Methods of proof. I. Multiply the quotient and divisor together, add in the remainder, and the result will be like the dividend when the work is right. II. Cast the nines out of the divisor and quotient, setting the remainders on opposite sides of a cross, as in Multiplication. Multiply these together, and, having cast out the nines from the product, set what is over at the top of the cross. Cast the nines out of the remainder; subtract what is over from the dividend, and cast the nines out of what remains; put the remainder at the bottom of the cross, and it will be like the top figure when the work is right. III. Add the remainder, and all the lower lines of figures together, and if the work is right, the sum will be like the dividend. First, I take as many (and no more) of the left hand figures of the dividend (viz. 550.) as contain the divisor 234; I then try how often 234 will go in 550, product of the divisor and quotient, with the remainder added, equals the divi dend; now the numbers here actually added are the remainder, and the products of the divisor into the several quotient figures, placed in order; and since the sum of the products of the parts into any number equals the product of the whole into that number, it follows that the sum of these products and the remainder will be equal to the dividend, if the operation be correct. Another method of proof is, to subtract the remainder from the dividend, and divide the result by the quotient; the resulting quotient will be like the divisor when the work is right. Thus in ex. 43. if the remainder 78 be taken from the dividend, and the result 550836 be divided by the quotient 2354, the quotient of this division will be 234, which is equal to the given divisor. and find it will go twice; I therefore put 2 in the quotient, (viz. on the right of the dividend,) and multiply the divisor 234 by it, the product 468 I then place under the 550. Next I subtract 468 from 550, and the remainder is 82; to this I bring down the 9, making 829; I try how often the divisor will go in this, and find it will go 3 times; I put 3 in the quotient, and multiply the divisor by it; the product 702 I place under the 829, and, subtracting the lower from the upper, the remainder is 127; I bring down 1 to this, and it becomes 1271; I try how often the divisor will go in this, and find it will go 5 times; I therefore put 5 in the quotient, and multiplying the divisor by it, I put the product 1170 under, and subtract it from 1271; to the remainder 101 I bring down the last figure in the dividend, viz. 4, making 1014; I find that the divisor is contained 4 times in this number; I put the 4 in the quotient, multiply the divisor by it, place the product under the 1014, and subtract as before; there being no more figures in the dividend to bring down, the 78 is the remainder. In the proof by Multiplication, the quotient 2354 is multiplied by the divisor 234, and the remainder 78 added to the product, and the result being like the dividend shews that the work is right. In the proof by Addition, I add the remainder and the lower lines of figures (not the upper) together, as they stand vertically, or under each other; thus 8 and 6 are 14; put down 4, and carry 1 to 7 are 8 and 3 are 11 and 0 are 11; put down 1, and carry 1 to the 9, 7, and 2, which make 19; put down 9, and carry 1 to the 1, 0, and 8, which make 10; put down 0, and carry 1 to the 1, 7, and 6, which make 15; put down 5, and carry 1 to the 4 is 5, which I put down. In the proof by the cross, I cast the nines out of the divisor 234, and the 0 which remains I place on the left of the cross; I then cast the nines out of the quotient 2354, and place the remainder 5 on the right. I next multiply 5 and 0 together, and put the result 0 at the top. I then cast the nines out of the remainder 78, and, subtracting the 6 that is over from the dividend, I cast the nines out of the remainder, and the result is 0 like the top; wherefore the work is right. Quot. 12345. rem. 6. Quot. 123456. 47. Divide 382701 by 31. 48. Divide 814483 by 23. 49. Divide 24753819 by 26. 50. Divide 80132457 by 29. 51. Divide 15185088 by 123. 52. Divide 85691764 by 243. 53. Divide 73671248 by 857. 54. Divide 175729358 by 3542. 55. Twelve men divide 20736 pounds equally among themselves; what is the share of each? Ans. 17281. Quot. 352641. rem. 1. Quot. 49613. rem. 112. 56. A baker has 5138 pounds of dough, of which he intends to make loaves weighing 14 pounds each; how many will he have? Ans. 367. 57. In a certain frigate there are 176 common sailors, and they share 308001. prize-money equally; what sum does each receive? Ans. 1751. 