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9. Reduce 54 gallons to the fraction of a logshead of
jus, 10. What part of a bogshead is 9 gallons ! Anis, i 11. What part of a pound troy is 1002. 10pwt. 10grs. ?
DECIMAL FRACTIONS. A Decimal Fruction is that whose denominator is an unit, with a cypher, or cyphers annexed to it, Thus, to, 767 1467, &c. &c.
The integer is alirays divided either into 10, 100, 1000, &c. equal parts; consequently the denominator of the fraction will always, be either 10, 100, 1000, or 10000, &c. which being understood, need not be expressed; for the true value of the fraction may be expressed by writing the numerator only with a point before it on the left hand thus, 16, is written ,5; 4 ,45; 102 ,725, &c.
But if the numerator has not so many places as the denominator has cyphers, put so many cyphers before it, viz. at the left hand, as will make up the defect; so write To thus, ,05; and too thus, ,006, &c.'
NOTE. The point prefixed is called the separatrix.
Decimals are counted from the left towards the right hand, and each figure takes its value by its distance from the unit's place; if it be in the first place after units, (or separating point) it, signifies tenths; if in the second, hundredt!is, &c. decreasing in each place in a tenfold proportion, as in the following
X. Thusandth parts.
. Cyphers placed at the right hand of a decimal fraction do not alter its value, since every significant figure continues to possess the same place : so ,5 ,50 and 9500 are all the saine value, and equal to or 1.
But cyphers placed at the left hand of decimals, decrease their value in a tenfold proportion, by removing tjera further from the decimal point. Thus, ,5 ,05 ,005, &c. are five tenth parts, five hundredth parts, five thousandth parts, &c. respectively. It is therefore evident that the magnitude of a decimal fraction, compared with another, does not depend upon the number of its figures, but upon the value of its first left hand figure : for in.. stance, a fraction beginning with any figure less than ,9 such as ,899229, &c. if extended to an infinite number of figures, will not equal 99. i
* ADDITION OF DECIMALS.
RULE. 1. Place the numbers, wnęther mixed or pure decimals, under each other, according to the value of their places..
2. Find their sum as in whole numbers, and point off so many places for the decimals, as are equal to the greatest number of decimal parts in any of the given numbers.
Sum, '103,512 which is 103 integers, and 3.1.2 parts of an unit. Or, it is 103 units, and 3 tenth parts, 1 hun. dredth part, and 2 thousandth parts of an unit, or 1. .
Hence we may observe, that decimals, and FEDERAL MONEY, are subject to one, and the same law of notation, and consequently of operation.
For since dollar is the money unit; and a dime being the tenth, a cent the hundredth, and a mill the thousandth part of a dollar, or unit, it is evident that any number of dollars, dimes, cents and mills, is simply the expression of dollars, and decimal parts of a dollar: Thus, 11 dollars, 6 dimes, 5 cents,=11,65 or 11.00 dol. &c. 2. Add the following mixed numbers together.
(4) : Yards. .. Ounces,
SUBTRACTION OF DECIMALS.
Place the numbers according to their value ; then subtract as in whole numbers, and point off the decimals as in Addition,
! Dollars. 1. From 125,64
Inches. 2. From 14,674
6. From 480 take 245,0075
Ans. 234,9925 7. From 236 dols. take ,549 dols. Ans. 8235,451 8. From ,145 take ,09684
- Ans. ,04816 9. From ,2754 take ,2371
Ans. ,0883 10. From 271 take 215,7
Ans. 55,3 11. From 270,2 take 75,4075
Ans. 194,7925 12. From 107 take ,0007
Ans. 106,9993 13. From an unit, or 1, subtract the millionth part of stself.
MULTIPLICATION OF DECIMALS.
RULE. 1. Whether they be mixed numbers, or pure decimals, place the factors and multiply them as in whole numbers.
2. Point off so many figures from the product as there are decimal places in both the factors; and if there be not so many places in the product, supply the defect by prefixing cyphers to the left hand..
To multiply by 10, 100, 1000, &c. remove the separaiting point so many places to the right hand, as the multiplier has cyphers.
Multiplied by 10, makes 4,25 80 ,425
by 100, makes 42,5 I
by 1000, is 425, For ,425 x 10 is 4,250, &c.
DIVISION OF DECIMALS.
RULE. 1. The places of the decimal parts of the divisor and quotent cuunted together, must always be equal to those