find the difference between those numbers, we subtract; but when we take one number from another once, or more, to find how many times one number can be subtracted from another, we divide. Although we subtract in Division as well as in Subtraction, the object to be obtained by dividing, is essentially different from that sought in Subtraction. In Subtraction we never subtract but once, the only object being to find how much larger one number is than the other; but in Division we may subtract once or more, the object being to find how many times one number contains another. If the difference between 8 and 4 be demanded, we ascertain it by subtracting the 4 once from the 8; thus, 8 4 4 difference. But if 8 is to be divided by 4, we subtract the 4 from the 8 as many times as it can be ; thus, 8 It appears that 4 can be taken twice from 8, therefore 2 is the answer. We may always obtain an answer to any question in Division, by subtracting the less number from the greater, and then the less number from the remainder, and so on, subtracting the less number every time from the last remainder, until the remainder. becomes less than the number we are subtracting ; and the number of times we subtract would be the answer. But it is easy to see that this method must be very tedious when the less number can be subtracted from the other a great number of times. therefore a shorter method has been devised. DEFINITIONS, 4 Dividend is a number given to be divided. 4 Divisor is a number given to divide by. A Quotient is a number showing how many times the divisor can be subtracted from the dividend. 4 Remainder is a number left after dividing; which must always be less than the divisor. [If it be required to divide 16 by 5, 16 is the dividend, 5 the divisor, and as we can subtract 5 3 times from 15, 3 is the quotient, and the 1, which is left, is the remainder.] Let the scholar answer the following questions. How many times can 3 be taken from 9 2 We found in the Multiplication Table, that 3 multiplied by 3, or 3 added together 3 times, made 9; consequently 3 can be taken from 9 as many times as there were 3’s put together to make 9. How many times can 3 be taken from 127 3 from 15 ! 4 from 12? 4 from 16? 5. from 107 5 from 20 ! 2 from 142 5 from 15 3 from 15 2 3 from 18 2 3 from 21 2 7 from 21 2 How many times does 24 contain 82 24 contain 6 7 32 contain 87 36 contain 62 42 contain 7 ? 42 contain 62 How many times can 9 be taken from 28, and what number will remain 2 7 from 302 7 from 462 9 from 402 9 from 60 7 8 from 36 7 8 from 47 ? 8 from 602 10 from 54 * 10 from 85 ° 9 from 100% RULE I. Place the divisor at the left hand of the dividend. Ascertain as near as you can in the mind, how many times the divisor can be taken from the dividend, and place a figure expressing that number at the right hand of the dividend for the quotient. Multiply the divisor by the quotient, and subtract the product from the dividend. Mote 1. If the remainder be as large or larger than the divisor, the quotient is too small for the divisor, in that case, could be subQuestion 1. How many times can 12 be taken from 48 2 Operation by subtracting. 4 8 dividend. 1 2 divisor. -- Illustration. In this ques1st subtraction 3 6 tion, as we can subtract 12 1 2 4 times from the dividend, 4 is the quotient. The 36 is tracted once or more from the remainder. - ient is too If the product be larger than the dividend, the quotient is e. 2d & 4 2 - 4 a part of the 48 left after 1 2 subtracting the 12 once; the - 24 is what remains of the 36 3d 44 1 2 after taking out 12; conse1 2 quently it is a part of the 48; and the 12, from which we subtract the last time, is that part of 24 remaining, after taking 12 from 24; therefore in each of the subtractions, we subtract from 48, or some part of it. Now were we to add the 12 together as many times as we have subtracted it, the sum would be 48. But were we to multiply 12, the divisor, by 4, the number of times we have subtracted, the product would be the same number that we found by adding the 12, 4 times together; and if we subtract . the 48, made by multiplying the 12 by 4, we should subtract 12 as many times as we did when we subtracted only 12 at a time, as may be seen by the following operation. Operation by Division. Divisor 12) 4 8 (4 quotient. 4 8 By comparing this operation — with the preceding one, we see O how much easier it is to perform questions by dividing than it would be were we obliged to perform the same operations by subtracting. 2. Divide 99 by 11. Quotient 9. 3. Divide 128 by 16. Quot. 8. 4. How many times can 13 be taken from 65? Ans. 5. *. What is the quotient of 176, divided by 18? Operation. 18) 1 7 6 (9 1 6 2 * 1 4 remainder. 7. Divide 115 by 19. Quotients, Remainders. 3. 4% 127 “ 25. {{ 5 4. 2. 13. A boy wishes to divide 35 pears equally be. tween 5 companions: how many must he give each? Operation. 5) 3 5 (7 Ans. Explanation. Were the 3 5 boy to give 1 pear to each com-- panion, and then I more to - each, and so on, until he had distributed the whole number, he would evidently give each companion an equal number. By taking 5 pears from the whole number, we take away enough to give one to each boy; by taking away 5 more, we take enough to give each boy another, then each boy will have 2; consequently, as many times as we can take 5 from 35, the whole number of pears, so many each boy will receive; and as 7 is the number of times we can subtract 5 from 35, 7 is the number which each boy is to receive. JNote 2. It appears from what has been shown in the explanation of the last question, that when any number or quantity, is to be divided into equal parts, or equally between several persons, that the number of parts into which such a number or quantity is to be. divided, must be the divisor, and that the quotient will show what one of those parts is. 14. If 66 be divided into 22 equal parts, what will be one of those parts? Ans. 3. 15. Divide 696 dollars equally between 116 men; how many dollars will each man receive 2 Ans. 6. 16. What is the 25th part of 225 ! Ans. 9. 17. What is the 9th part of 81 7 Ans. 9. 18. How many bushels of wheat, at 9 shillings a bushel, can be bought for 54 shillings 2 Ans. 6. 19. An army consisting of 97.02 men, receives 67914 pounds of bread per week; how many pounds does each man receive 2 Ans. 7. RULE. II. When it is found that the quotient is more than 9, ascertain how many times the divisor can be taken from an equal number of #. at the left hand of the dividend, or one more, if an equal number be not so large as the divisor; set a figure representing the number of times for the first figure of the quotient; multiply the divisor by this figure, and subtract the product from the figures taken at the left hand of the dividend. To the right hand of the remainder bring down the next figure of the dividend; divide this number as directed by the first rule; and continue the same process till all the remaining figures of the dividend are brought down. If the number made by the remainder and figure brought down, be not so large as the divisor, place a cipher in the quotient and bring down the next figure. 20. Required the quotient of 2341, divided by 15. Operation. 1 5 ), 2 3 4 1 (1 5 6 Explanation. In this I 5 question, we find that the divisor can be taken only 8 4 - once from the two first 7 5 figures of the dividend, *- therefore we place 1 for 9 1 . the first figure of the quo.9 O tient, multiply the divisor - by it, and subtract the I product from the 23. To 8, the remainder, we |