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When the divisor is 10, 100, 1000, &c. division is instantly performed, by cutting off from the right-hand of the dividend as many digits as there are ciphers in the divisors: the figures cut off, are the remainder, and the other figures the

quotient.
Cramples.
Let 478415 be divided by 10, and 675832 by 100.
47841, s 6758, #,

106. 38417874 –- 10. 107. 471861494.37 - 1000 108. 541181496 ... 100 109. 108149387425... 100 I10. 31.61800859 ... 1000 111. 24100864.184 ... 1000 112. 4810087108 ... 10000 113. 51493.175900 ... 10000 114, 57.16478102 ... 10000. 115. 684753869000... 100000

If the divisor is any digit with ciphers annexed, by employing the foregoing method, we may readily arrive at the quotient, by considering the digit of the divisor, changed for an unit and dividing the quotient of this division, by the digit of the divisor.

Čramples, 200) 474374,20 4000) 5785,840 237187 or 1446+$43 116. 31819.463, -i- 20 117. 473510634 -:-300 118. 4321094 ... 400 119. 31474.1750 .. 50

120. 24.18688.190 .. 500 121. 2341967.420 . , 60 122. 543109670 .. 700 123. 1494758360 .. 70 124. 751009476 .. 800 125. 54.19831096 . . . 80 126. 4104187470 ... 900 127. 63181953.8420. 90 : 128. 384.1931024 .. 40 129. 74181007040 . . .300. 130. 5321465700 .. 5000 131. 93418007607 ... 3000 132.60843200 .. 7000 133. 8324468.4900 . . 4000 134. 179425.960 - .. 6000 135. 2410083900 .. 5000

136. 834.100760 - .. 7000 13. 4810984.70 .. 8000 138, 4271860475 ... 9000 139, 846075084 ... 90000 - -

When any divisor has a cipher or ciphers at the right-hand thereof, cut off from the dividend as many digits as there are -eiphers in the divisor, and proceed with the figures of the divisor as before directed, neglecting the ciphers until the division is performed; when, if there is a remainder, the figures cut off are to be annexed thereto, and placed over the entire divisor in the form of a fraction. When there is not a remainder to division, the figures cut off are only to be so placed.

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73400 ) 6410958042 (87.342###3 5872 73400 • 5389 34937042 Second proof. 5138 262031 From 6410958042 25T5 611309 Take 55242 2202 58042 3T38 6410058042 6410902800 2936 87342) 6410902800 (73400 7020 611394 1468 - .296962 Remainder 55242 . . . 202026 - 349368 349368 - - - - - - 00 140. 8400 -- 280 141. , 54.964 -:- 2600

142. 759384 .. 3700 143. 1418072 . . 4300 144. 638490 . . 5300 145. 29.19600 . . 6100 146. 3207400 .. 7900 147. 83186404 ... 8700 148. 34078580 . . 9300 149. 32740082 . . 2340 150. 48087100 ... 376000 151. 617004700 .. 65300 152. 213958.404 .. 53900 153. 135043950 .. 79800 154. 418190073 .. 47800 155. 1431018070 ... 97.300 156, 32.7194800 .. 87600 157. 32714107408 ... 36500 158. 43580904. .. 586400 159. 32.10873984 .. 753800

When the divisor is large, it may sometimes save labour to make a tablet of the several products of the divisor, by which means division may be performed without the aid of multiplication, by making this tablet, by continual additions, as before noticed in multiplication,

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160. 31094.1085484+471867, 161. 47310973.1265+590739 162. 178496100864+6593.184 163.2196.14819645+438.1075 164. 921073407384+6310439 165. 813109865418+8968578. 166.930471076527+9481067.

Division is sometimes performed in which the figures of the products as they are found are subtracted, and the remainders only set down; and this is called Italian Division.

(framplcá. - Let 419508673 be divided by 4985

4985) 419508673 (84.1544, or
. .20708 Proof.
7686 From 419508673
27017. . Take 983
20923 84154)419507690 (4985

... 983 828916 .

