called? What is the RULE for multiplying simple numbers, when the multiplier does not exceed 12? What, when the multiplier exceeds 12? What is the method of proof? What, the first step? the second? the third ?the fourth? When ciphers occur between any of the significant figures of the multiplier, how do you proceed? What is a composite number? What are the parts of a composite number called? How do you multiply by a composite number? How do you proceed when the multiplier is 1, with any number of ciphers annexed? How, when there are ciphers on the right hand side of either or both the factors? 5. In one mile are 320 rods; how many rods in 57 miles? 6. It is 436 miles from Boston to the city of Washington; how many rods is that? Ans. 139520. 7. What will 131 yards of Irish linen come to, at 38 cents per yard? Ans. 4978 cents. Ans. $54096. 8. What cost 784 chests of tea at $69 per chest? 9. 10700 men take a prize, whereof each man receives $46; what was the sum of money taken? Ans. $492200. 10. There are 24 hours in a day;-if a ship sail 7 miles in an hour, how far will she sail in one day? 36 days? 365 days? Ans. to the last, 61320 miles. 11. If 46 men can do a piece of work in 60 days, how many will it take to do it in one day? Ans. 2760. 12. If 1851 men receive $758 apiece, how many dollars will they all receive? Ans. $1403058. 13. Two men depart from the same place, and travel in opposite directions, one at the rate of 27 miles a day, the other 31 miles a day; how far apart will they be at the end of 6 days? Ans. 348 miles. 14. A merchant bought 13 pieces of cloth, each piece containing 28 yards, at $6 a yard; how many yards were there, and what the whole cost? Ans. there were 364 yards, and it cost $2184. 15. If 18 men can do a piece of work in 90 days, how long will it take one man to do the same? Ans. 1620 days. 16. If the use of $100, for one year, be $6, what will be the use for 2 years? 7 years? 15 years? 3 years? $7? $15? 74 years? Ans, to the last, $444. 17. If the use of $1, for one year, be 6 cents, what will be the use of $2? $5? $25? (for the same time.) Ans. to the last, 150 cents. 18. An army has 150 wagons loaded with beef; each has 6 barrels, each barrel contains 2 hundred pounds, and each hundred pounds is worth $3; What is the value of the whole? Ans. $5400. 19. What sum of money must be divided among 27 men, that each may receive $115? Ans. $3105. 20. If a carriage wheel turn round 346 times in running 1 mile, how many times will it turn round in the distance from New York to Philadelphia, it being 95 miles? Ans. 32870. 21. Two men, A and B, start from the same place at the same time, and travel the same way; A travels 52 miles a day, and B 44; how far apart will they be at the end of 14 days? Ans. 112 miles: 22. There is a field full well I know, Ans. 58800 bushels. DIVISION OF SIMPLE NUMBERS, COMMONLY CALLED SIMPLE DIVISION. 43. SIMPLE DIVISION is finding how often one simple number is contained in another, and is a short method of performing SEVERAL SUBTRACTIONS. Illus. Master James divided 12 pears among 3 of his companions; how many pears did each one receive? By division 3)12(4 times. That is, 3 will go in 12, 4 times; therefore each boy will receive 4 pears. Ans. 12 9 6 3 3 3 3 3 96 30 operations become tedious. The number of subtractions TABLE. Hence it appears that questions in division can be performed by several subtractions. But when the numbers are large the indicate the quotient. 45 50 55 60? 54! 60 66 72! 5 10 15 20 25 30 35 40 6 12? 18? 24? 30 36 42 48 7 14 21 28? 35! 42? 49? 56? 8 16? 24? 32! 40? 48! 56 64? 9 18 27 36 45 54! 63? 72 10 20 34? 40 50 60 70 80 11 22 33 44? 55? 66? 77! 88! 12 24 36 48! 60! 72! 84? 96? 108? 120? 132? 144? 72! 80? 88? 96? 81? 90? 99? 108? 90? 100? 110? 120? 99? 110? 121? 132? 