28. 3 boys had 15 cents apiece, and they agreed to divide the whole among 5 poor persons. How many cents did they give each? 29. 9 boys had 12 marbles apiece, and they put them into 3 equal parcels. How many were in each place? 30. A man bought 5 yards of cloth at 12 shillings a yard. How many dollars did it come to, allowing 6 shillings to a dollar? 31. A man bought 4 loads of wood, at 3 shillings a load. How many dollars did it come to? 32. How many times 3, in 6 times 2? in 9 times 2? in 7 times 12? 33. How many times 7, in 3 times 14? in 2 times 21 ? 34. How many times 2, in 3 times 8? in 7 times 4? in 9 times 6? 35. 18 twos are how many 3s.? how many 4s.? how many 9s. ? 36. 12 threes are how many 9s.? how many 6s.? how 4s.? 37. 16 fours are how many 8s.? how many 32s.? how many 2s.? 38. 10 threes are how many 5s. ? how many 15s. ? how many 6s. ? 39. 8 nines are how many 3s.? how many 4s.? how many 12s. ? 40. 6 fives are how many 15s. ? how many 10s.? how many 3s. ? 41. 4 nines are how many 12s.? how many 25.? how many 3s. ? 42. 9 tens are how many 5s.? how many 3s.? how many 6s. ? 43. 3 sixteens are how many 4s. ? how many 12s.? how many 8s.? XXV. The learner will observe that, in the last section, the examples have required him either, I. TO SEPARATE A NUMBER INTO SEVERAL EQUAL PARTS, or II. TO FIND HOW OFTEN ONE NUMBER IS CONTAINED IN ANOTHER. The following examples are illustrations of these two operations. The same numbers are employed in both. I. A man had 12 acres of land, which he wished to fence off into 4 equal lots. How many acres might he have in each lot? 4 First, take one acre for each lot. This will make 4 acres. Subtract the 4 from 12, and 8 acres are left. Then take one more acre for each, making 4 more, and subtract as before. are left. Take 1 acre for each again, making 4 more, subtract again, and nothing is left. We have now distributed all the land into 4 equal lots. 12 4 8 4 4 4 0 This we did, by taking one acre, for each lot, 3 times successively. But one acre, taken three times is 3 acres. Therefore, each lot will contain 3 acres. II. A man had 12 acres of land, which he wished to divide into lots of 4 acres each. How many lots had he? First, take 4 acres for one lot, and 8 are left. Then, 4 acres for another, and 4 are left. Then, 4 acres for another, and nothing is left. 241841 We have now taken all the lots of 4 acres, which it is possible to take from 12 acres, and nothing is left. On looking back, we see that we have taken 4 acres away 3 times. Therefore there are 3 lots. From these illustrations, we see, that, in both these cases, the practical operation is the same; and consists, simply, in finding how often one number is contained in another. We see, moreover, that, we may find how often one number is contained in another, by repeated Subtraction. This however, would be, often, a long and tedious process. If, for instance, it were required to find how often 3 was contained in 3,729, we should be obliged to subtract 3, one thousand, two hundred and forty three times, in order to obtain the answer. For this reason, a shorter method has been contrived, called DIVISION. Then, DIVISION IS A CONCISE METHOD OF PERFORMING MANY SUBTRACTIONS OF THE SAME NUMBER. Of course, it is exactly the reverse of Multiplication. You see that two numbers are always given; one to be repeatedly subtracted: and another, from which the repeated subtractions are to be made: or, in other words, one to divide by, and another, to be divided. THE NUMBER, GIVEN, TO DIVIDE BY, IS CALLED THE DIVISOR. THE NUMBER, given, to be DIVIDED, IS CALLED THE DIVIDEND. THE RESULT, OBTAINED BY DIVISION, IS CALLED THE QUOTIENT. between two numbers signifies that the former is to be divided by the latter. Thus 8-4-2 signifies that S divided by 4, equals 2. But this sign is not often used. The most proper mode of indica ting Division, is, to write the Divisor, immediately under the Dividend, with a line between them. As the whole dividend ought to be divided, in order to render the Division complete, if there be any thing left, we must write it over the Divisor, and annex it to the Quotient; thereby indicating the Division of this part, because it cannot be actually performed. Thus 23÷7 is written 3 or the Division is performed thus 34. This will be treated of more fully, hereafter. In Subtraction, we have seen, that the difference, left, after taking the Subtrahend from the Minuend, is called the Remainder. So, likewise, in Division, which is a concise Subtraction, if there be any thing left, after taking away the Divisor as often as possible, it receives the same name. This excess must, of course, be less than the Divisor; for, were it not, the Divisor might be taken away again; which would be impossible, were the Division completely performed. Then, THE EXCESS Sometimes left, after perforMING DIVISION, IS CALLED THE REMAINDER, which must always be less than the Divisor. NOTE. It will be seen, that, the Division cannot be considered complete, until the Remainder is annexed to the quotient, as explained above. In order to understand division, we must observe, that, as the quotient shows, of how many times the divisor, the dividend consists, the quotient multiplied into the divisor will equal the dividend. The Dividend, therefore, corresponds to the Product, in Multiplication; and the Divisor and Quotient, to the two factors. This gives us a new definition. DIVISION IS FINDING A FACTOR, WHICH, MULTIPLIED BY THE DIVISOR, WILL PRODUCE THE DIVidend. EXAMPLES. 