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CASE III. To divide by a unit with ciphers annexed. Cut off as many figures from the right hand of the dividend as there are ciphers in the divisor, and the figures on the left hand of the separatrix will be the quotient, and those on the right hand the remainder.

Quotients. 1. Divide 123456789 by 10 12345678 9 2. Divide 987654321 by 100 9876543 21 3. Divide 122112347800 by 1000 122112347 800 4. Divide 89765432156 by 1000000 89765 432156

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CASE IV. To divide by a composite number, that is, a number produced by the multiplication of two or more numbers.

Divide the dividend by any one of the factors, and the quotient thus found by another, and thus proceed till every factor has been made a divisor, and the last quotient will be the true quotient required.

Note. – To find the true remainder, we multiply the last remainder by the last divisor but one, and to the product add the next preceding remainder; we multiply this sum by the next preceding divisor. and to the product add the next preceding remainder; and so on, till we have gone through all the divisors and remainders to the first.

This rule will be better understood by the pupil, after he has become acquainted with fractions.

EXAMPLES 1. Divide 47932 by 72.

As 72 is equal to 9 times 8, we first 9)47932

divide the dividend by 9, and the quotient

thence arising by 8; and to find the true 8)5325 7

remainder, we multiply the last remainder, 5, by the first divisor, 9, and to the product add the first remainder, 7; and find the amount to be 52, the true remainder. 2. Divide 5371 by 192.

We find 192 equal to the product of 4 4)5371

times 6 times 8,=4X 6 X8= 192. We 6)1342 — 3

therefore divide by these factors, as in the

last example. To find the true remainder, 8)223-4

we multiply the last remainder, 7, by the 27-7= 187 last divisor but one, 6; and to the product

add the last remainder but one, 4; this sum we multiply by the first divisor, 4; and to the product add the first remainder, 3; and find the amount to be 187.

Quotients. Rem. 3. Divide 7691 by 24 = 4 x 6

320 11 4. Divide 8317 by 27 = 3 x 9

308 5. Divide 3116 by 6. Divide 61387 by 121 = 11 x 11 507 40 7. Divide 19917 by 144 8. Divide 91746 by 336 = 6 X 7 X 8 273 18 9. Divide 376785 by 315 = X 7 X 9 1196

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SECTION VI.
CONTRACTIONS IN MULTIPLICATION.

I. To multiply by 25. Rule. — Annex two ciphers to the multiplicand, and divide it by 4, and the quotient is the product required.

Rationale. — By annexing two ciphers, we increase the multiplicand one hundred times, and by dividing this number by 4, the result will be an increase of the multiplicand only twenty. five times, because 25 is one fourth of 100. 1. Multiply 785643 by 25.

OPERATION. 4)78564300

19641075 Product. 2. Multiply 9876543 by 25.

Ans. 3. Multiply 47110721 by 25.

Ans. II. To multiply by 33. RULE. — Annex two ciphers to the multiplicand, and divide it by 3, and the quotient is the product required.

Rationale. — As in the last case, by annexing two ciphers, we increase the multiplicand one hundred times; and by dividing the number by 3, we only increase the multiplicand thirty-three and one third times, because 33} is one third of 100. 4. Multiply 87138942 by 33.

OPERATION.
3)8713894200

2904631400 Product.
5. Multiply 66666993 by 33}.
6. Multiply 12336723 by 33.

Ans.
Ans.

III. To multiply by 125. RÚLE. — Annex three ciphers to the multiplicand, and divide by 8, and the quotient is the product.

Note. — By annexing three ciphers, the number is increased one thousand times; and, by dividing by 8, the quotient will be only one eighth of 1000, that is, 125 times. 7. Multiply 12345678 by 125.

OPERATION.
8)12345678000

1543209750 Product.

IV. To multiply by any number of 9's. RULE. — Annex as many ciphers to the multiplicand as there are 9's in the multiplier, and from this number subtract the number to be multiplied, and the remainder is the product required.

8. Multiply 87654 by 999. OPERATION.

By annexing three ciphers, we make the 87654000

number one thousand times larger. If from 87654

this number, with the ciphers annexed, we

subtract the multiplicand, we make the prod. 87566346 Product. uct one thousandth part less; that is, the product will be only 999 times the multiplicand. Q. E. D.

9. Multiply 7777777 by 9999. Ans. 77769992223. 10. Multiply 5555 by 999999. Ans. 5554994445. Note. - To multiply by any number of 3's, proceed as above and divide the product by 3; but if it be required to multiply by 6's, proceed as above and then multiply the product by 2, and divide the result by 3, and the quotient is the product.

11. Multiply 987654 by 333333.

OPERATION. 987654000000

987654 3)987653012346

329217670782 Product, Ans. 12. Multiply 32567895 by 3333.

Ans. 13. Multiply 876543 by 66666.

Ans.
OPERATION.
87654300000

876543
87653423457

3)175306846914

58435615638 Product, Ans. 14. Multiply 345678 by 6666666.

Ans.

V. When the multiplier can be separated into periods, which are multiples of one another, the operation may be contracted in the following manner. 15. Multiply 112345678 by 288144486.

OPERATION. 112345678 288144486

674074068 = the product by 6. 5392592544 = the foregoing product X by 8 for 48. 16177777632 = the last product x by 3 for 144. 32355555264

= the last product X by 2 for 288. 32371787641631508 Product.

Section VII.

CONTRACTIONS IN DIVISION.

I. To divide by 5. Rule. — Multiply the dividend by 2, and the product, except the last figure at the right, is the quotient. Note. — The remainder will be tenths.

1. Divide 67895 by 5.

Ans. 13579.

OPERATION.

67895

13579,0 Quotient.

II. To divide by 25. RULE. – Multiply the dividend by 4, and the product, except the last two figures at the right, is the quotient. NOTE. — The two figures at the right are hundredths. 2. Divide 8765887 by 25.

Ans. 35063546 OPERATION. 8765887

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350635,48 Quotient.

III. To divide by 33}. RULE. — Multiply the dividend by 3, and the product, except the last two figures at the right, is the quotient, and the last two are hundredths. 3. Divide 876735 by 331.

Ans. 26302760 OPERATION. 876735

26302,05 Quotient.

IV. To divide by 125. Rule. — Multiply the dividend by 8, and the product, except the last three figures, is the quotient, and these last three figures will be thousandths. 4. Divide 1234567 by 125.

Ans. 987666
OPERATION.
1234567

8

9876,536 Quotient. 5. Divide 8786789 by 125.

Ans. 6. Divide 1234567 by 125

Ans.

V. A short method of performing Long Division. 7. Divide 16294896 by 21.

Ans. 678954

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