44. This sign÷signifies DIVISION, and is read divided by. As 155 3, that is to say, 15 divided by 5 is equal to 3. 45. The VINCULUM indicates that the numbers or quan: tities over which it is placed, should FIRST be united. As 15×3+6, indicates that 3 & 6 are to be united, previous to their being multiplied into 15: hence 15X3+6=135. Here, if it had not been for the vinculum, we should have multiplied 15 by 3, and added 6 to the product, and thus we should have obtained only 51. 46. There are FOUR terms to be noticed in division, the diridend, divisor, quotient, and remainder. I. The DIVIDEND, is the number given to be divided. III. The QUOTIENT is the number of times the dividend contains the divisor. IV. The REMAINDER is what is left of the dividend, after containing the divisor as many times as is possible. From this last definition it is evident 1st, That the remainder will always be of the same name as the dividend, because it is "what is left" of it; that is, it is a part of it. 2dly, That the remainder must always be less than the divisor, else it would be "possible" for the dividend to contain the divisor another time. Illus. If 13 men have equal shares in a cotton manufactory, and a dividend of 3579 dollars is to be made out between them; how much will each man receive? Here it is evident that if we divide 3579 dollars into 13 equal parts, one of these parts will be the sum which each man is to receive. Therefore, Divis. Dividend. Quot./ 13)3579* (275 26 97 69 65 Remainder 4 dols. 1st, We SEEK how often 13 will go into 35 (this being a sufficient number of the left hand figures of the dividend) which we find to be 2 times; therefore we put 2 in the quotient. 2dly, We MULTIPLY the divisor by this quotient figure, and set the product under that part of the dividend used, viz. 35 (00). 3dly, We SUBTRACT the product therefrom, (26 from 35.) 4thly, We BRING DOWN the next figure of the dividend (7) to the right hand of the remainder (9). Then SEEK, MULTIPLY, SUBTRACT, and BRING DOWN, as before. Hence 47. To divide SIMPLE NUMBERS, we have this RULE. I. SEEK how often the divisor will go in as many of the left hand figures of the dividend as are just necessary, and set the result in the quotient. II. MULTIPLY the divisor by this quotient figure, and set the product under that part of the dividend used. III. SUBTRACT it therefrom. IV. BRING DOWN the next figure of the dividend to the right hand of the remainder, and proceed as before. NOTE. If, after having brought down a figure to the remainder, it should still be less than the divisor, place a cipher in the quotient, and bring down another figure. [The foregoing method of performing the operation is called LONG DIVISION.} 4921 PROOF. Charles gave 3 boys 15 lemons, how many did he give each one? 15 3 how many? Ans. 5. Ans. 15. Charles gave 3 boys 5 lemons each, how many did he give all three? 3 x 5 how many? A man sold 6 swine for 18 dollars, how much was that apiece? 186 how many? Ans. 3 (dollars.) A man sold 6 swine for 3 dollars each, how much was that in all ? 6 x 3 how many? Hence it appears that Ans. 18 (dollars.) 49. Division is the REVERSE of multiplication, and may be proved by it. And indeed, the method we have given, although generally styled the proof by addition, is, in reality, proof by multiplication, as plainly appears from our illustration of it. EXAMPLES IN DIVISION. 1. Nineteen men drew a prize of 17576 dollars; how much was that apiece? Ans. 925 dollars, and 1 dollar left. 2. Divide 2408881 by 413. Quot. 5832, Rem. 265. 3. I would plant 2070 fruit trees in 14 rows; how many trees must place in each row? Ans. 155. dollars are to be raised Ans. 1275 dollars. 4. Ninety thousand five hundred and twenty-five by seventy-one men; how much is that per man? 5. A paymaster received 153033 dollars to be paid out among 87 men; how much must each man receive? Ans. 1759 dollars. 