; what is the number? Ans. 3162. REVIEW QUESTIONS. What is a Concrete Number? (6) An Abstract Number? (7) |