58. How many times will a watch, which goes 31 hours, require winding up in 28458 hours? Ans. 918 times. 59. The crop of wheat on an estate is 1164 quarters; what quantity was sown, supposing every corn to have produced 97 on an average? Ans. 12 quarters, 60. How many times is the number 999 contained in the ten digits, arranged in their natural order? Ans. 1235803 times, and 693 over. b REDUCTION. 42. Reduction teaches how to change numbers from one denomination to another, without altering their value; and is performed by multiplication and division. When numbers are brought from a higher denomination into a lower, the operation is called Reduction descending, and is performed by multiplication. When numbers are brought from a lower denomination into a higher, it is called Reduction ascending, and is performed by division. 43. To bring great names into small, that is, to reduce numbers from a higher denomination to a lower. RULE. Multiply by the number denoting how many of the lower denomination make one of the higher. Thus to bring pounds into shillings, I multiply the pounds by 20, because 20 shillings make one pound: to bring shillings into pence, I multiply the shillings by 12, because 12 pence make one shilling: and to bring pence into farthings, I multiply the pence by 4, because 4 farthings make one penny c. b From the Latin reduco, to restore, or bring back. • The reason of this rule will be readily understood. Suppose it were required to reduce any number of pounds into farthings; the most convenient method would evidently be, by reducing the given number first into shillings, then into pence, and next into farthings. Now 1 pound contains (once twenty, or) twenty times I shilling; 2 pounds contain (twice twenty, or) 20 times 2 shillings; in like manner 3 pounds contain 20 times 3 shillings; and in general any number of pounds will contain 20 times that number of shillings. By similar reasoning it appears, that any number of shillings contains 12 times that number of pence; and any number of pence, 4 times that number of farthings. Wherefore, since there are 20 shillings in 1 pound, and 12 pence in 1 shilling, it follows, that there are 20 times 12, or 240 pence, in 1 pound; and likewise 44. When there are intermediate denominations between the given denomination, and that to which you would reduce it. RULE. Reduce the given number step by step in order, through all the intermediate denominations, (by the foregoing rule,) until you have brought it down to the proposed denomination. Thus to bring pounds into farthings, I first reduce the pounds into shillings, then the shillings into pence, and lastly the pence into farthings d 45. When the given number consists of several denominations. RULE. Begin at the highest denomination, reduce it to the second, and to the result add the second denomination in the given number; reduce this sum to the third denomination, and to the result add the third denomination in the given number; proceed until you have arrived at the denomination required. Thus to bring pounds, shillings, pence, and farthings, into farthings; I begin with the pounds, reduce them into shillings, and add the given shillings to the result; I then reduce this number into pence, and take in the given pence; and lastly I reduce this last number into farthings, and take in the given farthings. 46. To bring small names into great; that is, to reduce numbers from a lower denomination to a higher. RULE. Divide by the number denoting how many of the lower denomination make one of the higher. Thus to bring farthings into pence, I divide the farthings by 4, because 4 farthings make one penny; to bring pence into shillings, I divide the pence by 12, because 12 pence make one shilling; and to bring shillings into pounds, 1 divide the shillings by 20, because 20 shillings make one pounde. (since 4 farthings make 1 penny) there will be 20 times 12 times 4, or 960 farthings, in 1 pound; consequently any number of pounds multiplied by 240 will produce the number of pence, and by 960, the number of farthings, in those pounds. d When there are shillings, pence, or farthings, connected with the given number, it is plain that these must be added in successively, each with its like; viz. shillings with shillings, pence with pence, and farthings with farthings. This being understood, the reason of the rule, as applied to weights and measures, will likewise be evident. e Because there are 4 farthings in 1 penny, 8 farthings in 2 pence, 12 farthings in 3 pence, and in general 4 times the number of farthings in any number of pence; it follows that there will be in any number of farthings one |