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with this new sum as before, placing the quotient as the next figure of the answer; and so on till all the figures of the dividend have been brought down and divided, when the operation is completed. If there is a remainder after the division is finished, it is marked down as part of the answer, with the divisor written below it, forming a fraction.
It will be found useful, on bringing down in succession the figures to the remainders, to place dots below them in the dividend, to prevent any mistakes as to what have been brought down.
When the divisor is large, and it cannot be seen at a glance what the quotient figure in any case should be, it will usually be a guide to the required figure (except when it is l) to take as a trial figure the number of times that the first figure of the divisor is contained in the first figure, or first two figures of the dividend. This will give an approximation, and will help to shew what the true figure should be.
Example.—Divide 494033 by 239. 239)494033(2067,20 In this example, finding that there are 478
two times 239 in 494, we place 2 in the 1603
quotient; then multiply 239 by 2, making 1434
478, which we subtract from 494. To the remainder, 16, we bring down 0, the next
figure in the dividend, making 160; and, 1673
finding that 239 is not contained in 160, 20 = 20
we place a 0 in the quotient, and bring down 3, the next figure in the dividend,
making 1603. In this sum, 239 being contained 6 times, we place 6 in the quotient, and multiply 239 by 6, making 1434, which we subtract from 1603. To the remainder, 169 we bring down 3, the last figure in the dividend, making 1693, in which 239 is contained 7 times; we place 7 in the quotient, and multiply 239 by 7, making 1673, which, being deducted from 1693, leaves a remainder of 20, written, with the divisor below it, as 0.
THE REASON for the rule of Division will appear from the following example:Divide 94608 by 73.
1. 73 )94608(1000 Here it will be seen, 73)94608 (1296 73000
on comparing No. 1
with No. 2, that the 21608( 200
process is virtually the
first to ascertain how 6570
many thousand times 657 438(
73 is contained in the 6
dividend; how many 438
hundred of times in 438 Total, 1296 times. the remainder ; how
many times ten in the
next remainder; and how many times one in the last remainder; and adding all these, we
have the total number of times. In this example, we have 73 contained in the dividend and the successive remainders, 1000, 200, 90, and 6 times-in all, 1296 times. It will be seen, that the nothings annexed to the quotients, &c., in No. 1, may, in practice, be left out, as the other figures have the same value without them, by being placed as in No. 2, according to their rank.
PROOF.—To prove that any question in division is correctly performed, multiply the quotient by the divisor, and add to the product any remainder. The answer will be the same as the dividend, if the working has been correct.
Example.- Divide 4966 by 37. 37)4966 (134
Here, on dividing 4966 by 37, 1348
ing the quotient by the divisor, 938
and adding the remainder, the 402 148
product is the same as the di 8 Remainder. vidend.
Exercises. Divide the following numbers : 1. 79512587 by 13 23 31 1 9. 12345678 by
79 2. 89659053 34 41 73 10. 90273139,
432 3. 19271873, 51 43 83 11. 87625432 199 843 4. 85296307 1 61 71 85 12. 17927618, 925 379 5. 41824680 1 62 52 74 13. 419352716 , 123 456 6. 36925814 1 82
14. 900416824 1 54321 98765 7. 70369257 , 91
15. 519387549 , 17297 2731 8. 84169273, 29 37 38 16. 183926157 , 37246 8799
16. 4938394291 209039900
NOTE.—WHEN THE DIVISOR is the product of two numbers, neither of which exceeds 12, short division may be employed, by dividing the dividend by one of the numbers, and the quotient of this by the other.
If there be any remainders, multiply the last remainder by the first divisor, and add the first remainder to the product; then write the whole original divisor below this, and the fraction thus formed is annexed to the quotient.
Example.—Divide 37255 by 63. 9)37255
Here the divisor, 63, may be resolved into the
two factors, 9 and 7; therefore divide first by 7)41394 9, and then the quotient, 4139, by 7: the quo59122 Ouot. tient, 591, is the answer required. Again, since
the second remainder is 2, multiply it by 9, the
first divisor, and to the product, 18, add 4, the first remainder; and the sum, 22, is the true remainder.
Exercises. Divide the following numbers :1. 38297546 by 14 15 16 | 3. 90127539 by 22 24 25 2. 12345678 i 18 20 21 4. 16254973 į 32 36 42
III. WHEN THE DIVISOR IS A NUMBER HAVING nothings ANNEXED
TO IT. RULE.-Point off the nothings from the divisor, and also point off an equal number of figures from the right of the dividend; then divide by the remaining figures of the divisor. If these do not exceed 12, the question is wrought by short division. If above 12, long division is employed. To any remainders after the division is completed, annex the figures pointed off from the dividend, and mark the whole in the quotient, with the divisor written below it, to form a fraction.
