Proof by Vulgar Fractions.

Using only the latter example, we have

Hence, decimals are said to be reduced to a common denominator, when ciphers are supplied so that there is the same number of decimal places in each.

II. SUBTRATION OF DECIMALS.

109. RULE. Place the less quantity under the greater as in Addition; suppose ciphers to be supplied if necessary, in the upper line; and the difference, found as in integers, will have as many decimal places as are contained in each, either expressed or understood.

Proof by Vulgar Fractions.

In the latter of these examples, we have

as before; and the necessity of supposing the cipher to be supplied is here shewn.

III. MULTIPLICATION OF DECIMALS.

110. RULE. Multiply together the quantities proposed as if they were integers: and the product will contain as many places of decimals, as there are decimal places in the multiplicand and multiplier together.

where the former product, found as if all the quantities were whole numbers would manifestly be ten thousand times too great, because 45 and 21 are a hundred times as great as .45 and .21 respectively; and therefore the true product is obtained by placing the decimal point four places towards the left hand, by Article (106).

Proof by Vulgar Fractions.

The latter product in these examples is

111. RULE. Supply the dividend with ciphers to the right hand, if necessary, and divide exactly as in integers: then the quotient will have a number of decimal places equal to the excess of the number of such places in the dividend above that in the divisor.

In these instances, the numbers of decimal places in the quotients are the excesses of the numbers of decimal places in the dividends above those in the divisors, because the divisors and quotients must together comprise as many places as the dividends, by the last Rule.

If the divisor and dividend have the same number of decimal places, the quotient will comprise no decimal places as there is no excess but if there be more places in the divisor than in the dividend, ciphers must be supplied so as to render the number in the dividend not less than that in the divisor, before the rule can be applied: and the reason of this will be seen in the following example:

625 25 625 1000 625 1000

× 100 = 2500, where there is annexed to the right of the quotient obtained as in integers, a number of ciphers equal to the excess of the number of decimal places in the divisor above that in the dividend, the quotient being the integral quantity 2500.

If the division do not terminate, three or four decimal places in the quotient are generally sufficient.

Examples for Practice.

(1) Express in decimals; One and Fifty four hundredths; Twenty four and Seventy nine thousandths; Three hundred and fifteen, Eight thousandths and Fifty millionths.

(2) Find the sum of .295, 3.086, 12.87, .0051, 729.54: also, of 3608.26, 360.826, 36.0826, 3.60826, .22314.

(3) Add together 36.053, .0079, .000952, 417, 85.5803, .0000501: and prove it by vulgar fractions.

(4) Find the difference of 27.903 and .054: of 7295.06 and 254.738 of 35.08989 and 3.508989.

(5) Required the excess of 2.057 above 1.0097 and of 3.025 above .003025: and prove the results.

(6) Required the product of .718 and .57 of 16.8 and .0024 of 144 and .0625: of 12.5 and .062216.

(7) Multiply 270.56 by .37025: .00579 by 3796.8: .384759375 by .00032: and prove them.

(8) Find the continued product of .275, 2.75 and 27.5: also, of 3.24, 215 and .0028.

(9) Required the quotient of 35.9424 by 7.02: of .278831 by .653: of 11.444495 by 4.735: of .020872522 by .08635 and prove them by vulgar fractions.

(10) Divide .0257 by .0041: 325.46 by .0187: .0719 by 27.53: to three or more places of decimals.

(11) Find the quotient of 1.68 by .024 of 971.7 by .123 of 142.025 by .0437 of 84.375 by .00375: and prove the results by vulgar fractions.

(12) Simplify the arithmetical expressions, 5-3.22 + 2.333 1.4444 and 75.012

.075012345.

7.50123.7501234

Answers: 2.6686 and 68.185881055.

(13) Express in the decimal notation, the value of

8.0625-6.00375 +1.09236

25679

10000

Answer: .54321.

(14) Simplify 1.26 of 663 +5 of 1.0375 and 3§ of .003-0011 of 71.

Answers: 89.395 and .0029.

(15) Reduce to decimal forms, the following expressions:

2.004 3.375 .0295 1.18

.167 4

and

3.04 .00152

3.25 −2.765 + 3.125 × 8.607095÷.027.

Answers: 10.125, .0000125 and 3.

REDUCTION OF DECIMALS.

112. A general view having now been taken of decimals, we proceed to shew how they may be made to change their denominations when they are considered as belonging to a particular unit; and in what ways they may be adapted to the practical computations in which they are most frequently employed.

113. A Decimal may be changed into another, whose denomination shall have a given relation to its own.

RULE. Multiply or divide the decimal by the number or numbers which connect the denominations in order, according as the denomination of the required decimal is lower or higher than its own.

For, from what has been said in the reduction of compound quantities, it is evident that

114. A compound quantity may be exhibited in the form of a Decimal whose denomination is given.

RULE. Divide the lowest denomination by the number which connects it with the next, and to the left of the quotient affix the number of this denomination: and continue the process till the required denomination is obtained.

Let us take 7 fur. 25 po., and express it as the decimal of a mile then,