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Division of DECIMALS.
RULE.* Divide as in whole numbers ; and to know how many decimals to point off in the quotient, observe the following rules :
1. There must be as many decimals in the dividend, as in both the divisor and quotient ; therefore point off for decimals in the quotient so many figures, as the decimal places in the dividend exceed those in the divisor.
2. If the figures in the quotient are not so many as the rule requires, supply the defect by prefixing cyphers.
3. If the decimal places in the divisor be more than those in the dividend, add cyphers as decimals to the dividend, till the number of decimals in the dividend be equal to those in the divisor, and the quotient will be integers till all these decimals are used. And, in case of a remainder, after all the figures of the dividend are used, and more figures are wanted in the quotient, annex cyphers to the remainder, to continue the division as far as necessary.
4. The first figure of the quotient will possess the same place of integers or decimals, as that figure of the dividend, which stands over the units place of the first product.
EXAMPLES, 1. Divide 3424'0056 by 43•6.
* The reason of pointing off as many decimal places in the quotient, as those in the dividend exceed those in the divisor, will
2. Divide 3877875 by 675. 3. Divide .0081892 by *347. 4. Divide 7'13 by '18.
I. If the divisor be an integer with any number of cyphers at the end ; cut them off, and remove the decimal point in the dividend so many places further to the left, as there were cyphers cut off, prefixing cyphers, if need be ; then proceed as before.
*04538, &c. Here I first divide by 3, and then by 7, because 3 times 7 is 21. 2. Divide 41020 by 32000.
Ans. I'281875 Note. Hence, if the divisor be i with cyphers, the quotient
will be the same figures with the dividend, having the deci. mal point so many places further to the left, as there are cyphers in the divisor,
EXAMPLES 21793 100 = 2:173.
419 by 10 = 419. 5:16 by 1000 = '00516. -21 by, 1000 = '00021. II . When the number of figures in the divisor is great, the operation may be contracted, and the necessary number of decimal places obtained.
RULE. 1. Having, by the 4th general rule, found what place of decimals or integers the first figure of the quotient will pos
easily appear ; for since the number of decimal places in the divi. dend is equal to those in the divisor and quotient, taken together, by the nature of multiplication ; it follows, that the quotient contains as many as the dividend exceeds the divisor.
sess; consider how many figures of the quotient will servo the present purpose ; then take the same number of the left-hand figures of the divisor, and as many of the dividend figures as will contain them (less than ten times); by these find the first figure of the quotient.
2. And for each following figure, divide the last remainder by the divisor, wanting one figure to the right more than before, but observing what must be carried to the first product for such omitted figures, as in the contraction of Multiplication ; and continue the operation till the divisor be exhausted.
3. When there are not so many figures in the divisor, as are required to be in the quotient, begin the division with all the figures as usual, and continue it till the number of figures in the divisor and those remaining to be found in the quotient be equal ; after which use the contraction.
EXAMPLES 1. Divide 2508.928065051 by 9241035, so as to have four decimals in the quotient... In this case, the quotiens will contain six figures. Hence
51544621 2. Divide 721017562 by 2.257432, so that the quotient may contain three decimals.
3. Divide 12-169825 by 3*14159, so that the quotient may contain five decimals.
4. Divide 87'076326 by 9-365407, and let the quotient contain seven decimals.
REDUCTION of DECIMALS.
RULE.* Divide the numerator by the denominator, annexing as many cyphers as are necessary; and the quotient will be the decimal required.
* Let the given vulgar fraction, whose decimal expression is required, be . Now since every decimal fraction has 10, 100, 1000, &c. for its denominator ; and, if swo fractions be equal,
EXAMPLES Į. Reduce to a decimal.
Ans. "375 Ans. '04:
•208333, &c. 2. Required the equivalent decimal expressions for and *
Ans. •25, '5 and 75. 3. What is the decimal of ? 4. What is the decimal of
? 5. What is the decimal of
Ans. '015625 6. Express 3:14. decimally. Ans. •071577, &c.
CASE II. To reduce numbers of different denominations to their equivalent
RULE. 1. Write the given numbers perpendicularly under each other for dividends, proceeding orderly from the least to, the greatest.
2. Opposite to each dividend, on the left hand, place such a number for a divisor, as will bring it to the next superior name, and draw a line between them,
it will be, as ine denominator of one is to its numerator, so is the denominator of the ot. 'er to its numerator; therefore 13:7:: 1000, &c. : 7x1000, &c. _bm0000, &c.
= '53846, the numerator of 13
13 the decimal required ; and is the same as by the rule.
+ The reason of the rule may be explained from the first ex. ample ; thus, three farthings is of a penny, which brought to a decimal is -75 ; consequently 91d. may be expressed 9.75d. but 9'75 is 47% of a penny =1166 of a shilling, which brought to a decimal is 8125 ; and therefore 155. 9 d. may be expressed 15.81255. In like manner 15.81255. is '
Too of a shilling = 188076 of a pound =, by bringing it to a decimal, +790625l. as by the rule.