12. A boy had two apples, and gave one half an apple to each of his companions; how many were his companions? 13. A boy divided four apples among his companions, by giving them one third of an apple each; among how many did he divide his apples? 14. How many quarters in 3 oranges? 15. How many oranges would it take to give 12 boys one quarter of an orange each? 16. How much is one half of 12 apples? 17. How much is one third of 12? 18. How much is one fourth of 12? 19. A man had 30 sheep, and sold one fifth of them; how many of them did he sell? 20. A man purchased sheep for 7 dollars apiece, and paid for ther.. all 63 dollars; what was their number? 21. How many oranges, at 3 cents each, may be bought for 12 cents? It is plain, that as many times as 3 cents can be taken from 12 cents, so many oranges may be bought; the object, therefore, is to find how many times 3 is contained in 12. 15. It is plain, that the cost of one orange, (3 cents,) multiplied by the number of oranges, (4,) is equal to the cost of all the oranges, (12 cents;) 12 is, therefore, a product, and 3 one of its factors; and to find how many times 3 is contained in 12, is to find the other factor, which, mu!tiplied into 3, wiil produce 12. This factor we find, by trial, to be 4, (4 X 3 12;) consequently, 3 is contained in 12 4 times. Ans. 4 oranges. 22. A man would divide 12 oranges equally among 3 children; how many oranges would each child have? Here the cbject is to divide the 12 oranges into 3 equal parts, and to ascertain the number of oranges in each of those parts. The operation is evidently as in the last example, and consists in finding a number, which, multiplied by 3, will produce 12. This number we have already found to be 4. Ans. 4 oranges apiece. As, therefore, multiplication is a short way of performing many additions of the same number; so, division is a short way of performing many subtractions of the same number; and may be defined, The method of finding how many times one number is contained in another and also of dividing a number into any number of equal parts. In all cases, the process of division consists in finding one of the factors of a given product, when the other factor is known.) The number given to be divided is called the dividend, and answers to the product in multiplication. The number given to divide by is called the divisor, and answers to one of the factors in multiplication. The result, or answer sought, is called the quotient, (from the Latin word quoties, how many?) and answers to the other factor. SIGN. The sign for division is a short horizontal line be tween two dots, It shows that the number before it is to be divided by the number after it. Thus 27 ÷ 93 is read, 27 divided by 9 is equal to 3; or, to shorten the expression, 27 by 9 is 3; or, 9 in 27 3 times. In place of the dots, the dividend is often written over the line, and the di visor under it, to express division; thus, 23, read as before. 4464 20 42442815225 525 530 532 5 18=6246 44 6366 = 6 21=7287 | 3 :7427 42 = 7 =8 28328428888 27930949=93=9 The reading used by the pupil in committing the table may be, 2 by 2 is 1, 4 by 2 is 2, &c. ; or, 2 in 2 one time, 2 in 4 two times, &c. DIVISION TABLE-CONTINUED. =1 =1 = 1 | 48=1 | H : 1 405 455 485 = 5 28 73 234 678 40=7627 13=7 H=7 # = how many? 3311, or how many? 108÷ 12, or 108 ¶ 16. 23. How many yards of cloth, at 4 dollars a yard, can be bought for 856 dollars? Here the number to be divided is 856, which therefore is the dividend; 4 is the number to divide by, and therefore the divisor. It is not evident how many times 4 is contained in so large a number as 856. This difficulty will be readily overcome, if we decompose this number, thus: 856800+40 +16. Beginning with the hundreds, we readily perceive that 4 is ⚫ contained in & 2 times; consequently, in 800 it is contained 200 times. Proceeding to the tens, 4 is contained in 4 1 time, and consequently in 40 it is contained 10 times Lastly, in 16 it is contained 4 times. We now have 200+10+4=214 for the quotient, or the number of times 4 is contained in 856. Ans. 214 yards. We may arrive to the same result without decomposing the dividend, except as it is done in the mind, taking it by parts, in the following manner: For the sake of convenience, we write down the dividend with the divisor on the left, and draw a line between them; we also draw a line underneath Then, beginning on the left hand, we seek how often the divisor (4) is contained in 8, (hundreds,) the left hand figure; finding it to be 2 times, we write 2 directly under the 8, which, falling in the place of hundreds, is in reality 200. Proceeding to tens, 4 is contained in 5 (tens) 1 time, which we set down in ten's place, directly under the 5 (tens.) But, after taking 4 times ten out of the 5 tens, there is 1 ten left. This 1 ten we join to the 6 units, making 16. Then, 4 into 16 goes 4 times, which we set down, and the work is done. This manner of performing the operation is called Short Division. The computation, it may be perceived, is carried on partly in the mind, which it is always easy to do when the divisor does not exceed 12.) RULE. From the illustration of this example, we derive this general rule for dividing, when the divisor does not exceed 12: I. Find how many times the divisor is contained in the first figure, or figures, of the dividend, and, setting it directly under the dividend, carry the remainder, if any, to the next figure as so many tens. II. Find how many times the divisor is contained in this dividend, and set it down as before, continuing so to do till all the figures in the dividend are divided. PROOF. We have seen, (¶ 15,) that the divisor and quotient are factors, whose product is the dividend, and we have also seen, that dividing the dividend by one factor is merely a process for finding the other. Hence division and multiplication mutually prove each other. To prove division, we may multiply the divisor by the quotient, and, if the work be right, the product will be the same as the dividend; or we may divide the dividend by the quotient, and if the work is right, the result will be the same as the divisor. To prove multiplication, we may divide the product by ons factor, and, if the work be right, the quotient will be the other factor. EXAMPLES FOR PRACTICE. 94. A man would divide 13,462,725 dollars among 5 men ; how many dollars would each receive? D* OPERATION. Dividend. Divisor, 5) 13,462,725 2,692,545 Quotient, PROOF. Quotient. 2,692,545 b divisor. In this example, as we cannot have 5 in the first figure, (1,) we take two figures, and say, 5 in 13 will go 2 times, and there are 3 over, which, joined to 4, the next figure, makes 34; and 5 in 34 will go 6 times, &c. In proof of this example, we multiply the quotient by the divisor, and, as the product is the same as the dividend, we conclude that the work is right. From a bare inspection of the above example and its proof, it is plain, as before stated, that division is the reverse of multiplication, and that the two rules mutually prove each other. 13,462,725 25. How many yards of cloth can be bought for 4,354,560 dollars, at 2 dollars a yard? 4 dollars? at 5 dollars? -at 9? at 3 dollars? at 6 dollars? at 10? at Note. Let the pupil be required to prove the foregoing, and all following examples. 26. Divide 1005903360 by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. 27. If 2 pints make a quart, how many quarts in 8 pints? in 12 pints? in 20 pints? in 248 pints? in 24 pints? in 47632 pints? 28. Four quarts make a gallon; how many gallons in 8 quarts? in 3764 pints? in 12 quarts? in 20 quarts? in 36 quarts? in 368 quarts? in 4896 quarts? in 5436144 quarts? 23. A man gave 86 apples to 5 boys; how many apples would each boy receive? Dividend. Divisor, 5) 86 Quotient, 17-1 Remainder. Here, dividing the number of the apples (86) by the number of boys, (5,) we find, that each boy's share would be 17 apples; but there is one apple left. 17. 5)86 176 In order to divide all the apples equally among the boys, it is plain, we must divide this one remaining apple into 5 equal |