Example.-Divide 73450 by 700. 7.00)734.50
Here the account being stated for short divi
sion, we point off the two nothings from the 104458 Ans. divisor, and 2 figures from the dividend. We
then divide by 7; the answer is 104, and 6
over. Annex to this remainder the two figures which were pointed off from the dividend, 50, and the whole remainder is 650, below which the divisor is written, forming the fraction 468
Exercises. 1. Divide 83473 by 900, . . . Answer 92678 2. I 1 1 260,
321, 3. 8506126, 7400,
al 1149 3526 4. 1 9013735 8000, . . .
11269736 5. 1273068 , 9600, .
132 5868 6. 3568405 1 8100, . .
440 4485 NOTE.—WHEN THE DIVISOR IS 10, 100, 1000, or 1 with any other number of nothings annexed, the division is accomplished merely by pointing off as many figures from the right of the dividend as there are nothings in the divisor: the remaining figures are the quotient, and the figures pointed off, with the divisor written below them, form a fraction to be annexed to the quotient; for example, To divide 1748 by 10, point off 1 figure, and the quotient is 174
1 2376, 100, 1 2 figures, 1 1 23 106
IV. WHEN THE Divisor CONTAINS A FRACTION-AS 73.
A number with a fraction annexed, such as 7, is called a mixed number. Before we can divide by such a number, it must be converted into a simple one; that is, into a nuinber having one denomination.
RULE.--Multiply the whole number, or integer of the divisor, by the lower figure of the fraction, and annex the upper figure to the product : multiply also the dividend by the lower figure of the fraction; then divide the one sum by the other.
Example.-Divide 5736 by 67. 6%) 5736
Here the divisor, 6, is multiplied by 7, the 77
lower figure of the fraction, and 3, the upper 45 )40152 (892
figure, is added to the product, making 45.; that
is, 62 are the same as 45 sevenths. 360
In order, however, to preserve the proportion 415
between the numbers, the sum to be divided 405
must also be multiplied by 7, reducing it in like, 102
manner to sevenths. The two numbers being
now both reduced to sevenths, the division takes 90
place: the answer is 892, and the remainder is 12 = 1? 12, which is expressed as 1.
COMPOUND NUMBERS are those which consist of two or more kinds: when we speak of a pound, a simple number is expressed, it is one kind of thing; but when we speak of a sum consisting of pounds, shillings and pence, then we refer to various denominations or kinds; in other words, a sum compounded, of various kinds of money, and hence termed a compound number.
All questions which refer to sums of money, and to calculations of weights, measures, &c., which consist of various kinds or denominations, are placed under the head of Compound Numbers.
Calculations in simple numbers are the same in every country, and must continue without change throughout all time; for example, that 2 and 2 are equal to 4, is a universal truth, which all mankind cannot alter. The rules of Simple Addition, Subtraction, Multiplication, and Division, are therefore in use in every country in the world. With calculations in compound numbers, the case is entirely different, as almost every country has its own standards of money, weights, and measures, and the arithmetical rules for working differ accordingly. These calculations, however, could be rendered as easy as those in simple numbers, if the standards of money, weights, and measures, were constructed on the same decimal principle of advancing by tens.
In Great Britain and Ireland, the standards of money, weights, and measures are mixed and various, as may be learned from the following tables.
It is necessary that the pupil commit these tables to memory, before proceeding with the calculations in compound numbers.
I farthing. 2 Farthings
= 1 half-penny. . . 4 Farthings.
. . .
. = 1 penny. 12 Pence. .
. = 1 shilling. 20 Shillings, or 240 pence, or 960 farthings = 1 pound.
A groat is 4 pence: a half-crown, 2 shillings and sixpence: a crown, 5 shillings: a half-sovereign, 10 shillings: a sovereign or pound, 20 shillings. The guinea, a coin now disused, but still often spoken of, consisted of 21 shillings.
All accounts are kept and reckoned by pounds, shillings, pence, and farthings. L. or £ is the initial letter of the Latin word libra, a pound, and is used to denote pounds; s., from the Latin word solidus, stands for shillings; and d., from denarius, for pence: £s. d. are therefore respectively placed over columns of pounds, shillings, and pence. The mark for a half-penny is ; for a farthing, *; and for three farthings, a.
The old Scottish pound was equal to Is. 8d. sterling : hence £100 Scots was == £8, 6s. 8d, of our present money.
NOTE.-For an account of a new system of reckoning money decimally, proposed to be substituted for the present mode of reckoning; see Appendix.