Dividend. Divisor, 8) 453 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 answer 56 dollars and § (five eighths) of another dollar, to each man. Quotient, 56 T 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 of a dollar. Hence, expresses the quotient of 5 divided by 8. Proof. 56 8 453 ģis 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 448 +5, 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 may be bought for 25 dollars? how many for 50 dol barrels for 487 dollars ? = for 7631 dollars? lars? 6404, or 640 160 dollars, Ans. 34. 6786, or 679 how many y? 35. 5040 how many? 36. 1234 how many? 37. 3464-how many? Ans. 113. Ans. 3848 38. 2764 how many? ¶ 19. 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 : OPERATION. Divisor. Dividend. Quotient. 21) 4370 (208. 42 170 168 2 Remainder. We may write the divisor and dividend as in short di vision, but, instead of writing the quotient under the dividend, it will be found more convenient to set it to the right hand. Taking the dividend by parts, we seek how often we can have 21 in 43 (hundreds ;) tinding it to be 2 times, we set down 2 on the right hand of the dividend for the highest figure in the quotient. The 43 being hundreds, it follows, that the 2 must also be hundreds. This, however, we need not regard, for it is to be followed by tens and units, obtained from the tens and units of the dividend, and will therefore, at the end of the operation, be in the place of hundreds, as it should be. It is plain that 2 (hundred) times 21 dollars ought now to be taken out of the dividend; therefore, we multiply the divisor (21) by the quotient figure 2 (hundred) now found, baking 42, (hundred,) which, written under the 43 ia the dividend, we subtract, and to the remainder, 1, (hundred,) bring down the 7, (tens,) making 17 tens. 170. We then seek how often the divisor is contained in 17, (tens;) finding that it will not go, we write a cipher in the quotient, and bring down the next figure, making the whole We then seek how often 21 can be contained in 170, and, finding it to be 8 times, we write 8 in the quotient, and, multiplying the divisor by this number, we set the product, 168, under the 170; then, subtracting, we find the remainder to be 2, which, written as a fraction on the right hand of the quotient, as already explained, gives 208 dollars, for the answer. This manner of performing the operation is called Long Division. It consists in writing down the whole computation. From the above example, we derive the following RULE. J. Place the divisor on the left of the dividend, separate them by a line, and draw another line on the right of the dividend to separate it from the quotient. II. Take as many figures, on the left of the dividend, as contain the divisor once or more; seek how many times they contain it, and place the answer on the right hand of the dividend for the first figure in the quotient. III. Multiply the divisor by this quotient figure, and write the product under that part of the dividend taken. IV. Subtract the product from the figures above, and to the remainder bring down the next figure in the dividend, and divide the number it makes up, as before. So continue to do, till all the figures in the dividend shall have been brought down and divided. Note 1. Having brought down a figure to the remainder, if the number it makes up be less than the divisor, write a cipher in the quotient, and bring down the next figure. Note 2. If the product of the divisor, by any quotient figure, be greater than the part of the dividend taken, it is an evidence that the quotient figure is too large, and must be diminished. If the remainder at any time be greater than the divisor, or equal to it, the quotient figure is too small, and must be increased. EXAMPLES FOR PRACTICE. 1. How many hogsheads of molasses, at 27 dollars a hogshead, may be bought for 6318 dollars? Ans. 234 hogsheads. 2. If a man's income be 1248 dollars a year, how much is that per week, there being 52 weeks in a year? Ans. 24 dollars per week. 3. What will be the quotient of 153598, divided by 29? Ans. 52961. 4. How many times is 63 contained in 30131 ? Ans. 47813 times; that is, 478 times, and of another time. 5. What will be the several quotients of 7652, divided by 16, 23, 34, 86, and 92? 6. If a farm, containing 256 acres, be worth 7168 dollars, what is that per acre ? 7. What will be the quotient of 974932, divided by 365 ? Ans. 2671 8. Divide 3228242 dollars equally among 563 men; how many dollars must each man receive? Ans. 5734 dollars. 9. If 57624 be divided into 216, 586, and 976 equal parts, what will be the magnitude of one of each of these equal parts' Ans. The magnitude of one of the last of these equal parts will be 59,4%. 10. How many times does 1030603615 contain 3215? Ans. 320561 times. 11. The earth, in its annual revolution round the sun, is said to travel 596088000 miles; what is that per hour, there being 8766 hours in a year? 12. 1234567890 = how many? 13. 40783920= how many? 14. 987649031 = 9124 how many? CONTRACTIONS IN DIVISION. I. When the DIVISOR is a COMPOSITE NUMBER. 1 20. 1. Bought 15 yards of cloth for 60 dollars; how much was that per yard? 15 yards are 3 × 5 yards. If there had been but 5 yards, the cost of one yard would be = 12 dollars; but, as there are 3 times 5 yards, the cost of one yard will evidently be but one third part of 12 dollars; that is, 2: 4 dollars. Ans. Hence, when the divisor is a composite number, we may, if we please, divide the dividend by one of the component parts, and the quotient, arising from that division, by the other; the last quotient will be the answer. 2. If a man can travel 24 miles a day, how many days will it take him to travel 264 miles? It will evidently take him as many days as 264 contains 24. 6. Divide 448 by 56. II. To divide by 10, 100, 1000, &c. ¶ 21. 1. A prize of 2478 dollars is owned by 10 men, what is each man's share? Each man's share will be equal to the number of tens contained in the whole sum, and, if one of the figures be cut off at the right hand, all the figures to the left may be considered so many tens; therefore, each man's share will be 247 dollars. It is evident, also, that if 2 figures had been cut off from the right, all the remaining figures would have been so many hundreds; if 3 figures, so many thousands, &c. Hence we derive this general RULE for dividing by 10, 100, 1000, &c. Cut off from the right of the dividend so many figures as there are ciphers in the divisor; the figures to the left of the point will express the quotient, and those to the right, the remainder. 2. In one dollar are 100 cents; how many dollars in 42400 cents? Ans. 424 dollars. 424/00 Here the divisor is 100; we therefore cut off 2 figures on the right hand, and all the figures to the left (424) express the dollars. 3. How many dollars in 34567 cents? Ans. 345 dollars. 4. How many dollars in 4567840 cents? 5. How many dollars in 345600 cents? 6. How many dollars in 42604 cents? in 25000 mills? Ans. 426-o in 845000? 7. 1000 mills make one dollar; how many dollars in 4000 mills? 8. How many dollars in 6487 mills? 9. How many dollars in 42863 mills? Ans. 6,487 dollars. in 368456 how many cents in 40 in 20 mills? in 34640 mills? in 468 III. When there are CIPHERS on the right hand of the divisor. T 22. 1. Divide 480 dollars among 40 men? 4|0)48|0 OPERATION. 12 dolls. Ans. In this example, our divisor, (40,) is a composite number, (10 X 440;) we may, there fore, divide by one component part, (10,) and that quotient by the other, (4;) but to divide by 10, we have seen, is but to eut off the right hand figure, leaving the figures to the left |