715309

420770 167. 458719425 -- 487 168. 5241078671994 + 373 169. 796541963 ... 397 170. 4871068104364 .. 645 171. 491.4865104 .. 535 172. 8161947845271 .. 847

173. 61418614765 ... 3425

174. 531095863978 .. 7854 When the divisor is the product of two numbers, neither of which exceeds 12.

First divide by one of those numbers, and the quotient divide by the other.”

When there is not a remainder in the second division, that of the first, if any, is the true remainder: when there is not a remainder in the first division, that of the second multiplied by the first divisor, will give the true remainder; and if there is a remainder in each division, the second remainder multiplied by the first divisor, and added to the first remainder, will give the same result as if the operation had been performed by long division.

- - tramples, ot f Let 476859 be divided by 56, and 8510875 by 72, 7)476859 - 9) 8510875 8) 68.122-5 8) 945652-7 #} - I18206-4 56) 476859 (85.15} 72) 8510875 (118206# . 288 131 - - .. 85 - . 590 289 - 148 TOT . . 475 43 175. 42194695 -- 24 176. 1758196 -- 28 177. 93.100964, .. 32 178. 4191694, .. 36 179. 74196742 . . 42 180. 7949468 . . 48 181. 510908675 . . 54 182. 8694840 .. 56 183. 71948634 ... 63 184. 4210964 .. 64 185. 5849249 . . 72 186. 3642867 .. 81 187. 2084648 . . 84 188. 4258948 . . 96

189. 47104842 . , 121 190, 10486.94 ... 108

* When the divisor consists of 3 or more parts, it is in general, easier and readier to employ long division.

+ In the above example the number 476859, is to be divided by 56, which is the product of 8 x 7 or of 7x8; first dividing by 7, the number contains 68122 sevens, and 5 over ; these being divided by 8, gives a quotient of 8515, and has two left, which are two sevens; therefore 2×7+5=19, the true remainder according to the rule, which is the same as is produced by the second operation of division.

If the divisor be an integer with a fraction annexed thereto, commonly called a mixed number.

Multiply the whole number by the denominator of the fraction, and to the product add the nnmerator for a new divisor, multiply also the given dividend by the same denominator, which will produce a new dividend; divide the new dividend by the new divisor, by any of the preceding methods which ever answers the purpose best, and the quotient is the answer.”

(Bramples.

Let 439145 be divided by 34, and 1731415 by 3}

4 4 3 : 3

15) 1756580 15 10) 5194245 10

117105.or 519424, s

191. 21319481 + 2 192. 314275242 + 2} 193. 37510964 . . 194. 619428024 ... 3; 195. 14219482 .. 4 196. 81647.1025 .. 4; 197. 34168045 .. 5; 198. 47529087 .. 6+

199, 58437642 .. 74 200. 73480718 .. 84

By the substitution of another number in place of the given divisor, the operation of division may be often advantageously shortened, by first performing the division with the substituted number, as if it were the true one, and afterwards correcting the quotient by adding to, or subtracting therefrom, aecording as the substituted divisor exceeds, or falls short of the true one; thus, if our divisor was 999, and dividend 78834, by the substitution of 1000, we may at once perceive that the quotient is 78 with a remainder of 912, for by this substitution we have subtracted too many, as the dividend contained 1000, 78 times with a remainder of 834; and as 1000 exceeds 999 by 1, therefore by adding 78 to the first remainder, we obtain the true one : again, if our dividend was 364964, and divisor 1001, by a similar substitution of 1000, we at once perceive the quotient to be 364, with a remainder of 600, for here we have subtracted 364 too few, which being subtracted from the first remainder 964, leaves 600 for the true remainder required. o

* Some may †. that the above more . belongs to fractions, this I grant, #. there are to any who may not have learned, or whomay never learn fractions, to whom it might be useful.

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