DIVISION TABLE-CONTINUED. 13 = 1 =2 =3 :4 :1 ៖ =18=1|4=1| 13 8| 옷을 ¶ 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 8 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 tiil 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 one factor, and, if the work be right, the quotient will be the other factor. EXAMPLES FOR PRACTICE. 24. A man would divide 13,462,725 dollars among 5 men ; many dollars would each receive? how D* OPERATION. Dividend. Divisor, 5) 13,462,725 2,692,545 Quotient, PROOF. Quotient. 2,692,545 5 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? 7? at 5 dollars? at 3 dollars? at 6 dollars? at 10? at 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 248 pints? in 24 pints in 47632 pints? 28. Four quarts make a gallon; how many gallons in 8 quarts? in 12 quarts? quarts? in 368 quarts? in 5436144 quarts? 29. 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. 117. 5)86 17+ In order to divide all the apples equally among the boys, it is plain, we must divide this one remaining apple into 5 equal parts, and give one of these parts to each of the boys. Then each boy's share would be 17 apples, and one fifth part of another apple; which is written thus, 17 apples. Ans. 17 apples each. The 17, expressing whole apples, are called integers, (that is, whole numbers.) The (one fifth) of an apple, expressing part of a broken or divided apple, is called a fraction, (that is, a broken number.) Fractions, as we here see, are written with two numbers, one directly over the other, with a short line between them, showing that the upper number is to be divided by the lower. The upper number, or dividend, is, in fractions, called the numerator, and the lower number, or divisor, is called the denominator. Note. A number like 17, composed of integers (17) and a fraction, (,) is called a mixed number. In the preceding example, the one apple, which was left after carrying the division as far as could be by whole numbers, is called the remainder, and is evidently a part of the dividend yet undivided. In order to complete the division, this remainder, as we before remarked, must be divided into 5 equal parts; but the divisor itself expresses the number of parts. If, now, we examine the fraction, we shall see, that it consists of the remainder (1) for its numerator, and the divisor (5) for its denominator. Therefore, if there be a remainder, set it down at the right hand of the quotient for the numerator of a fraction, under which write the divisor for its denominator. Proof of the last example. 17금 5 86 In proving this example, we find it necessary to multiply our fraction by 5; but this is easily done, if we consider, that the fraction expresses one part of an apple divided into 5 equal parts; hence, 5 times is=1, that is, one whole apple, which we reserve to be added to the units, saying, 5 times 7 are 35, and one we reserved makes 36, &c. 30. Eight men drew a prize of 453 dollars in a lottery; how many dollars did each receive? Dividend. Divisor, 8) 453 Quotient, 56ğ answer 56 dollars to each man. Here, after carrying the division as far as possible by whole numbers, we have a remainder of 5 dollars, which, written as above directed, gives for the and § (five eighths) of another dollar, ¶ 18. Here we may notice, that the eighth part of 5 dollars is the same as 5 times the eighth part of 1 dollar, that is, the eighth part of 5 dollars is g of a dollar. Hence, expresses the quotient of 5 divided by 8. Proof. 56 8 453 is 5 parts, and 8 times 5 is 40, that is, 40 = 5, which, reserved and added to the product of 8 times 6, makes 53, &c. Hence, to multiply a fraction, we may multiply the numerator, and divide the product by the denominator. Or, in proving division, we may multiply the whole number in the quotient only, and to the product add the remainder; and this, till the pupil shall be more particularly taught in fractions, will be more easy in practice. Thus, 56 × 8= 448, and 4485, the remainder, = 453, as before. 31. There are 7 days in a week; how many weeks in 365 days? Ans. 524 weeks. 32. When flour is worth 6 dollars a barrel, how many barrels may be bought for 25 dollars? how many for 50 dollars ? for 487 dollars? for 7631 dollars? T19. 41. Divide 4370 dollars equally among 21 men. When, as in this example, the divisor exceeds 12, it is evident that the computation cannot be readily carried on in the mind, as in the foregoing examples. Wherefore, it is more convenient to write down the computation at length, in the following manner: |