Practical Arithmetic, by Induction and Analysis

the purpose of explaining the principle on which the product is pointed.

The denominator of the product of two decimals will be 1, with as many ciphers annexed as there are ciphers in both the denominators. But the number of ciphers in each denominator is the same, Art. 169, as the number of places in the decimal.

Hence, the number of decimal places in the product, must equal the number of decimal places in both factors.

*5. Multiply 15 hundredths by 7 tenths. Ans. .105

General Rule for Multiplication. Multiply as in Simple Numbers; point off from the right of the product as many figures for decimals as there are decimal places in both mul tiplicand and multiplier; if there be not so many places in the product, supply the deficiency by prefixing ciphers.

PROOF. The same as in Multiplication of Simple Numbers.

ART. 182. The operations of multiplying by 10, 100,

REVIEW.-181. To what is the denominator of the product of two decimals equal? To what the number of ciphers in each denominator?

1000, &c., may be shortened by removing the decimal point as many places to the right, as there are ciphers in the multiplier: and,

If there be not so many figures on the right of the point, annex ciphers to supply the deficiency.

Thus, 2.07X10=20.7

For additional problems, see Ray's Test Examples.

DIVISION OF DECIMALS.

ART. 183. Decimals may be divided when the divisor, or dividend, or both, are decimals.

Since the dividend is equal to the product of the divisor and quotient, it must contain as many decimal places as there are decimals in both divisor and quotient. Art. 181: hence,

There must be as many decimals in the quotient as the decimal places in the dividend exceed those in the divisor. 1. Divide 2.125 by 5 tenths.

SOLUTION.-Divide as in Simple Numbers; then, since there are three decimal places in the dividend, and one in the divisór, point off two decimals in the quotient.

2. Divide 21 units by .5

OPERATION.

.5)2.125

Ans. 4.25

OPERATION.

.5)21.0

Ans. 42.

SOLUTION. In dividing a whole number by a fraction, the whole number is reduced to the same parts of a unit as the fraction, that the divisor and dividend may be of the same denomination.

So, in dividing a whole number by a decimal; reduce the dividend to the same denomination as the divisor, by annexing to it as many ciphers as there are decimal places in the divisor, and the quotient is a whole number.

3. Divide 83:1 by 4.

SOLUTION. Divide the figures of the dividend by the divisor, as in whole numbers, and a remainder occurs. Then, to continue

REVIEW.-181. To what is the number of decimal places in the product equal? What is the General Rule for multiplication?

182. How multiply a decimal by 10, 100, &c.? 183. When may decimals be divided? How many decimal places must the quotient contain? Why?

the division, annex ciphers to the dividend, which does not alter its value, (Art. 172), and divide as before. Continue the division until there is no remainder, or until the quotient is sufficiently exact.

OPERATION.

4)83.100

Ans. 20.775

As there are three places of decimals in the dividend, and none in the divisor, there must be three in the quotient.

4. Divide 2.11 by .3

OPERATION.

.3) 2.11000

In this example, the division win not terminate. In such cases, it is to be carried to sufficient exactness: the sign annexed to denote that the division is not complete.

*5. Divide 1.125 by .03

*6. Divide 2 by .008

Ans. 7.0333+

Ans. 37.5

Ans. 250.

Ans. 7.44

*7. Divide 37.2 by 5.

General Rule for Division.-Divide as in Simple Num bers, and point off from the right hand of the quotient as many places for decimals as the decimal places in the dividend exceed those in the divisor; if there be not so many places, supply the deficiency by prefixiny ciphers.

PROOF-The same as in Division of Simple Numbers.

NOTES.-1. When the divisor has more decimals than the dividend, annex ciphers to the dividend until its decimal places equal those of the divisor; the quotient will be a whole number.

2. After dividing all the figures of the dividend, if there be a remainder, annex ciphers to it, and continue the division till there is no remainder, or until the quotient is sufficiently exact. In pointing the quotient, regard the ciphers annexed as decimal places.

REVIEW.-183. What is the General Rule for Division? NOTE 1. When the number of places in the divisor exceeds those in the dividend, what is required? Why? What will be the quotient?

Ans. 1.

24. Divide 1 thousandth by 1 thousandth.

25. 1 ten-thousandth 1 ten-millionth. Ans. 1000.

26. 1 hundredth÷4 millionths.

Ans. 2500.

Ans. 2.142857+

27.

1.5.7

ART. 184. To divide a Decimal by 10, 100, 1000, &c., remove the decimal point as many places to the left as there are ciphers in the divisor :

And, if there are not so many figures on the left of the point, supply the deficiency by prefixing ciphers.

Thus, 18.3 divided by 10 =1.83

18.3 divided by 100 = .183

18.3 divided by 1000= .0183

ART. 185. REDUCTION OF DECIMALS. CASE I. To reduce a common fraction to a decimal.

1. Reduce the fraction to a decimal.

SOL. The numerator, 3, will not be changed by writing ciphers in the place of tenths, hundredths, &c. Since a fraction is expressed in the form of an unexecuted division (Art. 125), regard the operation as division of decimals, and perform it according to

the rule, (Art. 183). Since the divisor has no decimal places, the quotient must have as many places as there are ciphers annexed.

*2. Reduce to a decimal.

OPERATION,

4)3.00 Ans. .75

Ans. .125

Rule for Case I.-Annex ciphers to the numerator, divide by the denominator, and point off in the quotient as many places for decimals as there are ciphers annexed to the numerator.

NOTE. When common fractions can not be exactly expressed in decimals, continue to divide till the quotient is sufficiently exact. REDUCE THESE COMMON FRACTIONS TO 'DECIMALS: