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DECIMAL FRACTIONS.

Fractions whose denominators are 10, 100, 1000, etc., are rendered decimals of the same name by a little change in form; thus, a decimal point is placed on the left of the decimals, or on the right of the units, and the same relation exists between the successive orders, as in abstract numbers, but the orders themselves are reversed.

to = .1, Ito = .01, Too

Togo = .001,
Totoo = .0001,

=

and are read alike; thus,

Also,

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one tenth,

one thousandth,
one hundredth, one ten-thousandth.

B = .3, read three tenths;
To = .07, seven hundredths ;

.36, thirty-six hundredths;
= .456, four hundred and fifty-six

thousandths.

36

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4 5 6 1000

Hence, to enumerate a decimal fraction, read it as you would an integral number, adding to this the name of the denominator, when a common fraction, which will be expressed by 1 with as many zeros attached to it as there are numbers of decimal figures.

ADDITION AND SUBTRACTION.

EXAMPLES.
1. Add .1,.01, .001, .0001, and .0001; thus:
(1.)
(2.)

(3.)
.1
Add .0234

Add 5.634
.01
.213

21.321
.001
.3146

.654
.0001
.32

.012
.00001
.6

5.364
.11111
1.4710

32.985

(4.)
From 4.36215
Take 1.83754
Rem. 2.52461

(5.)
From 326159
Take 234573
Rem. .091586

COR.-As the relation of the orders are the same, and the decimals rise in value in the same direction, whilst in name they take the opposite direction; hence, addition and subtraction of decimals are performed as in Integral Numbers.

MULTIPLICATION.

THEOREM.

In the multiplication of decimals, the product will have as many places of decimals as both factors.

tt x t = Tto
70 x

.1 x.1 = .01,
and

Ito x to = Todo .01 X.1 = .001.

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1ST COL.

2D COL or,

1 x 1 = 1 and 1 x .1 = .1
.1 x 1 = .1 and .1' x1 = .01

.01 x 1 = .01 and .01 X 1 = .001 The first column of products is the same as the first column of multiplicands, as 1 is the multiplier. The multiplier in the second case is one-tenth, consequently the products of the second column must be one-tenth of the first.

Therefore the product of two decimal factors will have as many decimal places as both factors.

1 x1 = 1, units.
.1 x.1 = .01, hundredths.
.01 x .01 = .0001, ten thousandths.

1 x1 = 1, units.
10 x 10 = 100, hundreds.
100 x 100 = 10000, ten thousands.

REM.—Observe the correspondence in name, when the contrary orders are multiplied.

PROBLEMS.

1. Multiply

3.156
.215

2. Multiply

.534
.136

15780 3156 6312 .678540

3204 1602 534 .072624

REM. Each product must have six decimals, hence in the second example a zero must be prefixed.

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DIVISION. Corollaries to Theorem, Page 34. COR. 1.-As the product of the divisor and quotient is equal to the dividend, therefore the dividend has as many decimal figures as both divisor and quotient.

COR. 2.-If the divisor has decimal figures and the dividend has none, or less than the divisor, as many must be added to the dividend as to make the number equal to that of the divisor, and then the quotient will be integral. If more decimals are added to the dividend, the quotient will contain as many.

PROBLEMS.

1. Divide 21,4263 by 3.12. 3.12 ) 21.42|63 ( 6.86+ As the divisor has two 18 72

places of decimals, the 2 706

quotient will be integral 2 496

for two places of decimals 2103

in the dividend; after that

! 1872

the quotient will be deci231, remainder. mal. 2. Reduce the fraction 7 to a decimal. 4 ) 1.00

3 .25

5 ) 3.0

.6 COR.–Any common fraction may be reduced to a decimal by performing the division indicated by the terms.

a

EXAMPLES.

1. Multiply 1 by .1; by .01; by .001; by .0001. 2. Multiply .1 by .1; by .01; by .001 ; by .0001.

3. Multiply .2 by 2; .03 by.4; .05 x .04; .06 x .003; and .003 x .004.

4. Multiply 4.732 by .345. 5. Multiply 2.074 by .021. 6. Multiply 3.541 by .002. 7. Muitiply .002 by .3754. 8. Multiply 721.56 by 21.42. 9. Multiply 642.54 by 2162. 10. Multiply 756.48 by 4635. REM.-Prove the last seven examples by division.

PRACTICAL EXAMPLES.

1. A merchant sold 205 yards cotton cloth at $.125 per yard, 75 yards gray flannel at $.625 per yard, 12 pairs hose at $.375 per pair, 54 yards linen at $.555 per yard, What was the amount of the bill ? Ans. $106.97.

2. Bought five tracts of land ; viz., 237 acres at $57.43 per acre, 326 acres at $49.02 per acre, 431 acres at $31.21 per acre, 1274 acres at $12.48 per acre, and 21346 acres at $2.045 per acre. The whole is to be paid in three equal instalments; how much is each payment?

Ans. $34198.341. This table of aliquot parts enables us to shorten the operations of multiplication and division. 10 cts. = $1. 75 cts.

66 cts. = $5. 20 cts. =

121 cts. = 374 cts. = $3. 25 cts. = $1.

16 cts. = $1. 621 cts. = $ 50 cts. $1.

33} cts. = 871 cts. = $1.

$7.

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