DIVISION OF DECIMALS.

RULE.-Divide as in whole numbers; and point off in the quotient as many places for decimals, as the decimal places in the dividend exceed those in the divisor.*

When the places of the quotient are not so many as the rule requires, let the defect be supplied by prefixing ciphers.

When there happens to be a remainder after the division; or when the decimal places in the divisor are more than those in the dividend; then ciphers may be annexed to the dividend, and the quotient carried on as far as required.

WHEN the divisor is an integer, with any number of ciphers annexed; cut off those ciphers, and remove the decimal point in the dividend as many places farther to the left as there are ciphers cut off, prefixing ciphers if need be; then proceed as before.†

* The reason of this rule is evident; for, since the divisor multiplied by the quotient gives the dividend, therefore the number of decimal places in the dividend is equal to those in the divisor and quotient taken together, by the nature of Multiplication; and consequently the quotient itself must contain as many as the dividend exceeds the divisor.

This is no more than dividing both divisor and dividend by the same number, either IC, or 100, or 1000, &c., according to the number of ciphers cut off; which, leaving them in the same proportion, does not affect the quotient. And, in the same way, the decimal point may be moved the same num. ber of places in both the divisor and dividend, either to the right or left, whether they have ciphers or not.

CONTRACTION II.

HENCE, if the divisor be 1 with ciphers, as 10, or 100, or 1000, &c.; then the quotient will be found by merely moving the decimal point in the dividend so many places farther to the left as the divisor has ciphers; prefixing ciphers if need be.

So, 217-3
And 5.16

100 = 2.173.
100 =

and 419 ÷
and 21

10= 1000 =

CONTRACTION III.

WHEN there are many figures in the divisor; or only a certain number of decimals are necessary to be retained in the quotient, then take only as many figures of the divisor as will be equal to the number of figures, both integers and decimals, to be in the quotient, and find how many times they may be contained in the first figures of the dividend, as usual.

Let each remainder be a new dividend; and for every such dividend, leave out one figure more on the right hand side of the divisor; remembering to carry for the increase of the figures cut off, as in the 2d contraction in Multiplication. Note. When there are not so many figures in the divisor as are required to be in the quotient, begin the operation with all the figures, and continue it as usual till the number of figures in the divisor be equal to those remaining to be found in the quotient, after which begin the contraction.

EXAMPLES.

1. Divide 2508-92806 by 92-41035, so as to have only four decimals in the quotient, in which case the quotient will contain six figures.

2. Divide 4109-2351 by 230-409, so that the quotient may contain only four decimals.

3. Divide 37-10438 by 5713.96, that the quotient may contain only five decimals.

4. Divide 913-08 by 2137-2, that the quotient may contain only three decimals.

REDUCTION OF DECIMALS.

CASE I.

To reduce a vulgar fraction to its equivalent decimal.

RULE.-Divide the numerator by the denominator as in Division of Decimals, annexing ciphers to the numerator as far as necessary; so shall the quotient be the decimal required.

To find the value of a decimal in terms of the inferior denominations.

RULE. Multiply the decimal by the number of parts in the next lower denomination; and cut off as many places for a remainder, to the right hand, as there are places in the given decimal.

Multiply that remainder by the parts in the next lower denomination again, cutting off for another remainder as before.

Proceed in the same manner through all the parts of the integer; then the several denominations separated on the left hand, will make up the answer. Note. This operation is the same as Reduction Descending in whole numbers.

EXAMPLES.

1. Required to find the value of 775 pounds sterling.

To reduce integers or decimals to equivalent decimals of higher denominations.

RULE. Divide by the number of parts in the next higher denomination; continuing the operation to as many higher denominations as may be necessary, the same as in Reduction Ascending of whole numbers.

2. Reduce 9d. to the decimal of a pound.

3. Reduce 7 dr. to the decimal of a pound avoird. 4. Reduce 26d. to the decimal of a £.

5. Reduce 2-15 lb. to the decimal of a cwt.

6. Reduce 24 yards to the decimal of a mile.

7. Reduce 056 poles to the decimal of an acre.
8. Reduce 1-2 pints of wine to the decimal of a hhd.
9. Reduce 14 minutes to the decimal of a day.
10. Reduce 21 pints to the decimal of a peck.

Ans. 0375%. Ans. 02734375 lb. Ans. 0010833, &c. £. Ans. 019196 + cwt. Ans. 013636, &c. miles. Ans. 00035 ac. Ans. 00238+ hhd. Ans. 009722, &c. da. Ans. 013125 pec.

NOTE. When there are several numbers, to be reduced all to the decimal of the highest.

Set the given numbers directly under each other, for dividends, proceeding orderly from the lowest denomination to the highest.

Opposite to each dividend, on the left hand, set such a number for a divisor as will bring it to the next higher name; drawing a perpendicular line between all the divisors and dividends.

Begin at the uppermost, and perform all the divisions; only observing to set the quotient of each division, as decimal parts, on the right hand of the dividend next below it; so shall the last quotient be the decimal required.

RULE.-Prepare the terms by reducing the vulgar fractions to decimals, any compound numbers either to decimals of the higher denominations, or to integers of the lower, also the first and third terms to the same name: then multiply and divide as in whole numbers.

Note. Any of the convenient examples in the Rule of Three or Rule of Five in Integers, or Vulgar Fractions, may be taken as proper examples to the same rules in Decimals.—The following example, which is the first in Vulgar Fractions, is wrought here to show the method.

DUODECIMALS, or CROSS MULTIPLICATION, is a rule made use of by workmen and artificers, in computing the contents of their works.

Dimensions are usually taken in feet, inches, and quarters; any parts smaller than these being neglected as of no consequence. And the same in multiplying them together, or casting up the contents.

RULE.-Set down the two dimensions, to be multiplied together, one under the other, so that feet stand under feet, inches under inches, &c.

Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each straight under its corresponding term, observing to carry 1 for every 12, from the inches to the feet.

In like manner, multiply all the multiplicand by the inches and parts of the multiplier, and set the result of each term one place removed to the right hand of those in the multiplicand; omitting however what is below parts of inches, only carrying to these the proper number of units from the lowest denomination. Or, instead of multiplying by the inches, take such parts of the multiplicand as these are of a foot.

Then add the two lines together, after the manner of Compound Addition, carrying 1 to the feet for 12 inches, when these come to so many.