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Now, if this product be divided by one of the factors the quotient will be the other factor (see 35.) Hence, in division, the dividend must contain just as many decimal places as the divisor and quotient together. The quotient, therefore, will contain as many places as the dividend, less those of the divisor.

EXAMPLES.

1. Divide 1,38483 by 60,21.

OPERATION. There are five decimal places in the dividend, and two in the divi

60,21)1,38483(23

1 2042 sor: there must therefore be three

18063 places in the quotient: hence one 0

18063 must be prefixed to the 23, and the decimal point placed before it.

Ans. ,023. Hence, for the division of decimals we have the following

RULE, Divide as in simple numbers, and point of in the quotient, from the right hand, so many places for decimals as the decimal places in the dividend exceed those in the divisor; and if there are not so many, supply the deficiency by prefixing ciphers.

Q. If one decimal fraction be multiplied by another, how many decimal places will there be in the product? How does the number of decimal places in the dividend compare with those in the divisor and quotient? How do you determine the number of decimal places in the quotient ? If the divisor contains four places and the dividend six, how many in the quotient? If the divisor contains three places and the dividend five, how many in the quotient ? Give the rule for the division of decimals.

EXAMPLES.

1. Divide 2,3421 by 2,11.

Ans. 1,11. 2. Divide 12,82561 by 3,01.

Ans. 3. Divide 33,66431 by 1,01.

Ans. 33,331. 4. Divide ,010001 by ,01.

Ans. 5. Divide 8,2470 by ,002.

Ans. 4123,5. 6. What is the quotient of 37,57602, divided by 3? By ,3? By ,03 ? By ,003 ? By ,0003?

7. What is the quotient of 129,75896, divided by 8? By ,08? By ,008? By 90008? By ,00008 ?

8. What is the quotient of 187,29900, divided by 9? By ,9? By ,09? By ,009? By ,0009 ? By ,00009 ?

9. What is the quotient of 764,2043244, divided by 6? By ,06? By ,006 ? By ,0006 ? By ,00006? By ,000006?

$ 130. Note 1. When any decimal number is to be divided by 10, 100, 1000, &c. the division is made by removing the decimal point as many places to the left as there are O's in the divisor; and if there be not so many figures on the left of the decimal point, the deficiency must be supplied by prefixing ciphers.

10

(2,769

100 27,69 divided by

,2769 1000

,02769 10000 ,002769 10

64,289 100

6,4289 642,89 divided by { 1000

,64289 10000

,064289 100000

,0064289 Q. How do you divide a decimal number by 10, 100, 1000, &c. ? If there be not as many figures to the left of the decimal point as there are ciphers in the divisor, what do you do?

$ 131. Note 2. When there are more decimal places in the divisor than in the dividend, annex as many ciphers to the dividend as are necessary to make its decimal places equal to those of the divisor; all the figures of the quotient will then be whole numbers.

EXAMPLES.

1. Divide 4397,4 by 3,49.

We annex one ( to the dividend. Had it contained no decimal place we should have annexed two.

OPERATION. 3,49)4397,40(1260

349
907
698
2094
2094
Ans. 1260.

2. Divide 2194,02194 by ,100001. Ans. 21940. 3. Divide 9811,0047 by ,325947.

Ans. 4. Divide ,1 by ,0001.

Ans. 1000. 5. Divide 10 by ,l.

Ans. 6. Divide 6 by ,6. By ,06. By ,006. By ,2. By ,3. By ,003. By,5. By ,05. By ,005. By ,000012.

Q: If there are more decimal places in the divisor than in the dividend, what do you do? What will the figures of the quotient then be ?

§ 132. Note 3. When it is necessary to continue the division farther than the figures of the dividend will allow, we may annex ciphers and consider them as decimal places of the dividend.

EXAMPLES.

by 1.

OPERATION.

1. Divide 4,25 by 1,25.

OPERATION.

1,25)4,25(3,4 In this example we annex one 0

3,75 and then the decimal places in the

500 dividend will exceed those in the divisor

500

Ans. 3,4. 2. Divide , 2 by ,06. We see in this example that the

,06),20(3,33+ division will never terminate. In

18 such cases the division should be

20 carried to the third or fourth place, which will give the answer true enough for all practical purposes,

20 and the sign + should then be

18 written, to show that the division

20 may be still continued.

