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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.

1 x 1 = 1, units.

.1 x .1

=

.01 x .01

=

1 x 1 =

=

1. Multiply

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

10 x 10 100 x 100

and

and

and

3.156

.215

15780

3156

1 x .1

.1 × 1 =

.01 × 1 =

6312

.678540

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

.01, hundredths.

.0001, ten thousandths.

1, units.

100, hundreds.

10000, ten thousands.

PROBLEMS.

2D COL.

.1

.01

.001

2. Multiply

.534

.136

3204

1602

534

.072624

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

(3.)

.01

.01

.0001

2 706

2 496

DIVISION.

Corollaries to Theorem, Page 54.

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.

1. Divide 21.4263 by 3.12. 3.12) 21.42 63 ( 6.86+

1872

2103

1872

PROBLEMS.

(4.) .00001

.00001

.000000001

231, remainder. 2. Reduce the fraction

4) 1.00

.25

As the divisor has two places of decimals, the quotient will be integral for two places of decimals in the dividend; after that the quotient will be decimal.

to a decimal.

3

5) 3.0

.6

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

EXAMPLES.

1. Multiply 1 by .1;
2. Multiply .1 by .1;

by .01; by .001; by .0001.
by .01; by .001; by .0001.

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

4. Multiply 4.732 by .345.

5. Multiply 2.074 by .021.
6. Multiply 3.541 by .002.
7. Multiply .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. $29.97.

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. $34198.34.

This table of aliquot parts enables us to shorten the operations of multiplication and division. 10 cts. = $11. 75 cts. = $3. 66 cts. = $.

37 cts.

20 cts. = $1.
25 cts. =
50 cts. =

621 cts. = $8.

87 cts. = $7.

121 cts. $1.
16 cts. = $1.
33 cts. $1.

EXAMPLES.

1. Multiply 576 by 100.

2. Multiply 576 by 25.

576 × 100

3. Multiply 576 by 50.

4. Divide 576 by 100.

5. Divide 576 by 50.

.

=

576 × 100 = 28800.

6. Divide 576 by 25.

576 ÷ 10o = 576 × 2 ÷ 100 = 11.52.

14400.

7. Divide 67453.2645 by 47.215.

576 100 = 576 × 4100 23.04.

133226

94430

SOLUTION.

47.215) 67453.2645 (1428.2+

47215

202182

188860

387964

377720

Ans. 57600.

102445

94430

8015

Ans. 5.76.

There must be one decimal in the quotient; for 3 in divisor + 1 in quotient = 4 in dividend.

8. Divide 98637.42598 by 21.798.

9. Divide 7326.4873 by 86.324.

10. Divide 83465.987 by 4365.3315.
Add 1 decimal to dividend.

11. What is the cost of 576 yards at .12?
576 × 1
= $72.

12. What is the cost of 576 yards at .16?
576 × = $96.
13. What is the cost of 576 yards at .621?
576 × 9 = 576 × 5 ÷ 8 = $360.

14. What is the cost of 576 yards at .871?
576 × 3 = 576 × 78 $504.

CIRCULATING DECIMALS.

PROBLEMS.

1. In the reduction of common fractions to decimals, when the denominator has no other factor than 2 or 5, or 2 and 5, the decimal will terminate with the number of figures equal to the greatest number of factors of 2 or 5 in the denominator.

2. When the common fraction has any other denominator, the decimal fraction will not terminate; and at some point in the division, the quotient will begin to repeat the same figures; each period of which is called a repetend, and the repetends are called Circulating Decimals.

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