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2. How many steps of 2 feet and 5 inches each, will it require a man to take, going from Leominster to Boston, it being 43 miles.

Ans. 93948 steps; the last step will carry him into the town 12 inches.

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3. LET 70 dollars be distributed among three men in such a manner that as often

48 and the third 9/ What will each one re0793948 as the first has 5/ the second shall have 7/

ceive?

Ans. First 16 dolls. 4f Second

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4. Ir a vinter be desirous to draw off a pipe of Canary into bottles containing pints, quarts, and 2 quarts, of each an equal number, how many must he have? Ans. 144 of each.

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5. THERE are three fields; one contains 7 acres, another 10 acres and the other 12 acres and 1 rood; how many shares of 76 perches each, are contained in the whole? Ans. 61 shares and 44 perches over.

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6. There are 106 lbs. of silver, the property of 3 men ; of which A receives 17lb. 10oz. 19 pwts. 19grs. of what remains B shares loz. 7grs. so often as C shares 13/wts. What are the shares of B. and C?

Answer, B's share 53 lb. 8oz. 5pwis. 5grs. C's share 34lb. 4oz. 15payts.

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§ 2. Fractions.

WHEN the thing or things signified by figures are whole ones, then the figures which signify them are called Integers, or whole numbers. But when only some parts of a thing are signified by figures, as two thirds of any thing, five sixths, seven tenths, &c. then the figures which signify these parts of a thing, being the expression of some quantity less than one, are called FRACTIONS. FRACTIONS are of two kinds, Vulgar and Decimal; they are distinguished by the manner of representing them; they also differ in their modes of operation.

'VULGAR FRACTIONS.

To understand Vulgar Fractions, the learner must suppose an integer (or the number 1) divided into a number of equal parts; then any number of these parts being taken, would make a fraction, which would be represented by two numbers placed one directly over the other, with a short line between them, thus two thirds, three fifths, seven eighths, &c.

EACH of these figures have a different name and a different signification. The figure below the line is called the Denominator and shews into how many parts an integer, or one individual of any thing is divided.....the figure above the line is called the Numerator and shews how many of those parts are signified by the fraction.

FOR illustration, suppose a silver plate to be divided into nine equal parts. Now, one or more of these parts make a fraction, which will be represented by the figure 9 for a denominator placed underneath a short line, shewing the plate to be divided into nine equal parts; and supposing two of those parts to be taken for the fraction, then the figure 2 must be placed directly above the 9 and over the line (3) for a Numerator, shewing that two of those parts are signified by the fraction, or two ninths of the plate. Now let 5 parts of this plate, which is divided into 9 parts, be given to John, his fraction would be five ninths; let 3 other parts be given to Harry, his fraction would be three ninths; there would then be one part of the plate remaining still (5 and 3 are 8) and this fraction would be expressed thus one ninth.

In this way are all vulgar fractions written; the Denominator, or number below the line shewing into how many parts any thing is divided, and the nuinerator, or number above the line, shewing how many of those parts are taken, or signified by the fraction.

To ascertain whether the Learner understands what has now been taught him of fractions, let us again suppose a dollar to be cut into 13 equal parts ;--let 2 of those parts be given to A; 4 to B; and 7 to C,

A's fractionREQUIRED of the Learner that he should write B's fraction

C's fraction

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Ir is from Division only that fractions arise in Arithmetical operations: the remainder after division is a portion of the Dividend undivided; and is always the Numerator to a fraction of which the Divisor is the Denominator. The Quotient is so many integers.

THE Arithmetic of Vulgar Fractions is tedious and even intricate to beginners. Besides, they are not of necessary use. We shall not, therefore, enter into any further consideration of them here. This difficulty arises chiefly from the variety of denominators; for when numbers are divided into differ

ent kinds, or parts they cannot be easily compared. This consideration gave rise to the invention of

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DECIMAL Fractions are also expressions of parts of an integer; or, are in value something less than one of any thing, whatever it may be, which is signied by them.

IN decimals, an integer, or the number one, as 1 foot, 1 dollar, 1 year, &c. is conceived to be divided into ten equal parts, (in vulgar fractions an integer may be divided into any number of parts) and each of these parts is subdivided into ten lesser parts, and so on. In this way, the denominator to a decimal fraction in all cases, will be either 10, 100, 1000, or unity (1) with a number of cyphers annexed; and this number of cyphers will always be equal to the number of places in the numerator. Thus, 82% 850 Thus, 8 are Decimal Fractions, of which the cyphers in the denominator of each are equal to the number of places in its own numerator.

100 1000

"As the denominator of a decimal fraction is always 10, 100, 1000, &c. the "denominators need not to be expressed; for the numerator only may be made "to express the true value; for this purpose it is only required to write the "numerator, with a point (,) before it, called a separatrix, at the left hand t "distinguish it from a whole number; thus, is written,6; 20,27; 1005 ,685 &c.

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WHEN integers and decimals are expressed together in the same sum, that sum is called a mixed number; Thus, 25,63 is a mixed number; 25, or all the figures to the left hand of the separatrix being integers, and ,63 or all the figures to the right hand of the same point being decimals.

THE first figure on the right hand of the decimal point signifies tenth parts, the next hundredth parts, the next thousandth parts, and so on.

,7 signifies seven tenth parts.

,07- -seven hundredth parts.

,27---two tenth parts and seven hundredth parts; or twentyseven hundredths.

,357————three tenth parts, five hundredth parts, and seven thousandth parts; or, 357 thousandths.

5,7 five and seven tenth parts.

5,007-five and seven thousandths.

THE value of each figure from unity, and the decrease of decimals, toward the right hand, may be seen in the following

TABLE.

4

CYPHERS placed to the right hand of decimals do not alter their value, placed at the left hand, they diminish their value in a tenfold proportion.

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