the sum may always find the remainder arising from such division by adding together the figures of the number and repeating the process, if necessary, upon till we get a number of one figure. Thus in the number 37698897, the sum of the figures is 57, the figures 5 and 7 added together give 12, and these added together 3, which is the remainder after dividing 37698897 by 9, In adding the figures of a number we may omit any nines it may contain without altering the remainder; for in the above number, omitting the nines, the sum of the other figures is 39, here again omitting the 9, the remainder is three as before. SIGNS. We have explained that multiplication is an easy method of performing several additions, and since these operations are somewhat similar, so also are the signs which denote them. We have already seen that the sign of addition is an upright cross +; the sign of multiplication is also a cross, but oblique instead of upright, thus x. The expression 8 * 3 = 24; is read 8 multiplied by 3 equals 24. Examples.-Multiply 6786794 by 38006 and test the accuracy of the result. Place the multiplier under the multiplicand and proceed as directed in the rule. 6786794 The figures in the 38006 ... 8 multiplicand except 9 when added together 40720764 7 give 38; now say 3 and 54294352 20360382 8 are 11, and again i and I are 2; place 2 257938892764 7 to the right of the mul 2 (2) (6) (10) (3) (7) by 13. (8) 67 by 19. G. (1) (5) (9) (0) tiplicand. Next the sum of the figures in the multiplier is 17; and since 1 +7=8, put 8 under the 2, draw a line, and multiply these two remainders ; the result is 16; now I + 6 = 7, therefore we put 7 under the line. Again the sum of the figures in the product (omitting the nines) is 52 ; now 5 + 2 = 7, put this in a line with the product; and, since it is the same as the remainder previously obtained, we conclude that the sum is right. Multiply- 58 by 17. 86 by 18. 793 by 58. 367 by 31. 586 by 53. (10) 698 by 76. 438 by 23. 359 by 34. (8) 847 by 65. 11.15439 by 231. (8) 4267 by 314. (6) 8679 by 536. (9) 3876 by 132. 7958 by 432. 7483 by 708. (3) 8607 by 312. 6957 by 253. (6 8465 by 243 8076 by 403. 1. (1) 67835 by 4301.1) 36879 by 5008. (2) 28673 by 3042. 56387 by 6234. 15749 by 3602. 37498 by 5608. (6) 28674 by 4326. 27493 by 6798. 47538 by 4205. 85674 by 3897. f. (1) 46378 by 4508. 39700 by 4700. 63598 by 4630. 86570 by 4080. (3) 86750 by 3750. (8) 63970 by 8070. 27940 by 3800. 47936 by 8600. 57400 by 6890. (3) (1) 687 by 42. (6) (10) (8) (3) (8) (3) (6) (2) K. (1) 674385 by 28040. (6) 374659 by 79082. (2) 369750 by 48060. 476538 by 34705. 397460 by 35060. (8) 548739 by 64072. (1) 875460 by 34500. (m) 867000 by 30840. (5) 260504 by 86045. (10) 756500 by 84060. Find the product ofL. (1) 6483758 by 47306. 8237150 by 86004. 3867952 by 38067. 9367400 by 36090. 5867930 by 35804. 1379450 by 72060. 7389476 by 57008. 4723950 by 84006. (5) 3976850 by 40608. 2357946 by 64075. M."4873925 by 37896. 7389560 by 47806. (2) 5846793 by 48795. 3758975 by 46080. (8) 6738794 by 89574 58793&o by 570800. (96583976 by 9999. 6395748 by 50080. (5) 8679586 by 58069. ! (10) 6738500 by 99900. (3) (4) (9) (10) (6) (7) (8) (9) CHAPTER V. DIVISION. -SECTION I. If we were required to find how many pennies there are in a quantity of pence, we should probably take away one penny at a time and say as we did so, one, two, three, &c., till all of them were taken, when the number counted, or which is the same thing, the number of times one penny was taken, would be the same as the number of pence. In the same way the number of twopences in the sum may be found by taking away two pennies at a time and counting as before; the number of times twopence is taken will be the same as the number of twopences in the sum. So with abstract numbers; we may find, for D example, how many times 4 is contained in 12 hy taking away or subtracting 4 as often as possible; thus taking 4 from 12 the remainder is 8; taking 4 from 8 the remainder is 4, and taking 4 from 4 there is no remainder; hence 4 is contained in 12 three times. But if we were required to find how many times 4 is contained in 17584, this process would be very tedious, for we should have to perform some thousands of subtractions. In fact it would be just as absurd to employ this method as to empty a bag of wheat by taking away each grain singly instead of in handfuls at a time. Instead therefore of taking away one 4 at a time from 17584, let us take 4000 fours or 16000, this will leave 1584; now take away 300 fours or 1200, the remainder is 384 ; next take away 90 fours, the remainder is 24, lastly take away 6 fours and there is no remainder. The operation may be exhibited thus : 17584 itiro 16000 ... 4000 Total SH Str Bd 1584 ST 55 Na Fru 1200 300 ato 384 fours. bib sy 360 unter de stad 24 24 6 in nood tada ste Hence the number of times 4 is contained in 17584 is 4 thousands 3 hundreds 9 tens 6 units or 4396 ; so that taking away several fours at a time we have obtained by four subtractions what, by the former method, would have required no fewer than 4396 subtractions. The process just explained for finding how often 4 is contained in 17584 is called division; the pupil, therefore, will understand that division is a brief method of finding how often one number is contained in another number. The number to be divided (in this case 17584) is called the dividend. The number by which we divide (in this case 4) is called the divisor. The number of times the divisor is contained in the dividend (in this case 4396) is called the quotient. If a number be left after taking the divisor as many times as possible from the dividend, it is called the remainder. SIGNS. The operation of division is expressed by a colon, thus -, the dots being separated or divided by a straight line. The expression 21 = 7 = 3, is read 21 divided by 7 equals 3. In practice, when the divisor does not exceed 12, the division is effected as follows :- Taking the same numbers as before, we place the 4 before the dividend with a curve between them, and draw a line under the dividend thus- 4) 17584 4396 Take the first figure, or, if necessary, as is this case, the number expressed by the first two figures of the |