1. A man bought 3 barrels of cider for 9 dollars. give a barrel? What did he 3) 9 (3 9 Set down the dividend, 9, with the divisor, 3, on the left of it, and a line between; thus. Then, think what number you must multiply the divisor, 3, by, to produce the dividend, 9. This is 3, which place on the right of the dividend, for a quotient. Multiply the divisor by it, and place the product, 9, under the dividend, 9. Subtract it from the dividend, and nothing remains. Therefore 3 is the answer. In like manner perform the following. 0 2. A farmer paid 16 dollars for ploughs, giving 8 dollars apiece. How many ploughs did he buy? 3. If you can travel 40 miles in 8 hours, how far can you travel in 1 hour? 4. With 63 dollars, how many yards of cloth can you buy at 7 dollars a yard? 5 One man is 24 days performing a piece of work. How long would 4 men be about it? 6. Divide 18 by 6 68 16 66 66 25 66 8 5 Divide 28 by 7 66 33 66 3 If the dividends in the last example were to have each a cypher on the right, it is plain, we should have the same answers, with a cypher on the right of each. 7. Divide 180 by 6 66 160 66 8 ละ : 739 Divide 280 by 7 8. Examples of dividends with cyphers on the right. Divide 8,000 by 4 Divide 35,000 by 5 66 64,000 66 587 It is plain, however, that if the dividend consist of many significant figures, we cannot find the quotient in this manner. But, as in Multiplication, we multiplied each order separately, so, in Division, we may divide each order separately, and unite the several results. Thus, 9. A man travelled 369 miles, at the rate of 3 miles an hour. How many hours was he travelling? 369 consists of 300+60+9. Divide 300 by 3 9 66 3 The same is thus performed, without separating the orders. The quotient is found for each order, and then multiplied by the divisor, and subtracted from the dividend. the right, cyphers are annexed to the quotients and products, to show their real value. And, on the left, these cyphers are rejected. as unnecessary in the operation. The quotient figures are likewise written, one after another, in the order of their values, instead of separately. In subtracting, also, instead of bringing down all the figures of the dividend, which remain, we bring down only that, which is next to be divided. 9 9 Of course, if any order of the dividend is a cypher, the divisor is contained in it, No times, and a cypher must be placed in the quotient, so that the quotient figures, to the left of it, may stand in their proper places. How Ans. 401. 10. A man paid 802 dollars for land, at 2 dollars an acre. many acres did he buy? 11. I buy flour to the amount of 884 dollars, giving 4 dollars a barrel. How many barrels do I buy? Ans. 221. [Thus far each order has been divisible by the divisor, without any remain der. The pupil should recite the process of division, in these and the following examples, aloud, giving the names of the several orders, in succession.] In the preceding examples, it is unimportant on which side we begin to divide, if the quotient figures are correctly arranged. Here. after, we shall see, that it will be necessary to begin on the left. It may happen that the figures of some of the orders will leave a remainder, when divided. For example, 13. At $4.00, a box, how many boxes of oranges can I buy for $72.00? Here the figure in the highest order is 7. 4 is in 7, only once. Therefore, 1 is the greatest quotient figure I can obtain for that order. There is a remainder of 3. I join this 3 (tens,) that is, 30 units, to the 2 units, making 32 units, and divide both together. 4 is in 32, 8 times. Therefore 18 is the answer. 4) 72 (18 4 32 32 1881 Ans. 1,567. 14. In 3,134 pints, how many quarts? 16. In 9,872 lbs. of cheese, how many cheeses of 8 Ans. 2,459. pounds each? Ans. 1,234. How many Ans. 136. (3 barley-corns Ans. 261,318. 17. I put 816 lbs. of butter in boxes of 6 lbs. each. boxes are there? 18. In 783,954 barley-corns, how many inches? make 1 inch.) 19. At 7 dollars a barrel, how many barrels of flour can I buy for 525 dolls. ? 20. A merchant sold 7 cwt. of rice for $22.75. a cwt.? 21. Divide 125 by 5. 3,375 by 9. 5,274 by 9. What was that 83,136 by 6. Hence, when there is a remainder left, on dividing any order, consider it as so many tens of the next lower order, to which prefix it, and divide the whole together. This is the opposite of carrying, and may be called carrying downwards. If it should happen, that any order of the dividend is too small to contain the divisor, a cypher must be placed in the quotient, for that order; and the order itself must |