6. What number multiplied by 57 will produce the same as 143 multiplied by 71, plus 50? Ans. 179. * The several methods of proof may be found in Pike's large Arithmetic, or in Hutton's Mathematics. The one here given is the most practical and may be thus illustrated: Divisor. Dividend Product, &c. 367) 168696 (459 Divisor X 4 (00) = 1468.. .2189. Here it plainly appears, that the Dividend in division answers to the Product in multiplication, and the Divisor to one of the factors; and that the process of findg the Quotient is merely the process of finding the other factor. Here we find that, after we have brought down 7, the divisor 23 will not go in therefore we place 0 in the quotient and bring down another figure. See Note: 7. How many times will 517 go in 30380471 ? Ans. 58763. 8. I would fence in a piece of ground for the purpose of setting out 3045 hoice fruit trees, so as to have 35 rows, and the trees 33 feet apart; how Jarge a piece must I enclose? Ans. 2871 feet long, by 1155 feet wide. Multiplying the divi- | 1 Divis. 1 Divid. X 2. 1 Quot. X 2. 4 ) 32 ( 8 Divis. Divid. Quot. 4 ) 16 ( 4 dend by 2 we have 1 Divis. 1 Divid.÷2. Quot. 2 Dividing the dividend by 2, we have 4) 8 (2 If the DIVISOR remain the SAME, Therefore, 50. Multiplying the DIVIDEND by any number is multiply. ing the QUOTIENT by that number, and dividing the DIVIDEND dividing the QUOTIENT. For, the larger the dividend the OFTENER the same divisor will be contained in it and the smaller the FEWER TIMES. Divis. Divid. Quot. 4) 16. Multiplying the divi sor by 2 we have 1 Divis. X 2. 1 Divid. 1 Quot. 2. 8 16 ( 2 1 Divisor÷2. 1 Divid. 1 Quot. X 2 Dividing the divisor by 2, we have 2 ) 16 ( 8 If the DIVIDEND remain the SAME, Therefore, 51. Multiplying the DIVISOR by any number, is dividing the QUOTIENT by that number, and dividing the DIVISOR is multiplying the QUOTIENT. For, the larger the divisor, the FEWER TIMES it will be contained in the same di vidend and the smaller the OFTENER. Divid. & Divis. (4 1 Divis. X 2. 1 Divid. X 2.1 Quot. Multiplying BOTH 16 and 4 by 2, we have 8 ) 32 2. 1 Quot. 1 Divis. 2. 1 Divid. Divid. & Divis. Dividing BOTH 16 and 4 by 2, we have 2 ) 8 (4 Therefore, 52. If the dividend and divisor be вOTH multiplied or BOTH divided by the same number, the quotient will NOT BE ALTERED, This is an evident consequence of the combination of the two preceding articles; in which the effect of the multiplication and division by the same number evident ly destroy each other. See Art. 30 c. CONTRACTIONS IN DIVISION. 53. When the divisor does not exceed 12, a shorter way is generally made use of, called SHORT DIVISION. It consists in performing the ope rations in the mind, setting down the quotient figures directly under the respective figures of the dividend, and carrying the remainder to the right. EXAMPLE. Divide the number 357 by 2. 2)357 First, we consider how often the divisor 2, will go in 3; Rem. which we find to be once, and that 1 remains. We set the Quot. 178-1 quotient figure 1 underneath, and conceive* the remainder 1, * Conceiving the remainder to be prefixed to the next figure of the dividend is only giving to it its proper value, or supposing it placed where it actually belongs; for, it is very evident, from our notation, that every unit of any place or order is 2)200(100 2)140 70 2) 17( 8 R equal to TEN units of the next right hand place, and the only way to express the remainder in connection with the next figure of the dividend, is, to prefix it to that next figure. This may be more easily seen from the annexed analysis: by which, we also 2)357(178 & 1 see, that the first quotient figure [1] is not barely 1, but 100; and that the second quotient figure [7] is not barely 7, but 70; as 'PIAIP 'onb may likewise be seen from what we have before observed. See Note to Illus. Art. 46, bottom of the page. |