Ans. 3,33+ 3. Divide 37,4 by 4,5.

Ans. 8,3111+ 4. Divide 586,4 by 375.

Ans. 5. Divide 94,0369 by 81,032.

Ans. 1,160+ Q: How do you continue the division after you have brought down all the figures of the dividend? What sign do you place after the quoticnt? What does it show?

18

APPLICATIONS IN THE FOUR PRECEDING RULES.

1. A merchant sold 4 parcels of cloth, the first contained 127 and 3 thousandths yards; the 2nd, 6 and 3 tenths yards; the 3rd, 4 and one hundredth yards; the 4th, 90 and one millionth yards : how many yards did he sell in all ?

Ans. 227,313001yd. 2. A merchant buys three chests of tea, the first contains 60 and one thousandth lb.; the second, 39 and one ten thousandth lb.; the third, 26 and one-tenth lb.: how much did he buy in all ?

Ans.

76. 3. What is the sum of $20 and three hundredths; and one-tenth, $6 and one thousandth, and $18 and one hundredth ?

Ans. $48,141. 4. A puts in trade $504,342; B puts in $350,1965; C puts in $100,11; D puts in $99,334; and E puts in $9001,32: what is the whole amount put in?

Ans. * 5. B has $936, and A has $1, 3 dimes and 1 mill: how much more money has B than A? Ans. $934,699.

6. A merchant buys 37,5 yards of cloth, at one dollar twenty-five cents per yard : how much does the whole come to?

Ans. $46,875. 7. A farmer sells to a merchant 13,12 cords of wood at $4,25 per cord, and 13 bushels of wheat at $1,06 per bushel : he is to take in payment 13 yards of broadcloth at $4,07 per yard, and the remainder in cash: how much money did he receive?

Ans. $16,63. 8. If 12 men had each $339 one dime 9 cents and 3 mills, what would be the total amount of their money?

Ans. ** 9. If one man can remove 5,91 cubic yards of earth in a day, how much could nineteen men remove ?

Ans. 112,29yd. 10. What is the cost of 8,3 yards of cloth at $5,47 per yard?

Ans. $45,401. 11. If a man earns one dollar and one mill per day, how much will he earn in a year? Ans. *

12. What will be the cost of 375 thousandths of a cord of wood, at $2 per cord ?

Ans. $0,75.

13. A man leaves an estate of $1473,194 to be equally divided among 12 heirs : what is each one's portion ?

Ans. $122,7661 REDUCTION OF VULGAR FRACTIONS TO DECIMALS.

§ 133. The value of every vulgar fraction is equal to the quotient arising from dividing the numerator by the denominator (see § 44.)

EXAMPLES. 1. What is the value in decimals of ..

We first divide 9 by 2 which I OPERATION. gives a quotient 4, and í for a re

{=41, but, mainder. Now 1 is equal to 10 tenths. If then we add a cipher, 41=4 =4,5 2 will divide 10, giving the quotient 5 tenths. Hence the true quotient is 4,5

2. What is the value of 23.

We first divide by 4 which gives OPERATION. a quotient 3 and a remainder 1.

x=31; but But 1 is equal to 100 hundredths. If then we add two ciphers, 4 will

31=3100=3,25. divide the 100, giving a quotient of 25 hundredths.

Hence, to reduce a vulgar fraction to a decimal, we have the following

RULE. I. Annex one or more ciphers to the numerator and then divide by the denominator.

II. If there is a remainder, annex a cipher or ciphers, and divide again, and continue to annex ciphers and to divide until there is no remainder or until the quotient is sufficiently exact : the number of decimal places to be pointed off in the quotient is the same as the number of ciphers used ; and when there are not so many, ciphers must be prefixed.

Q. What is the value of a fraction equal to? What is the value of four-halves ? What is the decimal value of one-half? Of threehalves? Of six-fourths ? Of nine-halves? Of seven-halves? Of five-fourths ?. Of one-fourth? Give the rule for reducing a vulgar fraction to a decimal.

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