Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση
[blocks in formation]

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.

[blocks in formation]

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.

1. Divide 47932 by 72.

[blocks in formation]

EXAMPLES.

As 72 is equal to 9 times 8, we first divide the dividend by 9, and the quotient thence arising by 8; and to find the true 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.

4)5371
6)1342 3
8) 223-4

We find 192 equal to the product of 4 times 6 times 8,4 x 6 x 8192. We therefore divide by these factors, as in the last example. To find the true remainder, we multiply the last remainder, 7, by the 27-7187 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.

[blocks in formation]

RULE.

[ocr errors]

I. To multiply by 25.

-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 twentyfive times, because 25 is one fourth of 100.

[blocks in formation]

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.

[blocks in formation]

RULE. 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.

RULE.

OPERATION.

8)12345678000

1543209750 Product.

IV. To multiply by any number of 9's.

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.

87654000

87654

By annexing three ciphers, we make the number one thousand times larger. If from 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. 10. Multiply 5555 by 999999.

Ans. 77769992223.
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.

[blocks in formation]

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 × by 8 for 48.

16177777632

32355555264

the last product x by 3 for 144.

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.

[blocks in formation]

1. Divide 67895 by 5.

RULE.

OPERATION.
67895
2

13579,0 Quotient.

II. To divide by 25.

Ans. 13579.

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.

[ocr errors]
[blocks in formation]

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 33}. Ans. 263028O.

OPERATION.

876735

3

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.

[blocks in formation]
« ΠροηγούμενηΣυνέχεια »