CASE V.

To find the value of any decimal of a pound by inspection.

RULE.

Double the first figure, or place of tenths, for shillings, and if the second be 5, or more than 5, reckon another shilling; then call the figures in the second and third places, after 5, if contained, is deducted, so many farthings; abating 1, when they are above 12; and 2, when above 37; and the result is the answer.

EXAMPLES.

1. Find the value of 7851. by inspection.
14s. double 7.

1s. for 5 in the place of tenths.
835 farthings.

4 for the excess of 12, abated.

15s. 84d. the answer.

2. Find the value of 8751. by inspection. Ans. 17s. 6d. 3. Value the following decimals by inspection, and find their sum, viz. 9271. + ·3511. + ·2031. +0611. 021. +oogl. Ans. 11. 11s. 5d.

FEDERAL MONEY.*

THE denominations of Federal Money, as determined by an Act of Congress, Aug. 8, 1786, are in a decimal ratio; and, therefore, may be properly introduced in this place.

A mill,

*The coins of federal money are two of gold, four of silver, and two of copper. The gold coins are called an eagle and haifeagle; the silver, a dollar, half-dollar, double dime and dime; and the copper, a cent and half-cent. The standard for gold and silver is eleven parts fine and one part alloy. The weight of fine gold

in

A mill, which is the lowest money of account, is 'oo1 of

a dollar, which is the money unit.

in the eagle is 246.268 grains; of fine silver in the dollar, 375.64 grains; of copper in 100 cents, 24lb. avoirdupois. The fine gold in the half-eagle is half the weight of that in the eagle; the fine silver in the half-dollar, half the weight of that in the dollar, &c. The denominations less than a dollar are expressive of their values thus, mill is an abbreviation of mille, a thousand, for 1000 mills are equal to 1 dollar; cent, of centum, a hundred, for 100 cents are equal to 1 dollar; a dime is the French of tithe, the tenth part, for 10 dimes are equal to 1 dollar.

The mint-price of uncoined gold, 11 parts being fine and I part alloy, is 209 dollars, 7 dimes and 7 cents per lb. Troy weight; and the mint-price of uncoined silver, 11 parts being fine and I part alloy, is 9 dollars, 9 dimes and 2 cents per lb. Troy.

In Mr. PIKE'S "Complete System of Arithmetic," may be seen "RULES for reducing the Federal Coin, and the Currencies of the several United States; also English, Irish, Canada, NovaScotia, Livres Tournois and Spanish milled dollars, each to the par of all the others." It may be sufficient here to observe respecting the currencies of the several States, that a dollar is equal to 6s. in New-England and Virginia; 8s. in New-York and NorthCarolina; 7s. 6d. in New-Jersey, Pennsylvania, Delaware and Maryland; and 4s. 8d. in South-Carolina and Georgia.

The English standard for gold is 22 carats of fine gold, and 2 carats of copper, which is the same as II parts fine and 1 part alloy. The English standard for silver is 18 oz. 2dwt. of fine silver, and 18dwt. of copper; so that the proportion of alloy in their sil ver is less than in their gold. When either gold or silver is finer or

coarser

A number of dollars, as 754, may be read 754 dollars, or 75 eagles, 4 dollars; and decimal parts of a dollar, as 365, may be read 3 dimes, 6 cents, 5 mills, or 36 cents, 5 mills, or 365 mills; and others in a similar manner.

Addition, subtraction, multiplication and division of federal money are performed just as in decimal fractions; and consequently with more ease than in any other kind of money.

EXAMPLES.

1. Add 2 dollars, 4 dimes, 6 cents, 4D. 2d., 4d. 9c., 1E. 3D. 5c. 7m., 3c. 9m., 1D. 2d. 8c. 1m.,

and 2E. 4D,

coarser than standard, the variation from standard is estimated by carats and grains of a carat in gold, and by penny-weights in silver. Alloy is used in gold and silver to harden them.

NOTE.-Carat is not any certain weight or quantity, but of any weight or quantity; and the minters and goldsmiths divide it into 4 equal parts, called grains of a carat.

7. Multiply 3D. 4d. 5c. 1m. by ID. 2d. 3c. 2m.

Ir has already been observed, that when an infinite decimal repeats always one figure, it is a single repetend; and when more than one, a compound repetend; also that a point

is

is set over a single repetend, and a point over the first and last figures of a compound repetend.

It

may be further observed, that when other decimal figures precede a repetend, in any number, it is called a

mixed repetend: as 23, or 104123: otherwise it is a pure, or..

simple, repetend: as 3 and 123.

Similar repetends begin at the same place: as 3 and 6, or 1*341 and 2.156.

Dissimilar repetends begin at different places: as 253 and

*4752.

Conterminous repetends end at the same place as 125 and ⚫009.

Similar and conterminous repetends begin and end at the same place : as 2°9104 and •6613.

REDUCTION of CIRCULATING DECIMALS.

CASE I.

To reduce a simple repetend to its equivalent vulgar fraction.

RULE.*.

r. Make the given decimal the numerator, and let the denominator be a number consisting of as many nines as there are recurring places in the repetend.

2. If

* If unity, with cyphers annexed, be divided by 9 ad infinitum, the quotient will be 1 continually; i. e. if be reduced to a decimal, it will produce the circulate' I; and since I is the decimal

I 91

equivalent to 2 will, 3, and so on till 9==1. Therefore, every single repetend is equal to a vulgar fraction, whose numerator is the repeating figure, and denominator 9.